Finite Group Databasebeta

← #33149097

Subgroups — #33149097

Order 18688.

828 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988891C211yesmaximal, characteristic, complemented
2C2×C4×#11924016C211yesmaximal, characteristic
3C4⋊C4×Dic146C211yesmaximal, characteristic, complemented
4#22611459C211yesmaximal, characteristic
5#22611407C211yesmaximal, characteristic, complemented
6#22611319C211yesmaximal, characteristic
7#22611266C211yesmaximal, characteristic, complemented
8#22610661C211yesmaximal, characteristic, complemented
9#22610490C211yesmaximal, characteristic, complemented
10#22610289C211yesmaximal, characteristic
11#22610289C211yesmaximal, characteristic
12#22610192C211yesmaximal, characteristic
13#22610184C211yesmaximal, characteristic, complemented
14#22610184C211yesmaximal, characteristic, complemented
15C73×#2454C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20#17988891—24no—
21C4×#11924016—24no—
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×#14556512C2211yescharacteristic
34C2×C4×#10596831C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988485C2211yescharacteristic
37#17988484—21no—
38#17988484—21no—
39#17988457C2211yescharacteristic
40#17988456—21no—
41C4⋊C4×Dic73—21no—
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×#125×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52#399×C73—21no—
53#396×C73—21no—
54#390×C73C2211yescharacteristic
55#390×C73C2211yescharacteristic
56#389×C73C2211yescharacteristic
57#389×C73C2211yescharacteristic
58#385×C73—21no—
59#380×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C2×#11924016—22no—
63C2×#11924016—22no—
64C2×#11924016Q814yes—
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C4×Dic146—22no—
70C4×Dic146—22no—
71C4×Dic146—22no—
72C4×Dic146—22no—
73C4×Dic146—41no—
74C4×Dic146—41no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88#14556524—22no—
89#14556524—22no—
90#14556524—41no—
91#14556522—22no—
92#14556522—22no—
93C4⋊#10596831—22no—
94#14556514—22no—
95#14556514—22no—
96#14556514D414yes—
97#14556512—22no—
98#14556512—22no—
99#14556512—41no—
100C4×#10596831—22no—
101#14556498—21no—
102#14556498—21no—
103C22×D4×C73C2311yescharacteristic
104C23×C4×C73—21no—
105#125×C73—22no—
106#125×C73—41no—
107C4×D4×C73—22no—
108C4×D4×C73—41no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73D411yescharacteristic
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C22⋊C4×C73—21no—
119C2×C22⋊C4×C73—21no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—21no—
122C2×C22⋊C4×C73—21no—
123C2×C42×C73C2311yescharacteristic
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126#95×C73—21no—
127#95×C73—21no—
128#95×C73D411yescharacteristic
129#95×C73D411yescharacteristic
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146C2×D412yes—
134C2×Dic146C2×Q812yes—
135C2×Dic146#4112yes—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140#11924016—28no—
141C2×#10596831—21no—
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831C2×D412yes—
152C2×#10596831C2×Q812yes—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—42no—
159C4⋊Dic73—81no—
160C4⋊Dic73#4112yes—
161#11924007—24no—
162#11924007—42no—
163#11924007—42no—
164#11924007—81no—
165#11924007—81no—
166C4×Dic73—24no—
167C4×Dic73—42no—
168C4×Dic73—42no—
169C24×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73C2×Q812yes—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—42no—
180C22×C4×C73—42no—
181C22×C4×C73—81no—
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189C4⋊C4×C73—22no—
190C4⋊C4×C73—22no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—42no—
197C4⋊C4×C73—42no—
198C4⋊C4×C73—81no—
199C4⋊C4×C73—81no—
200C4⋊C4×C73C2×D412yes—
201C22⋊C4×C73—42no—
202C22⋊C4×C73—42no—
203C22⋊C4×C73—81no—
204C22⋊C4×C73—81no—
