Finite Group Databasebeta

← #33127247

Subgroups — #33127247

Order 18688.

868 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988887C211yesmaximal, characteristic, complemented
2C2×#17988644C211yesmaximal, characteristic, complemented
3#22618241C211yesmaximal, characteristic, complemented
4#22611367C211yesmaximal, characteristic, complemented
5#22611357C211yesmaximal, characteristic, complemented
6#22611117C211yesmaximal, characteristic
7#22611110C211yesmaximal, characteristic
8#22610888C211yesmaximal, characteristic, complemented
9#22610699C211yesmaximal, characteristic, complemented
10#22610671C211yesmaximal, characteristic, complemented
11#22610666C211yesmaximal, characteristic, complemented
12#22610631C211yesmaximal, characteristic, complemented
13#22610631C211yesmaximal, characteristic, complemented
14#22609941C211yesmaximal, characteristic, complemented
15C4×#121×C73C211yesmaximal, characteristic
16C2×#14556583C2211yescharacteristic
17#17988887—24no—
18#17988644—24no—
19C2×#14556524C2211yescharacteristic
20C2×#14556524C2211yescharacteristic
21C2×#14556522C2211yescharacteristic
22C2×#14556522C2211yescharacteristic
23C2×C4⋊#10596831C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×C22⋊#10596831C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×#14556502C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988471—21no—
36#17988471—21no—
37#17988460—21no—
38#17988460—21no—
39#17988453C2211yescharacteristic
40#17988447C2211yescharacteristic
41#17988447C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×#121×C73C2211yescharacteristic
51C2×C4×D4×C73—21no—
52C2×C4×D4×C73C2211yescharacteristic
53C22×C42×C73C2211yescharacteristic
54#390×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#388×C73—21no—
57#386×C73—21no—
58#380×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60C4×C22⋊C4×C73—21no—
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65#14556583—22no—
66#14556583—22no—
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×C4⋊Dic73C2311yescharacteristic
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×C22⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77#14556524—22no—
78#14556524—22no—
79#14556522—22no—
80#14556522—22no—
81C4⋊#10596831—22no—
82C4⋊#10596831D414yescomplemented
83#14556512—22no—
84#14556512—22no—
85#14556512—22no—
86#14556512—22no—
87C22⋊#10596831—22no—
88C22⋊#10596831—22no—
89#14556502—22no—
90#14556502—22no—
91#14556498—21no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C22×D4×C73—21no—
98C22×D4×C73C2311yescharacteristic
99C23×C4×C73—21no—
100C23×C4×C73D411yescharacteristic
101C23×C4×C73D411yescharacteristic
102C23×C4×C73C2311yescharacteristic
103#121×C73—22no—
104#121×C73—22no—
105C4×D4×C73—22no—
106C4×D4×C73—22no—
107C4×D4×C73—44no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2311yescharacteristic
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73—21no—
115C2×C22⋊C4×C73—21no—
116C2×C22⋊C4×C73—21no—
117C2×C42×C73—22no—
118C2×C42×C73—22no—
119C2×C42×C73—21no—
120C2×C42×C73D411yescharacteristic
121C2×C42×C73D411yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#95×C73—21no—
125#95×C73—21no—
126#95×C73—21no—
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831C2×D412yes—
142C2×#10596831C2×D412yes—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151#11924007—42no—
152#11924007—42no—
153#11924007—42no—
154#11924007—42no—
155C22⋊Dic73—42no—
156C22⋊Dic73—42no—
157C24×C73—21no—
158C24×C73C2×D411yescharacteristic
159C2×D4×C73—24no—
160C2×D4×C73—21no—
161C2×D4×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—42no—
166C2×D4×C73—42no—
167C2×D4×C73#4112yes—
168C22×C4×C73—22no—
169C22×C4×C73—21no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—41no—
181C22×C4×C73—41no—
182C22×C4×C73—41no—
183C22×C4×C73—42no—
184C22×C4×C73—42no—
185C22×C4×C73—42no—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2×D411yescharacteristic
