Finite Group Databasebeta

← #29667908

Subgroups — #29667908

Order 15616.

720 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456941C211yesmaximal, characteristic
2#21456763C211yesmaximal, characteristic, complemented
3#21456380C211yesmaximal, characteristic
4#498×D61C211yesmaximal, characteristic, complemented
5#21456214C211yesmaximal, characteristic
6#21456196C211yesmaximal, characteristic, complemented
7#21445482C211yesmaximal, characteristic, complemented
8#21445476C211yesmaximal, characteristic, complemented
9#21445466C211yesmaximal, characteristic, complemented
10#21445464C211yesmaximal, characteristic, complemented
11#21445421C211yesmaximal, characteristic, complemented
12#21445420C211yesmaximal, characteristic, complemented
13#21445414C211yesmaximal, characteristic, complemented
14C8×#11758189C211yesmaximal, characteristic, complemented
15C61×#1707C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840313C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840160C2211yescharacteristic
22#16840159—22no—
23#16840158C2211yescharacteristic
24#16840158C2211yescharacteristic
25#16840141C2211yescharacteristic
26#16840135C2211yescharacteristic
27#16840129—22no—
28#16840123—22no—
29#16840114C2211yescharacteristic
30#16840114C2211yescharacteristic
31#16840113C2211yescharacteristic
32#16840113C2211yescharacteristic
33#16840112C2211yescharacteristic
34#16840112C2211yescharacteristic
35#16840112C2211yescharacteristic
36#16840110C2211yescharacteristic
37#16840110C2211yescharacteristic
38#16840110C2211yescharacteristic
39#16840109C2211yescharacteristic
40#16840109C2211yescharacteristic
41#16840077—22no—
42#16840041—22no—
43#16839992—22no—
44#16839970—22no—
45#16839916—22no—
46#16839849—22no—
47#16839813C2211yescharacteristic
48#106×D61—22no—
49#16839769—22no—
50#16839762—22no—
51C4×#11758189C2211yescharacteristic
52C4×C8×D61C2211yescharacteristic
53#16839484C2211yescharacteristic
54#16839483C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#500C2211yescharacteristic
57C61×#498C2211yescharacteristic
58C61×#498C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#487C2211yescharacteristic
62C82×C61C2211yescharacteristic
63C2×#11758226—21no—
64C2×Q8×D61—21no—
65C2×Q8×D61—21no—
66C2×#11758202—22no—
67C2×#11758190—22no—
68C2×#11758189D412yes—
69C2×C8×D61D412yes—
70#14519777C2311yescharacteristic
71#14519777C2311yescharacteristic
72#14519777C2311yescharacteristic
73#14519776—21no—
74#14519776—21no—
75#14519776—21no—
76#14519776—21no—
77#14519775—21no—
78#14519775—21no—
79#14519774C2311yescharacteristic
80#14519768—21no—
81#14519766—21no—
82#14519757—21no—
83#14519753—21no—
84#14519752—21no—
85#14519752—21no—
86Q8×Dic61—21no—
87#14519742—21no—
88#14519739—21no—
89#14519739—21no—
90#14519735—21no—
91C4⋊C4×D61—21no—
92C4⋊C4×D61—21no—
93C4⋊C4×D61—21no—
94C4⋊C4×D61—21no—
95C42×D61C2311yescharacteristic
96#14519710—22no—
97#14519709—22no—
98#14519708—22no—
99#14519708—22no—
100#14519706D412yes—
101#14519702—22no—
102#14519702—22no—
103#14519701—22no—
104#14519701—22no—
105#14519701—22no—
106#14519700—22no—
107#14519700—22no—
108#14519699—22no—
109#14519699—22no—
110#14519697—22no—
111#14519697—22no—
112#14519697—22no—
113#14519689D412yes—
114#14519687D412yes—
115C4×#92418C2311yescharacteristic
116C4×#92418C2311yescharacteristic
117C4×#92418C2311yescharacteristic
118C8×Dic61D412yes—
119C2×Q16×C61—22no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#107×C61—22no—
125#106×C61—22no—
126#106×C61—22no—
127#103×C61—22no—
128#103×C61—22no—
129C4×C8×C61C2311yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136#11758226—44no—
