Finite Group Databasebeta

← #29718945

Subgroups — #29718945

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455759C211yesmaximal, characteristic
2C2×#16839903C211yesmaximal, characteristic
3C2×#16839770C211yesmaximal, characteristic, complemented
4#21447185C212yesmaximal, complemented
5#21447082C212yesmaximal, complemented
6#21446722C212yesmaximal, complemented
7#21446474C212yesmaximal, complemented
8#21446280C211yesmaximal, characteristic
9#21446026C211yesmaximal, characteristic, complemented
10#21445742C211yesmaximal, characteristic, complemented
11C61×#2924C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C22×#11758190C2212yes—
14C22×#11758194C2211yescharacteristic
15C2×#14519751C2211yescharacteristic
16#16840086—21no—
17#16840086—21no—
18#16840080—21no—
19#16840075—21no—
20#16839903—21no—
21#16839903—21no—
22#16839903—22no—
23C2×#14519713C2212yes—
24#16839805C2211yescharacteristic
25#16839805C2211yescharacteristic
26#16839770—21no—
27#16839770—21no—
28#16839770C2212yes—
29#16839770C2212yes—
30#16839766C2211yescharacteristic
31#16839766C2211yescharacteristic
32C2×#14519702C2212yes—
33C2×#14519700—21no—
34C2×#14519700—21no—
35C2×#14519695C2212yes—
36C2×#14519695C2212yes—
37C2×#14519693C2212yes—
38#16839726C2211yescharacteristic
39#16839530C2212yes—
40#16839527C2212yes—
41C2×#117×C61C2211yescharacteristic
42C61×#433—22no—
43C61×#432—21no—
44C61×#431—21no—
45C61×#424C2212yes—
46C61×#407C2211yescharacteristic
47C2×C22⋊C8×C61C2211yescharacteristic
48C61×#405C2212yes—
49C22×Dic122—21no—
50C22×Dic122—21no—
51C22×Dic122—24no—
52C22×Dic122—21no—
53C22×Dic122—21no—
54C22×Dic122—21no—
55C22×Dic122—42no—
56C22×Dic122C2312yes—
57C22×D244C2311yescharacteristic
58C22×C22⋊D61—21no—
59C23×Dic61—21no—
60C2×#11758190—24no—
61C2×#11758190—22no—
62C2×#11758190—22no—
63C2×#11758190—42no—
64C2×#11758194—21no—
65C2×#11758194—42no—
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758194C2312yes—
69C22×#92418—21no—
70#14519751—21no—
71#14519751—21no—
72#14519751—21no—
73#14519751—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758198—21no—
76#14519713—42no—
77#14519710—21no—
78#14519710—21no—
79#14519708—21no—
80#14519708—21no—
81#14519702—22no—
82#14519702—22no—
83#14519700—22no—
84#14519700—22no—
85#14519700—22no—
86#14519700—22no—
87#14519697—21no—
88#14519697—21no—
89#14519697—21no—
90#14519697—21no—
91#14519695—24no—
92#14519695—42no—
93#14519693—42no—
94C22⋊#92418—21no—
95C22⋊#92418—21no—
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61C2312yes—
98C22×C8×C61C2312yes—
99#117×C61—21no—
100#117×C61—21no—
101#117×C61D411yescharacteristic
102#117×C61D411yescharacteristic
103#117×C61C2311yescharacteristic
104#117×C61C2311yescharacteristic
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C42×C61—21no—
109C4⋊C8×C61—22no—
110C4⋊C8×C61—22no—
111C22⋊C8×C61—21no—
112C22⋊C8×C61—21no—
113C22⋊C8×C61D411yescharacteristic
114C22⋊C8×C61D411yescharacteristic
115C22⋊C8×C61D412yes—
116#97×C61—22no—
117C4×C8×C61—22no—
118C24×C61#4111yescharacteristic
119C23×D61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61#4111yescharacteristic
129C22×C4×C61C2411yescharacteristic
130C2×C22⋊D61—41no—
131C2×C22⋊D61—41no—
132C2×C22⋊D61—82no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—44no—
136C22×Dic61—41no—
137C22×Dic61—41no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—22no—
141C2×D244—22no—
142C2×Dic122—24no—
143C2×Dic122—24no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—44no—
150C2×Dic122—44no—
151C2×Dic122—44no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—41no—
155C2×Dic122—41no—
156C2×Dic122#4114yes—
157M16×C61—22no—
158M16×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C2×C8×C61C2×D412yes—