191C22×C4×C73C2411yescharacteristic
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—42no—
198C4⋊C4×C73#4112yes—
199C22⋊C4×C73—24no—
200C22⋊C4×C73—42no—
201C22⋊C4×C73—42no—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—42no—
204C42×C73—22no—
205C42×C73—22no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—42no—
209C42×C73C2×D412yes—
210C42×C73#4112yes—
211C23×C73—21no—
212C23×C73—21no—
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—22no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73AGL(1,ℤ/8)12yes—
225C23×C73C22×D411yescharacteristic
226D4×C73—22no—
227D4×C73—22no—
228D4×C73—48no—
229D4×C73—41no—
230D4×C73—41no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—22no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—44no—
246C2×C4×C73—44no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73—84no—
260C2×C4×C73#12112yes—
261C2×C4×C73#12112yes—
262C2×C4×C73C22×D411yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C732−1+411yescharacteristic
267C2×C4×C732−1+411yescharacteristic
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280#10596831—22no—
281#10596831—22no—
282#10596831—41no—
283#10596831—41no—
284C22×C73—22no—
285C22×C73—22no—
286C22×C73—22no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73—41no—
291C22×C73—42no—
292C22×C73—42no—
293C22×C73—42no—
294C22×C73—81no—
295C22×C73—81no—
296C22×C73—81no—
297C22×C73—81no—
298C22×C73C2×#12111yescharacteristic
299C22×C73#52511yescharacteristic
300C22×C73#53611yescharacteristic
301C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C73#57511yescharacteristic
303C22×C73#57511yescharacteristic
304C22×C73#57511yescharacteristic
305C4×C73—22no—
306C4×C73—22no—
307C4×C73—42no—
308C4×C73—41no—
309C4×C73—41no—
310C4×C73—41no—
311C4×C73—81no—
312C4×C73—81no—
313C4×C73#54711yescharacteristic
314C4×C73#54811yescharacteristic
315C4×C73#55211yescharacteristic
316C4×C73#55611yescharacteristic
317Dic73—81no—
318Dic73—81no—
319Dic73—81no—
320Dic73—81no—
321#24706—731nomaximal, Sylow
322C2×C73—42no—
323C2×C73—82no—
324C2×C73#241511yescharacteristic
325C2×C73#305411yescharacteristic
326C2×C73#305511yescharacteristic
327C2×C73#311111yescharacteristic
328C2×C73#311411yescharacteristic
329C2×C73#323411yescharacteristic
330C2×C73#324011yescharacteristic
331C2×#478—731no—
332C2×#432—731no—
333#2851—731no—
334C4×#121D7311yescharacteristic
335#2077—731no—
336#2075—731no—
337#2034—731no—
338#2033—731no—
339#1989—731no—
340#1949—731no—
341#1944—731no—
342#1943—731no—
343#1936—731no—
344#1936—731no—
345#1804—731no—
346C73#2470611yescharacteristic, complemented, Sylow
347C2×#125—731no—
348C2×#121C2×D7311yescharacteristic
349C2×C4×D4—21no—
350C2×C4×D4C2×D7311yescharacteristic
351C22×C42C2×D7311yescharacteristic
352#478—1464no—
353#432—1464no—
354C2×#107—731no—
355C2×#107—731no—
356C2×#106—731no—
357C2×#106—731no—
358C2×C4⋊C8—731no—
359C2×C4⋊C8—731no—
360C2×#102—731no—
361C2×#102—731no—
362C2×#102—731no—
363C2×#102—731no—
364C2×C22⋊C8—731no—
365C2×C22⋊C8—731no—
366C2×#97—731no—
367C2×#97—731no—
368#390C2×D7311yescharacteristic
369#389—731no—
370#389—731no—
371#389C2×D7311yescharacteristic
372#388—21no—
373#387—1461no—
374#387—1461no—
375#386—21no—
376#385—1461no—
377#385—1461no—
378#382—731no—
379#380—731no—
380#380—731no—
381#380C2×D7311yescharacteristic
382C4×C4⋊C4C2×D7311yescharacteristic
383C4×C22⋊C4—21no—
384#340—731no—
385#340—731no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340—731no—
390#340—731no—
391#340—731no—
392C22×D4—21no—
393C22×D4C22×D7311yescharacteristic
394C23×C4—21no—
395C23×C4C22⋊D7311yescharacteristic
396C23×C4C22⋊D7311yescharacteristic
397C23×C4C22×D7311yescharacteristic
398C22×C8—731no—
399C22×C8—731no—
400C22×C8—731no—
401C22×C8—731no—
402#125—1462no—
403#125—1462no—
404#121—22no—
