Finite Group Databasebeta

← #29718923

Subgroups — #29718923

Order 15616.

976 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455767C211yesmaximal, characteristic, complemented
2C2×#16839915C211yesmaximal, characteristic
3C2×#16839766C211yesmaximal, characteristic
4#21447178C212yesmaximal, complemented
5#21447082C212yesmaximal, complemented
6#21446722C212yesmaximal, complemented
7#21446481C212yesmaximal, complemented
8#21446285C211yesmaximal, characteristic, complemented
9#21446026C211yesmaximal, characteristic, complemented
10#21445739C211yesmaximal, characteristic
11C61×#2924C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×#11758190C2212yes—
14C22×#11758194C2211yescharacteristic
15C2×#14519756C2211yescharacteristic
16#16840103—21no—
17#16840103—21no—
18#16840097—21no—
19#16840092—21no—
20#16839915—21no—
21#16839915—21no—
22#16839915—22no—
23C2×#14519713C2212yescomplemented
24#16839805C2211yescharacteristic, complemented
25#16839805C2211yescharacteristic, complemented
26#16839770C2211yescharacteristic, complemented
27#16839770C2211yescharacteristic, complemented
28#16839766—21no—
29#16839766—21no—
30#16839766C2212yes—
31#16839766C2212yes—
32C2×#14519699C2212yes—
33C2×#14519697—21no—
34C2×#14519697—21no—
35C2×#14519695C2212yescomplemented
36C2×#14519693C2212yes—
37C2×#14519693C2212yes—
38#16839726C2211yescharacteristic
39#16839530C2212yescomplemented
40#16839527C2212yescomplemented
41C2×#117×C61C2211yescharacteristic
42C61×#433—22no—
43C61×#432—21no—
44C61×#431—21no—
45C61×#424C2212yescomplemented
46C61×#407C2211yescharacteristic
47C2×C22⋊C8×C61C2211yescharacteristic
48C61×#405C2212yescomplemented
49C24×D61—21no—
50C22×Dic122C2311yescharacteristic
51C22×D244—21no—
52C22×D244—21no—
53C22×D244—24no—
54C22×D244—21no—
55C22×D244—21no—
56C22×D244—21no—
57C22×D244—42no—
58C22×D244C2312yes—
59C2×#11758190—24no—
60C2×#11758190—22no—
61C2×#11758190—22no—
62C2×#11758190—42no—
63C2×#11758194—21no—
64C2×#11758194—42no—
65C2×#11758194C2311yescharacteristic
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2312yes—
68C22×#92418—21no—
69#14519756—21no—
70#14519756—21no—
71#14519756—21no—
72#14519756—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×C22⋊Dic61—21no—
76#14519713—42no—
77#14519710—21no—
78#14519710—21no—
79#14519708—21no—
80#14519708—21no—
81#14519700—21no—
82#14519700—21no—
83#14519700—21no—
84#14519700—21no—
85#14519699—22no—
86#14519699—22no—
87#14519697—22no—
88#14519697—22no—
89#14519697—22no—
90#14519697—22no—
91#14519695—42no—
92#14519693—24no—
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—22no—
120C23×D61—44no—
121C23×D61—41no—
122C23×D61—41no—
123C23×D61—82no—
124C23×D61—82no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61#4111yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C2×D244—24no—
138C2×D244—24no—
139C2×D244—22no—
140C2×D244—22no—
141C2×D244—22no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—44no—
145C2×D244—44no—
146C2×D244—44no—
147C2×D244—42no—
148C2×D244—42no—
149C2×D244—41no—
150C2×D244—41no—
151C2×D244#4114yes—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156M16×C61—22no—
157M16×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61C2×D412yes—
162C2×C8×C61C2×D412yes—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C2×D411yescharacteristic
166C4⋊C4×C61C2×D411yescharacteristic
167C4⋊C4×C61C2×D411yescharacteristic
168C4⋊C4×C61C2×D411yescharacteristic
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181#11758196—41no—
182#11758196—41no—
183#11758194—22no—
184#11758194—22no—
185#11758194—22no—
186#11758194—22no—
187#11758194—42no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—22no—
191C2×#92418—42no—
192#11758190—44no—