163C2×C8×C61C2×D412yes—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C2×D411yescharacteristic
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C4⋊C4×C61C2×D411yescharacteristic
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178#11758198—41no—
179#11758198—41no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182#11758194—22no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—42no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—22no—
190C2×#92418—42no—
191#11758190—44no—
192#11758190—44no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—41no—
196C23×C61#12111yescharacteristic
197C23×C61#12111yescharacteristic
198C23×C61C2×#4111yescharacteristic
199C22×D61—41no—
200C22×D61—42no—
201C22×D61—81no—
202C22×D61—81no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—42no—
215C2×C4×C61—81no—
216C2×C4×C61#12111yescharacteristic
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C2×#4112yes—
221C2×C4×C61H(5,𝔽2)12yes—
222C22⋊D61—82no—
223C22⋊D61—82no—
224C2×Dic61—24no—
225C2×Dic61—44no—
226C2×Dic61—44no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—81no—
231D244—41no—
232D244—41no—
233D244—42no—
234Dic122—24no—
235Dic122—24no—
236Dic122—44no—
237Dic122—44no—
238Dic122—42no—
239Dic122—42no—
240C8×C61—22no—
241C8×C61—22no—
242C8×C61—42no—
243#92418—41no—
244#92418—42no—
245#92418—41no—
246#24523—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—44no—
255C22×C61#47112yes—
256C22×C61C2×#12111yescharacteristic
257C22×C61#55112yes—
258C2×D61—81no—
259C2×D61—81no—
260C2×D61—81no—
261C2×D61—162no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—41no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61#52511yescharacteristic
269C4×C61#53511yescharacteristic
270C4×C61#54612yes—
271Dic61—44no—
272Dic61—81no—
273#3094—611no—
274C2×#448—611no—
275#2924D6111yescharacteristic, complemented
276C2×#419—611no—
277#2064—612no—
278#2046—612no—
279#1977—612no—
280#1930—612no—
281#1900—611no—
282#1855—611no—
283#1784—611no—
284C2×C61—22no—
285C2×C61—22no—
286C2×C61—41no—
287C2×C61#241111yescharacteristic
288C2×C61C2×#47111yescharacteristic
289C2×C61#306911yescharacteristic
290D61—161no—
291C23×Q8—611no—
292C22×SD16—612no—
293C22×M16—611no—
294C2×#121—611no—
295C2×#117C2×D6111yescharacteristic
296#482—1221no—
297#482—1221no—
298#481—1221no—
299#480—1221no—
300#448—1221no—
301#448—1221no—
302#448—1222no—
303#433—22no—
304#432—21no—
305#431—21no—
306C2×#108—612no—
307#428—611no—
308#428—611no—
309#424C2×D6112yes—
310#419—612no—
311#419—612no—
312#419—1221no—
313#419—1221no—
314#418—611no—
315#418—611no—
316C2×#103—612no—
317C2×#103—1221no—
318C2×#103—1221no—
319C2×#101—612no—
320C2×#101—612no—
321C2×#100—612no—
322#407—611no—
323#407C2×D6111yescharacteristic
324C2×C22⋊C8C2×D6111yescharacteristic
325#405C2×D6112yes—
326#343—612no—
327#341—612no—
328C61#2452311yescharacteristic, complemented, Sylow
329C22×Q8—612no—
330C22×Q8—1224no—
331C22×Q8—1221no—
332C22×Q8—1221no—
333C22×Q8—1221no—
334C22×Q8—1221no—
335C22×Q8—1221no—
336C22×Q8—2442no—
337C22×D4—611no—
338C22×D4—1221no—
339C23×C4—1221no—
340C23×C4C22×D6111yescharacteristic
341C2×SD16—1224no—
342C2×SD16—1222no—
343C2×SD16—1222no—
344C2×SD16—2442no—
345C2×M16—612no—
346C2×M16—611no—
347C2×M16—611no—
348C2×M16—1221no—
349C2×M16—2442no—
350C2×M16C22×D6112yes—
351C22×C8—1221no—
352C22×C8C22×D6112yes—
353#121—1221no—
354#121—1221no—
355#121—1221no—
356#121—1221no—
357#117—21no—
358#117—21no—
359#117D24411yescharacteristic
360#117D24411yescharacteristic
361#117C22×D6111yescharacteristic
362#117C22×D6111yescharacteristic
363C2×C4⋊C4—611no—
364C2×C4⋊C4D24411yescharacteristic
365C2×C4⋊C4D24411yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1221no—
368C2×C42—21no—
369#108—2442no—
370#107—1221no—
371#107—1221no—
372#106—1221no—
373#106—1221no—
