Finite Group Databasebeta

← #33127368

Subgroups — #33127368

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988891C211yesmaximal, characteristic, complemented
2C2×#17988648C211yesmaximal, characteristic
3#22612613C211yesmaximal, characteristic
4#22611367C211yesmaximal, characteristic
5#22611357C211yesmaximal, characteristic
6#22611134C211yesmaximal, characteristic
7#22611134C211yesmaximal, characteristic
8#22610889C211yesmaximal, characteristic, complemented
9#22610687C211yesmaximal, characteristic, complemented
10#22610666C211yesmaximal, characteristic, complemented
11#22610661C211yesmaximal, characteristic, complemented
12#22610620C211yesmaximal, characteristic, complemented
13#22610606C211yesmaximal, characteristic, complemented
14#22609933C211yesmaximal, characteristic, complemented
15C73×#2629C211yesmaximal, characteristic
16C2×#14556570C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18#17988891—24no—
19#17988648—24no—
20C2×#14556524C2211yescharacteristic
21C2×#14556524C2211yescharacteristic
22C2×#14556524C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×C4⋊#10596831C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×C22⋊#10596831C2211yescharacteristic
31C2×C22⋊#10596831C2211yescharacteristic
32C2×#14556502C2211yescharacteristic
33C2×C4×#10596831C2211yescharacteristic
34#17988493C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988482—21no—
37#17988481—21no—
38#17988475—21no—
39#17988468—21no—
40#17988447C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50C2×#125×C73C2211yescharacteristic
51C2×#117×C73C2211yescharacteristic
52#398×C73—21no—
53#398×C73—21no—
54#396×C73—21no—
55#396×C73—21no—
56#389×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#384×C73C2211yescharacteristic
59#380×C73C2211yescharacteristic
60#380×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556570—22no—
67#14556570—22no—
68C4×Dic146—22no—
69C4×Dic146—22no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×#11924007—21no—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C2×C22⋊Dic73—21no—
77C2×C22⋊Dic73—21no—
78C2×C4×Dic73—21no—
79C2×C4×Dic73—21no—
80#14556524—22no—
81#14556524—22no—
82#14556524—22no—
83#14556522—22no—
84C4⋊#10596831—22no—
85C4⋊#10596831D414yes—
86#14556514—22no—
87#14556514—22no—
88#14556514—22no—
89#14556514—22no—
90C22⋊#10596831—22no—
91C22⋊#10596831—22no—
92#14556502—22no—
93C4×#10596831—22no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99C23×C4×C73C2311yescharacteristic
100#125×C73—22no—
101#125×C73—22no—
102#117×C73—22no—
103#117×C73—22no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73D411yescharacteristic
111C2×C4⋊C4×C73D411yescharacteristic
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C4⋊C4×C73C2311yescharacteristic
114C2×C4⋊C4×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C22⋊C4×C73—21no—
117C2×C22⋊C4×C73—21no—
118C2×C22⋊C4×C73D411yescharacteristic
119C2×C22⋊C4×C73D411yescharacteristic
120C2×C42×C73C2311yescharacteristic
121C2×C42×C73C2311yescharacteristic
122#95×C73—21no—
123#95×C73—21no—
124#95×C73—21no—
125#95×C73—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146#4112yes—
132C22×Dic73—21no—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C2×#10596831—21no—
137C2×#10596831—21no—
138C2×#10596831—21no—
139C2×#10596831—21no—
140C2×#10596831—21no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831C2×D412yes—
147C2×#10596831C2×D412yes—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73#4112yes—
154#11924007—24no—
155#11924007—42no—
156#11924007—81no—
157C22⋊Dic73—24no—
158C22⋊Dic73—81no—
159C4×Dic73—42no—
160C4×Dic73—81no—