193#11758190—44no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—41no—
197C23×C61#12111yescharacteristic
198C23×C61#12111yescharacteristic
199C23×C61C2×#4111yescharacteristic
200C22×D61—24no—
201C22×D61—44no—
202C22×D61—44no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—82no—
206C22×D61—82no—
207C22×D61—82no—
208C22×D61—816no—
209C22×D61—82no—
210C22×D61—82no—
211C22×D61—82no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—42no—
224C2×C4×C61—81no—
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61#12111yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4112yes—
230C2×C4×C61H(5,𝔽2)12yes—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—81no—
235D244—24no—
236D244—24no—
237D244—44no—
238D244—44no—
239D244—42no—
240D244—42no—
241Dic122—41no—
242Dic122—41no—
243Dic122—42no—
244C8×C61—22no—
245C8×C61—22no—
246C8×C61—42no—
247#92418—41no—
248#92418—42no—
249#92418—41no—
250#24521—611nomaximal, Sylow
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—44no—
259C22×C61#47112yes—
260C22×C61C2×#12111yescharacteristic
261C22×C61#55112yes—
262C2×D61—44no—
263C2×D61—84no—
264C2×D61—88no—
265C2×D61—88no—
266C2×D61—82no—
267C2×D61—82no—
268C2×D61—84no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61#52511yescharacteristic
276C4×C61#53511yescharacteristic
277C4×C61#54612yes—
278Dic61—81no—
279Dic61—81no—
280#3095—611no—
281C2×#450—611no—
282#2924D6111yescharacteristic, complemented
283C2×#418—611no—
284#2063—612no—
285#2046—612no—
286#1977—612no—
287#1931—612no—
288#1901—611no—
289#1855—611no—
290#1783—611no—
291C2×C61—22no—
292C2×C61—22no—
293C2×C61—41no—
294C2×C61#241111yescharacteristic
295C2×C61C2×#47111yescharacteristic
296C2×C61#306811yescharacteristic
297D61—84no—
298C23×D4—611no—
299C22×SD16—612no—
300C22×M16—611no—
301C2×C22⋊Q8—611no—
302C2×#117C2×D6111yescharacteristic
303#485—1221no—
304#485—1221no—
305#484—1221no—
306#483—1221no—
307#450—1221no—
308#450—1221no—
309#450—1222no—
310#433—22no—
311#432—21no—
312#431—21no—
313C2×#108—612no—
314#428—611no—
315#428—611no—
316#424C2×D6112yescomplemented
317#419—611no—
318#419—611no—
319#418—612no—
320#418—612no—
321#418—1221no—
322#418—1221no—
323C2×#102—612no—
324C2×#102—1221no—
325C2×#102—1221no—
326C2×#101—612no—
327C2×#100—612no—
328C2×#100—612no—
329#407—611no—
330#407C2×D6111yescharacteristic
331C2×C22⋊C8C2×D6111yescharacteristic
332#405C2×D6112yescomplemented
333#343—612no—
334#341—612no—
335C61#2452111yescharacteristic, complemented, Sylow
336C25—1221no—
337C22×Q8—611no—
338C22×D4—612no—
339C22×D4—1224no—
340C22×D4—1221no—
341C22×D4—1221no—
342C22×D4—1221no—
343C22×D4—1221no—
344C22×D4—1221no—
345C22×D4—2442no—
346C23×C4C22×D6111yescharacteristic
347C2×SD16—1224no—
348C2×SD16—1222no—
349C2×SD16—1222no—
350C2×SD16—2442no—
351C2×M16—612no—
352C2×M16—611no—
353C2×M16—611no—
354C2×M16—1221no—
355C2×M16—2442no—
356C2×M16C22×D6112yes—
357C22×C8—1221no—
358C22×C8C22×D6112yes—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8—1221no—
362C22⋊Q8—1221no—
363#117—21no—
364#117—21no—
365#117D24411yescharacteristic
366#117D24411yescharacteristic
367#117C22×D6111yescharacteristic
368#117C22×D6111yescharacteristic
369C2×C4⋊C4—611no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4D24411yescharacteristic
372C2×C4⋊C4D24411yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1221no—
375C2×C42—21no—
376#108—2442no—
377#107—1221no—
378#107—1221no—
379#106—1221no—
380#106—1221no—
381C4⋊C8—22no—
382C4⋊C8—22no—
383#103—1221no—
384#103—1221no—
385#103—1221no—
386#103—1221no—
387#102—1222no—
388#102—1222no—
389#102—1222no—
390#102—1222no—
391#102—1222no—
392#102—1222no—
393#101—2442no—
394#100—1224no—
395#100—2442no—
396C22⋊C8—21no—
397C22⋊C8—21no—
398C22⋊C8—1221no—
399C22⋊C8—1221no—
400C22⋊C8C22⋊D6111yescharacteristic