374C4⋊C8—22no—
375C4⋊C8—22no—
376#103—1222no—
377#103—1222no—
378#103—1222no—
379#103—1222no—
380#103—1222no—
381#103—1222no—
382#102—1221no—
383#102—1221no—
384#102—1221no—
385#102—1221no—
386#101—1224no—
387#101—2442no—
388#100—2442no—
389C22⋊C8—21no—
390C22⋊C8—21no—
391C22⋊C8—1221no—
392C22⋊C8—1221no—
393C22⋊C8C22⋊D6111yescharacteristic
394C22⋊C8C22⋊D6111yescharacteristic
395C22⋊C8C22⋊D6112yes—
396#97—22no—
397C4×C8—22no—
398C24—1221no—
399C24#1175822211yescharacteristic
400C2×Q8—614no—
401C2×Q8—1224no—
402C2×Q8—1224no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—2444no—
409C2×Q8—2444no—
410C2×Q8—2444no—
411C2×Q8—2442no—
412C2×Q8—2442no—
413C2×Q8—2441no—
414C2×Q8—2441no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4—1222no—
418C2×D4—1222no—
419C2×D4—4882no—
420C2×D4—2441no—
421C2×D4—2441no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—1222no—
427C22×C4—1221no—
428C22×C4—2444no—
429C22×C4—2441no—
430C22×C4—2441no—
431C22×C4C2×D24411yescharacteristic
432C22×C4C2×D24411yescharacteristic
433C22×C4#1175822211yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437SD16—2444no—
438SD16—2444no—
439M16—22no—
440M16—22no—
441M16—1222no—
442M16—1222no—
443M16—1222no—
444M16—1222no—
445M16—2442no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1222no—
452C2×C8—2442no—
453C2×C8C2×C22⋊D6112yes—
454C2×C8C2×C22⋊D6112yes—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4C2×D24411yescharacteristic
460C4⋊C4C2×D24411yescharacteristic
461C4⋊C4C2×D24411yescharacteristic
462C4⋊C4C2×D24411yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—2441no—
468C22⋊C4—2441no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—4881no—
477C23—4881no—
478C23—2442no—
479C23—2441no—
480C23#1451975111yescharacteristic
481C23#1451975111yescharacteristic
482C23C2×#1175822211yescharacteristic
483Q8—1224no—
484Q8—1224no—
485Q8—2444no—
486Q8—2444no—
487Q8—2442no—
488Q8—2442no—
489D4—4882no—
490D4—4882no—
491D4—2441no—
492D4—2441no—
493D4—2442no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—42no—
506C2×C4—81no—
507C2×C4—1224no—
508C2×C4—4881no—
509C2×C4—2441no—
510C2×C4—2444no—
511C2×C4—2444no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4#1451975111yescharacteristic
515C2×C4#1451975111yescharacteristic
516C2×C4C22×C22⋊D6111yescharacteristic
517C2×C4C22×D24411yescharacteristic
518C2×C4#41×D6112yes—
519C2×C4#1451982312yes—
520C8—22no—
521C8—22no—
522C8—42no—
523C8—2441no—
524C8—2442no—
525C8—2441no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—44no—
534C22—9762no—
535C22—4881no—
536C22—4881no—
537C22—4881no—
538C22#1684002512yes—
539C22C2×#1451975111yescharacteristic
540C22#1684043612yes—
541C4—21no—
542C4—21no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—4881no—
548C4—2444no—
549C4#1684027411yescharacteristic
550C4#1684033111yescharacteristic
551C4#1684040012yes—
552C2—22no—
553C2—22no—
554C2—41no—
555C2—9761no—
556C2#2144919811yescharacteristic
557C2C2×#1684002511yescharacteristic
558C2#2145555611yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2924C2
socleC22×C61C2×#121
prime radicalC2×#14519751C22
hypercenter#2924D61
hypocenterC61#24523
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2924

Complements

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

NH ≅ G/Ntype
3. C2×#16839770C2semidirect
4. #21447185C2semidirect
5. #21447082C2semidirect
6. #21446722C2semidirect
7. #21446474C2semidirect
9. #21446026C2semidirect
10. #21445742C2semidirect
11. C61×#2924C2semidirect
275. #2924D61semidirect
328. C61#24523semidirect

Maximal subgroups

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

subgroupindex
#214557592
C2×#168399032
C2×#168397702
#214471852
#214470822
#214467222
#214464742
#214462802
#214460262
#214457422
C61×#29242
#2452361