405#121—22no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—44no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1461no—
416C2×C4⋊C4—1461no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4C22×D7311yescharacteristic
419C2×C4⋊C4C22×D7311yescharacteristic
420C2×C4⋊C4C22×D7311yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—1461no—
427C2×C22⋊C4—1461no—
428C2×C42—22no—
429C2×C42—22no—
430C2×C42—21no—
431C2×C42C22⋊D7311yescharacteristic
432C2×C42C22⋊D7311yescharacteristic
433C2×C42C22×D7311yescharacteristic
434C2×C42C22×D7311yescharacteristic
435#107—1462no—
436#107—1462no—
437#106—1462no—
438#106—1462no—
439C4⋊C8—734no—
440C4⋊C8—1462no—
441#102—1462no—
442#102—1462no—
443#102—1462no—
444#102—1462no—
445C22⋊C8—1462no—
446C22⋊C8—1462no—
447#97—1462no—
448#97—1462no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458C24—21no—
459C24C2×C22⋊D7311yescharacteristic
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4#1192402812yes—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—41no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4C2×C22⋊D7311yescharacteristic
492C22×C4C2×C22⋊D7311yescharacteristic
493C22×C4C2×C22⋊D7311yescharacteristic
494C22×C4C2×C22⋊D7311yescharacteristic
495C22×C4C2×C22⋊D7311yescharacteristic
496C22×C4C23×D7311yescharacteristic
497C2×C8—732no—
498C2×C8—732no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—2922no—
516C4⋊C4—2922no—
517C4⋊C4—2922no—
518C4⋊C4—2922no—
519C4⋊C4—1461no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4—1461no—
523C4⋊C4—1461no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4#1192402812yes—
528C22⋊C4—24no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—2922no—
534C22⋊C4—2922no—
535C42—22no—
536C42—22no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42D4×D7312yes—
541C42#1192402712yes—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23#1455661712yes—
556C23C22×C22⋊D7311yescharacteristic
557D4—22no—
558D4—22no—
559D4—48no—
560D4—41no—
561D4—41no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4#1455656312yes—
604C2×C4#1455656312yes—
605C2×C4C2×D4×D7311yescharacteristic
606C2×C4C2×#1192402711yescharacteristic
607C2×C4C2×#1192402811yescharacteristic
608C2×C4C2×#1192402811yescharacteristic
609C2×C4#1455663811yescharacteristic
610C2×C4#1455663811yescharacteristic
611C8—2921no—
612C8—2921no—
613C8—1462no—
614C8—1462no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×#1455656311yescharacteristic
630C22#1798909811yescharacteristic
631C22#1798915811yescharacteristic
632C22C2×#1455661711yescharacteristic
633C22#1798938511yescharacteristic
634C22#1798938511yescharacteristic
635C22#1798938511yescharacteristic
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—5841no—
645C4—5841no—
646C4—5841no—
647C4—5841no—
648C4#1798924211yescharacteristic
649C4#1798924611yescharacteristic
650C4#1798927711yescharacteristic
651C4#1798930711yescharacteristic
652C2—42no—
653C2—82no—
654C2#2261336011yescharacteristic
655C2#2261953711yescharacteristic
656C2#2261954611yescharacteristic
657C2#2262004011yescharacteristic
658C2#2262006911yescharacteristic
659C2#2262111711yescharacteristic
660C2#2262116711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC4×#121×C73C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenterC4×#121D73
hypocenterC73#24706
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#17988887C2semidirect
2. C2×#17988644C2semidirect
3. #22618241C2semidirect
4. #22611367C2semidirect
5. #22611357C2semidirect
8. #22610888C2semidirect
9. #22610699C2semidirect
10. #22610671C2semidirect
11. #22610666C2semidirect
12. #22610631C2semidirect
13. #22610631C2semidirect
14. #22609941C2semidirect
82. C4⋊#10596831D4semidirect
346. C73#24706semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179888872
C2×#179886442
#226182412
#226113672
#226113572
#226111172
#226111102
#226108882
#226106992
#226106712
#226106662
#226106312
#226099412
C4×#121×C732
#2470673