401C22⋊C8C22⋊D6111yescharacteristic
402C22⋊C8C22⋊D6112yes—
403#97—22no—
404C4×C8—22no—
405C24—1222no—
406C24—4882no—
407C24—4882no—
408C24—2444no—
409C24—2441no—
410C24—2441no—
411C24#1175822211yescharacteristic
412C2×Q8—1221no—
413C2×Q8—1221no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×D4—614no—
417C2×D4—1224no—
418C2×D4—1224no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—1222no—
424C2×D4—2444no—
425C2×D4—2444no—
426C2×D4—2444no—
427C2×D4—2442no—
428C2×D4—2442no—
429C2×D4—2441no—
430C2×D4—2441no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4C2×D24411yescharacteristic
438C22×C4C2×D24411yescharacteristic
439C22×C4#1175822211yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C23×D6111yescharacteristic
443SD16—2444no—
444SD16—2444no—
445M16—22no—
446M16—22no—
447M16—1222no—
448M16—1222no—
449M16—1222no—
450M16—1222no—
451M16—2442no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1222no—
458C2×C8—2442no—
459C2×C8C2×C22⋊D6112yes—
460C2×C8C2×C22⋊D6112yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4C2×D24411yescharacteristic
468C4⋊C4C2×D24411yescharacteristic
469C4⋊C4C2×D24411yescharacteristic
470C4⋊C4C2×D24411yescharacteristic
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—2441no—
476C22⋊C4—2441no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C23—22no—
482C23—22no—
483C23—41no—
484C23—1224no—
485C23—4882no—
486C23—4882no—
487C23—4882no—
488C23—4882no—
489C23—4882no—
490C23—48816no—
491C23—4882no—
492C23—2444no—
493C23—2444no—
494C23—2442no—
495C23—2442no—
496C23#1451975111yescharacteristic
497C23#1451975111yescharacteristic
498C23C2×#1175822211yescharacteristic
499Q8—2441no—
500Q8—2441no—
501Q8—2442no—
502D4—1224no—
503D4—1224no—
504D4—2444no—
505D4—2444no—
506D4—2442no—
507D4—2442no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—81no—
521C2×C4—4881no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4#1451975111yescharacteristic
526C2×C4#1451975111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4C22×D24411yescharacteristic
529C2×C4#41×D6112yes—
530C2×C4#1451982312yes—
531C8—22no—
532C8—22no—
533C8—42no—
534C8—2441no—
535C8—2442no—
536C8—2441no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—44no—
545C22—4884no—
546C22—4888no—
547C22—4884no—
548C22—4888no—
549C22—4882no—
550C22—4882no—
551C22—2444no—
552C22#1684002512yes—
553C22C2×#1451975111yescharacteristic
554C22#1684043612yes—
555C4—21no—
556C4—21no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—4881no—
562C4—4881no—
563C4#1684027411yescharacteristic
564C4#1684033111yescharacteristic
565C4#1684040012yes—
566C2—22no—
567C2—22no—
568C2—41no—
569C2—4884no—
570C2#2144919811yescharacteristic
571C2C2×#1684002511yescharacteristic
572C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2924C2
socleC22×C61C2×#121
prime radicalC23×D244C22
hypercenter#2924D61
hypocenterC61#24521
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
1. #21455767C2semidirect
4. #21447178C2semidirect
5. #21447082C2semidirect
6. #21446722C2semidirect
7. #21446481C2semidirect
8. #21446285C2semidirect
9. #21446026C2semidirect
11. C61×#2924C2semidirect
23. C2×#14519713C22semidirect
24. #16839805C22semidirect
25. #16839805C22semidirect
26. #16839770C22semidirect
27. #16839770C22semidirect
35. C2×#14519695C22semidirect
39. #16839530C22semidirect
40. #16839527C22semidirect
45. C61×#424C22semidirect
48. C61×#405C22semidirect
282. #2924D61semidirect
316. #424C2×D61semidirect
332. #405C2×D61semidirect
335. C61#24521semidirect

Maximal subgroups

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

subgroupindex
#214557672
C2×#168399152
C2×#168397662
#214471782
#214470822
#214467222
#214464812
#214462852
#214460262
#214457392
C61×#29242
#2452161