137Q8×D61—44no—
138Q8×D61—82no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—41no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—41no—
153C2×Dic122—41no—
154Q16×C61—82no—
155C2×C8×C61C2×D412yes—
156C2×C8×C61C2×D412yes—
157C2×C8×C61C2×D412yes—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C42×C61C2411yescharacteristic
167#11758202—82no—
168#11758198—41no—
169#11758198—41no—
170#11758198—41no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—41no—
178#11758196—41no—
179#11758196—41no—
180#11758196—41no—
181#11758196—41no—
182#11758196—41no—
183C4×Dic61—41no—
184C4×Dic61C2×D411yescharacteristic
185C4×Dic61C2×D411yescharacteristic
186C4×Dic61C2×D411yescharacteristic
187C2×#92418C2×D412yes—
188C2×#92418C2×D412yes—
189C2×#92418C2×D412yes—
190#11758190—44no—
191#11758189—24no—
192C8×D61—24no—
193C22×D61—41no—
194C22×D61#12711yescharacteristic
195Q8×C61—42no—
196Q8×C61—42no—
197Q8×C61—42no—
198Q8×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61#12711yescharacteristic
210C2×Dic61#12711yescharacteristic
211C2×Dic61#12711yescharacteristic
212D244—42no—
213D244—42no—
214D244—42no—
215D244—82no—
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—82no—
219C4×D61—82no—
220C4×D61—82no—
221C4×D61#13514yes—
222Dic122—42no—
223Dic122—42no—
224Dic122—42no—
225Dic122—82no—
226Dic122—82no—
227C8×C61—22no—
228C8×C61—22no—
229C8×C61#13714yes—
230#92418—22no—
231#92418—22no—
232#92418—22no—
233#19626—611nomaximal, Sylow
234C22×C61C2×#12711yescharacteristic
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61#49512yes—
241C4×C61—81no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61C2×#13512yes—
246C4×C61C2×#13712yes—
247C4×C61#57712yes—
248Dic61—21no—
249Dic61—21no—
250Dic61—81no—
251Dic61—81no—
252Dic61—81no—
253Dic61#49512yes—
254#3226—611no—
255#3209—611no—
256#3167—611no—
257C2×#498—611no—
258#3141—611no—
259#3139—611no—
260#1720—611no—
261#1720—611no—
262#1718—611no—
263#1718—611no—
264#1708—611no—
265#1708—611no—
266#1707—611no—
267#1707D6111yescharacteristic, complemented
268C8×M16—611no—
269C2×C61C2×#49511yescharacteristic
270C2×C61#315211yescharacteristic
271C2×C61#315411yescharacteristic
272D61—22no—
273D61—161no—
274#533—611no—
275#532—611no—
276C2×C4⋊Q8—611no—
277C2×C4⋊Q8—611no—
278#500—611no—
279#500C2×D6111yescharacteristic
280#498—611no—
281#498—611no—
282#498—611no—
283#498—1222no—
284#498C2×D6111yescharacteristic
285#498C2×D6111yescharacteristic
286#494—611no—
287#494C2×D6111yescharacteristic
288#492—611no—
289#491—1222no—
290#489—1222no—
291#487—611no—
292#487—611no—
293#487—611no—
294#487—611no—
295#487—611no—
296#487—611no—
297#487—611no—
298#487C2×D6111yescharacteristic
299#487C2×D6111yescharacteristic
300#486—611no—
301#486—611no—
302#486—611no—
303#486—611no—
304#486—611no—
305#480—1222no—
306#474—1222no—
307#465—1222no—
308#462—1222no—
309#451—1222no—
310C4×SD16—1222no—
311#431—611no—
312C2×#106—1222no—
313#419—1222no—
314#417—1222no—
315C4×M16—611no—
316C2×C4×C8—611no—
317#322—611no—
318#322—611no—
319C82—611no—
320C82C2×D6111yescharacteristic
321C61#1962611yescharacteristic, complemented, Sylow
322C2×#41—1221no—
323C22×Q8—1221no—
324C22×Q8—1221no—
325C2×Q16—22no—
326C2×Q16—1222no—
327C2×SD16—1222no—
328C2×M16—612no—
329C22×C8—612no—
330C4⋊Q8—611no—
331C4⋊Q8—611no—
332C4⋊Q8—611no—
333C4⋊Q8—1221no—
334C4⋊Q8—1221no—
335C4⋊Q8—1221no—
336C4⋊Q8—1221no—
337C4⋊Q8—1221no—
338C4⋊Q8—1221no—
339C4⋊Q8C22×D6111yescharacteristic
340C4⋊Q8C22×D6111yescharacteristic
341C4⋊Q8C22×D6111yescharacteristic
342C4⋊Q8C22×D6111yescharacteristic
343#127—611no—
344#125—1221no—