161C24×C73C2×D411yescharacteristic
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2411yescharacteristic
178C4⋊C4×C73—22no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—42no—
189C4⋊C4×C73—42no—
190C4⋊C4×C73—81no—
191C4⋊C4×C73—81no—
192C4⋊C4×C73—81no—
193C4⋊C4×C73—81no—
194C22⋊C4×C73—22no—
195C22⋊C4×C73—22no—
196C22⋊C4×C73—81no—
197C22⋊C4×C73—81no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73C2×D412yes—
201C42×C73#4112yes—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73C22×D411yescharacteristic
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73#12112yes—
240C2×C4×C73#12112yes—
241C2×C4×C73C22×D411yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C732−1+411yescharacteristic
246C2×C4×C732−1+411yescharacteristic
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259Dic146—22no—
260Dic146—22no—
261Dic146—41no—
262Dic146—41no—
263#10596831—22no—
264#10596831—22no—
265#10596831—41no—
266#10596831—41no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73C2×#12111yescharacteristic
275C22×C73#52511yescharacteristic
276C22×C73#53611yescharacteristic
277C22×C73#57611yescharacteristic
278C22×C73#57711yescharacteristic
279C22×C73#57811yescharacteristic
280C22×C73#57811yescharacteristic
281C4×C73—22no—
282C4×C73—22no—
283C4×C73—41no—
284C4×C73—41no—
285C4×C73—41no—
286C4×C73—81no—
287C4×C73—81no—
288C4×C73—81no—
289C4×C73—81no—
290C4×C73#54711yescharacteristic
291C4×C73#54811yescharacteristic
292C4×C73#55211yescharacteristic
293C4×C73#55611yescharacteristic
294Dic73—42no—
295Dic73—42no—
296Dic73—81no—
297Dic73—81no—
298#24715—731nomaximal, Sylow
299C2×C73—81no—
300C2×C73#241511yescharacteristic
301C2×C73#306111yescharacteristic
302C2×C73#306311yescharacteristic
303C2×C73#311911yescharacteristic
304C2×C73#312711yescharacteristic
305C2×C73#324211yescharacteristic
306C2×C73#324311yescharacteristic
307C2×#479—731no—
308C2×#433—731no—
309#2629D7311yescharacteristic
310#2323—731no—
311#2077—731no—
312#2075—731no—
313#2036—731no—
314#2036—731no—
315#1989—731no—
316#1947—731no—
317#1943—731no—
318#1942—731no—
319#1935—731no—
320#1933—731no—
321#1803—731no—
322C73#2471511yescharacteristic, complemented, Sylow
323C2×#125C2×D7311yescharacteristic
324C2×C22⋊Q8—731no—
325C2×C4×Q8—731no—
326C2×#117C2×D7311yescharacteristic
327#479—1464no—
328#433—1464no—
329C2×#107—731no—
330C2×#107—731no—
331C2×#107—731no—
332C2×#106—731no—
333C2×C4⋊C8—731no—
334C2×C4⋊C8—731no—
335C2×#103—731no—
336C2×#103—731no—
337C2×#103—731no—
338C2×#103—731no—
339C2×C22⋊C8—731no—
340C2×C22⋊C8—731no—
341C2×#97—731no—
342C2×C4×C8—731no—
343#398—21no—
344#398—21no—
345#396—21no—
346#396—21no—
347#391—731no—
348#389—731no—
349#389—1461no—
350#389C2×D7311yescharacteristic
351#389C2×D7311yescharacteristic
352#388—1461no—
353#388—1461no—
354#387—1461no—
355#384C2×D7311yescharacteristic
356#380—731no—
357#380C2×D7311yescharacteristic
358#380C2×D7311yescharacteristic
359C4×C4⋊C4—731no—
360#340—731no—
361#340—731no—
362#340—731no—
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368C22×Q8—731no—
369C23×C4C22×D7311yescharacteristic
370C22×C8—731no—
371C22×C8—731no—
372C22×C8—731no—
373C22×C8—731no—
374#125—22no—
375#125—22no—
376C22⋊Q8—1462no—
377C22⋊Q8—1462no—
378C4×Q8—1462no—
379C4×Q8—1462no—
380#117—22no—
381#117—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—731no—
389C2×C4⋊C4—731no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4—1461no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4C22⋊D7311yescharacteristic
395C2×C4⋊C4C22⋊D7311yescharacteristic
396C2×C4⋊C4C22×D7311yescharacteristic
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1461no—