205C22⋊C4×C73—81no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—21no—
212C23×C73—22no—
213C23×C73—41no—
214C23×C73—41no—
215C23×C73—41no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73C22×D411yescharacteristic
220D4×C73—22no—
221D4×C73—22no—
222D4×C73—41no—
223D4×C73—41no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—22no—
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—21no—
240C2×C4×C73—21no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—84no—
250C2×C4×C73#13512yes—
251C2×C4×C73#13712yes—
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C22×Q811yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73H(5,𝔽2)11yescharacteristic
257C2×C4×C732−1+411yescharacteristic
258C2×Dic73—22no—
259C2×Dic73—22no—
260C2×Dic73—22no—
261C2×Dic73—22no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270Dic146—22no—
271Dic146—22no—
272Dic146—22no—
273Dic146C22⋊Q814yes—
274#10596831—22no—
275#10596831—22no—
276#10596831—41no—
277#10596831—41no—
278C22×C73—42no—
279C22×C73—42no—
280C22×C73—42no—
281C22×C73—81no—
282C22×C73—81no—
283C22×C73—81no—
284C22×C73—81no—
285C22×C73#52511yescharacteristic
286C22×C73D4211yescharacteristic
287C22×C73D4×Q811yescharacteristic
288C22×C73C2×#13511yescharacteristic
289C22×C73C2×#13711yescharacteristic
290C22×C73#57611yescharacteristic
291C22×C73#57711yescharacteristic
292C4×C73—22no—
293C4×C73—22no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73—42no—
297C4×C73—41no—
298C4×C73—41no—
299C4×C73—81no—
300C4×C73—81no—
301C4×C73C2×C22⋊Q811yescharacteristic
302C4×C73#53911yescharacteristic
303C4×C73#54711yescharacteristic
304C4×C73#55411yescharacteristic
305Dic73—24no—
306Dic73—42no—
307Dic73—81no—
308Dic73—81no—
309#26379—731nomaximal, Sylow
310C2×C73—82no—
311C2×C73#244911yescharacteristic
312C2×C73#310811yescharacteristic
313C2×C73#311911yescharacteristic
314C2×C73#326611yescharacteristic
315C2×C73#328711yescharacteristic
316C2×C73#338111yescharacteristic
317C2×C73#338511yescharacteristic
318C2×#479—731no—
319C2×C4×SD16—731no—
320#2454D7311yescharacteristic
321Q8×C4⋊C4—731no—
322#2093—731no—
323#2084—731no—
324#2067—731no—
325#2059—731no—
326#1942—731no—
327#1916—731no—
328#1880—731no—
329#1880—731no—
330#1864—731no—
331#1863—731no—
332#1863—731no—
333C73#2637911yescharacteristic, complemented, Sylow
334C22×SD16—731no—
335C2×#125C2×D7311yescharacteristic
336C2×C4×Q8—731no—
337C2×C4×Q8—731no—
338C2×C4×Q8—731no—
339C2×C4×D4C2×D7311yescharacteristic
340#479—1464no—
341C4×SD16—1464no—
342C2×#107—731no—
343C2×#107—731no—
344C2×#107—731no—
345C2×#106—731no—
346C2×#106—731no—
347C2×C4⋊C8—731no—
348C2×#103—731no—
349C2×#103—731no—
350C2×#103—731no—
351C2×#102—731no—
352C2×#102—731no—
353C2×#102—731no—
354C2×C4×C8—731no—
355#399—21no—
356#396—21no—
357#390C2×D7311yescharacteristic
358#390C2×D7311yescharacteristic
359#389—731no—
360#389—731no—
361#389—1461no—
362#389—1461no—
363#389C2×D7311yescharacteristic
364#389C2×D7311yescharacteristic
365#385—21no—
366#384—731no—
367#384—1461no—
368#380—21no—
369C4×C4⋊C4—1461no—
370C4×C4⋊C4C2×D7311yescharacteristic
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#336—731no—
376#336—731no—
377#336—731no—
378#336—731no—
379C22×Q8—731no—
380C22×D4C22×D7311yescharacteristic
381C23×C4—21no—
382C2×SD16—734no—
383C2×SD16—1462no—
384C2×SD16—1462no—
385C22×C8—731no—
386C22×C8—731no—
387C22×C8—731no—
388C22×C8—731no—
389#125—22no—
390#125—41no—
391C4×Q8—2921no—
392C4×Q8—2921no—
393C4×Q8—1462no—
394C4×Q8—1462no—
395C4×Q8—1462no—
396C4×Q8—1462no—
397C4×D4—22no—
398C4×D4—41no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—731no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1461no—
410C2×C4⋊C4—1461no—
411C2×C4⋊C4—1461no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4C22⋊D7311yescharacteristic