403C2×C22⋊C4—1461no—
404C2×C22⋊C4C22⋊D7311yescharacteristic
405C2×C22⋊C4C22⋊D7311yescharacteristic
406C2×C42—1461no—
407C2×C42—1461no—
408C2×C42C22×D7311yescharacteristic
409C2×C42C22×D7311yescharacteristic
410#107—1462no—
411#107—1462no—
412#107—1462no—
413#106—1462no—
414C4⋊C8—734no—
415C4⋊C8—1462no—
416#103—1462no—
417#103—1462no—
418#103—1462no—
419#103—1462no—
420C22⋊C8—1462no—
421C22⋊C8—1462no—
422#97—1462no—
423C4×C8—1462no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1461no—
429#95—1461no—
430#95—1461no—
431#95—1461no—
432#95—1461no—
433C24C2×C22⋊D7311yescharacteristic
434C2×Q8—732no—
435C2×Q8—1461no—
436C2×Q8—1461no—
437C2×Q8—1461no—
438C2×Q8—1461no—
439C2×Q8—1461no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1461no—
454C22×C4C2×C22⋊D7311yescharacteristic
455C22×C4C2×C22⋊D7311yescharacteristic
456C22×C4C2×C22⋊D7311yescharacteristic
457C22×C4C2×C22⋊D7311yescharacteristic
458C22×C4C2×C22⋊D7311yescharacteristic
459C22×C4C23×D7311yescharacteristic
460C2×C8—732no—
461C2×C8—732no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C2×C8—1461no—
468C2×C8—1461no—
469C2×C8—1461no—
470C2×C8—1461no—
471C2×C8—1461no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—2922no—
489C4⋊C4—5841no—
490C4⋊C4—732no—
491C4⋊C4—1464no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1461no—
496C4⋊C4—1461no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—5841no—
502C22⋊C4—1464no—
503C42—21no—
504C42—21no—
505C42—2922no—
506C42—5841no—
507C42D4×D7312yes—
508C42#1192402712yes—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C22×C22⋊D7311yescharacteristic
517Q8—2921no—
518Q8—2921no—
519Q8—1462no—
520Q8—1462no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—1462no—
561C2×C4—1462no—
562C2×C4#1455656312yes—
563C2×C4#1455656312yes—
564C2×C4C2×D4×D7311yescharacteristic
565C2×C4#41×D7311yescharacteristic
566C2×C4#41×D7311yescharacteristic
567C2×C4C2×#1192402711yescharacteristic
568C2×C4#1455663911yescharacteristic
569C2×C4#1455663911yescharacteristic
570C8—2921no—
571C8—2921no—
572C8—1462no—
573C8—1462no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1455656311yescharacteristic
582C22#1798909411yescharacteristic
583C22#1798916111yescharacteristic
584C22#1798939011yescharacteristic
585C22#1798939811yescharacteristic
586C22#1798940211yescharacteristic
587C22#1798940211yescharacteristic
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—2922no—
598C4—2922no—
599C4—5841no—
600C4—5841no—
601C4#1798923911yescharacteristic
602C4#1798924711yescharacteristic
603C4#1798927111yescharacteristic
604C4#1798930811yescharacteristic
605C2—81no—
606C2#2261336111yescharacteristic
607C2#2261959811yescharacteristic
608C2#2261962411yescharacteristic
609C2#2262012411yescharacteristic
610C2#2262022611yescharacteristic
611C2#2262119911yescharacteristic
612C2#2262121411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2629C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#2629D73
hypocenterC73#24715
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2629

Complements

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

NH ≅ G/Ntype
1. C2×#17988891C2semidirect
8. #22610889C2semidirect
9. #22610687C2semidirect
10. #22610666C2semidirect
11. #22610661C2semidirect
12. #22610620C2semidirect
13. #22610606C2semidirect
14. #22609933C2semidirect
322. C73#24715semidirect

Maximal subgroups

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

subgroupindex
C2×#179888912
C2×#179886482
#226126132
#226113672
#226113572
#226111342
#226108892
#226106872
#226106662
#226106612
#226106202
#226106062
#226099332
C73×#26292
#2471573