414C2×C4⋊C4C22⋊D7311yescharacteristic
415C2×C4⋊C4C22×D7311yescharacteristic
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C42—1461no—
424C2×C42—1461no—
425C2×C42—1461no—
426C2×C42C22×D7311yescharacteristic
427C2×C42C22×D7311yescharacteristic
428C2×C42C22×D7311yescharacteristic
429#107—2921no—
430#107—1462no—
431#107—1462no—
432#106—1462no—
433#106—1462no—
434C4⋊C8—1462no—
435#103—734no—
436#103—1462no—
437#103—1462no—
438#102—2921no—
439#102—1462no—
440#102—1462no—
441C4×C8—1462no—
442#95—21no—
443#95—21no—
444#95—1461no—
445#95—1461no—
446#95C22⋊D7311yescharacteristic
447#95C22⋊D7311yescharacteristic
448C24—21no—
449C2×Q8—732no—
450C2×Q8—732no—
451C2×Q8—732no—
452C2×Q8—1461no—
453C2×Q8—1461no—
454C2×Q8—1461no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4Q8×D7312yes—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—81no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4C2×C22⋊D7311yescharacteristic
472C22×C4C2×C22⋊D7311yescharacteristic
473C22×C4C2×C22⋊D7311yescharacteristic
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C2×C22⋊D7311yescharacteristic
476C22×C4C2×C22⋊D7311yescharacteristic
477C22×C4C23×D7311yescharacteristic
478SD16—1468no—
479C2×C8—732no—
480C2×C8—732no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—2922no—
503C4⋊C4—2922no—
504C4⋊C4—2922no—
505C4⋊C4—5841no—
506C4⋊C4—5841no—
507C4⋊C4—5841no—
508C4⋊C4—732no—
509C4⋊C4—1464no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4D4×D7312yes—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—2922no—
528C42—2922no—
529C42—1464no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C22×C22⋊D7311yescharacteristic
538Q8—734no—
539Q8—1462no—
540Q8—1462no—
541Q8—1462no—
542D4—22no—
543D4—22no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—84no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—1462no—
581C2×C4—1462no—
582C2×C4—1462no—
583C2×C4—1462no—
584C2×C4#1455661112yes—
585C2×C4#1455662112yes—
586C2×C4C2×D4×D7311yescharacteristic
587C2×C4C2×Q8×D7311yescharacteristic
588C2×C4#41×D7311yescharacteristic
589C2×C4#41×D7311yescharacteristic
590C2×C4#1455663611yescharacteristic
591C2×C4#1455663911yescharacteristic
592C8—2921no—
593C8—2921no—
594C8—1462no—
595C8—1462no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22#1798909411yescharacteristic
604C22D4×C22⋊D7311yescharacteristic
605C22Q8×C22⋊D7311yescharacteristic
606C22C2×#1455661111yescharacteristic
607C22C2×#1455662111yescharacteristic
608C22#1798939011yescharacteristic
609C22#1798939811yescharacteristic
610C4—22no—
611C4—22no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—2922no—
620C4—5841no—
621C4—5841no—
622C4—1464no—
623C4C22⋊Q8×D7311yescharacteristic
624C4#1798918311yescharacteristic
625C4#1798923911yescharacteristic
626C4#1798929111yescharacteristic
627C2—82no—
628C2#2261373711yescharacteristic
629C2#2262000711yescharacteristic
630C2#2262012411yescharacteristic
631C2#2262143711yescharacteristic
632C2#2262168711yescharacteristic
633C2#2262280611yescharacteristic
634C2#2262284111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2454C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2454D73
hypocenterC73#26379
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2454

Complements

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

NH ≅ G/Ntype
1. C2×#17988891C2semidirect
3. C4⋊C4×Dic146C2semidirect
5. #22611407C2semidirect
7. #22611266C2semidirect
8. #22610661C2semidirect
9. #22610490C2semidirect
13. #22610184C2semidirect
14. #22610184C2semidirect
333. C73#26379semidirect

Maximal subgroups

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

subgroupindex
C2×#179888912
C2×C4×#119240162
C4⋊C4×Dic1462
#226114592
#226114072
#226113192
#226112662
#226106612
#226104902
#226102892
#226101922
#226101842
C73×#24542
#2637973