345#124—1221no—
346C22⋊Q8—1221no—
347C22⋊Q8—1221no—
348#121—1221no—
349#121—1221no—
350C4×Q8—1221no—
351C4×Q8—1221no—
352C4×D4—1221no—
353C4×D4—1221no—
354#117—1221no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C4⋊C4—1221no—
359C2×C42—611no—
360#107—22no—
361#107—1222no—
362#106—22no—
363#106—22no—
364#106—1222no—
365#106—1222no—
366#106—1222no—
367C4⋊C8—612no—
368#103—22no—
369#103—22no—
370#103—1222no—
371#103—1222no—
372#103—1222no—
373#103—1222no—
374#103—1222no—
375#103—1222no—
376#103—1222no—
377#102—1222no—
378#102—1222no—
379#102—1222no—
380#102—1222no—
381#102—1222no—
382C22⋊C8—612no—
383#97—612no—
384C4×C8—612no—
385C4×C8—611no—
386C4×C8—611no—
387C4×C8—611no—
388C4×C8C22×D6111yescharacteristic
389C4×C8C22×D6111yescharacteristic
390C4×C8C22×D6111yescharacteristic
391#41—2444no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—1221no—
397C2×Q8—1221no—
398C2×Q8—1221no—
399C2×Q8—4882no—
400C2×Q8—2441no—
401C2×Q8—2441no—
402C2×Q8—2444no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—2441no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—2441no—
411C22×C4—2441no—
412C22×C4—2441no—
413Q16—82no—
414Q16—4882no—
415SD16—2444no—
416M16—1224no—
417C2×C8—612no—
418C2×C8—612no—
419C2×C8—612no—
420C2×C8—1224no—
421C2×C8D4×D6112yes—
422C2×C8D4×D6112yes—
423C2×C8D4×D6112yes—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—1221no—
437C4⋊C4—1221no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C4⋊C4—2441no—
442C4⋊C4—2441no—
443C4⋊C4—2441no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C22⋊C4—2441no—
447C42—611no—
448C42—611no—
449C42—611no—
450C42—2441no—
451C42C23×D6111yescharacteristic
452C23—611no—
453C23—2441no—
454Q8—42no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—4882no—
459Q8—4882no—
460Q8—2442no—
461Q8—2442no—
462Q8—2442no—
463D4—4882no—
464D4—2442no—
465D4—2442no—
466D4—2442no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—611no—
472C2×C4—614no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—1222no—
476C2×C4—1222no—
477C2×C4—4882no—
478C2×C4—4882no—
479C2×C4—4882no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4C2×D4×D6111yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C2×D4×D6111yescharacteristic
486C8—22no—
487C8—22no—
488C8—1222no—
489C8—1222no—
490C8—1222no—
491C8#1451980914yes—
492C22—612no—
493C22—1221no—
494C22—1221no—
495C22—4881no—
496C22—4881no—
497C22—4881no—
498C22#127×D6111yescharacteristic
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—612no—
504C4—1221no—
505C4—1221no—
506C4—4881no—
507C4—4881no—
508C4—4881no—
509C4#135×D6112yes—
510C4C2×#1451980912yes—
511C4#1684056812yes—
512C2—1222no—
513C2—9761no—
514C2#495×D6111yescharacteristic
515C2#2145627711yescharacteristic
516C2#2145628811yescharacteristic
subgroupG/H
centerC22#127×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1707C2
socleC22×C61C2×#127
prime radical#16840313C22
hypercenter#1707D61
hypocenterC61#19626
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1707

Complements

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

NH ≅ G/Ntype
2. #21456763C2semidirect
4. #498×D61C2semidirect
6. #21456196C2semidirect
7. #21445482C2semidirect
8. #21445476C2semidirect
9. #21445466C2semidirect
10. #21445464C2semidirect
11. #21445421C2semidirect
12. #21445420C2semidirect
13. #21445414C2semidirect
14. C8×#11758189C2semidirect
15. C61×#1707C2semidirect
267. #1707D61semidirect
321. C61#19626semidirect

Maximal subgroups

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

subgroupindex
#214569412
#214567632
#214563802
#498×D612
#214562142
#214561962
#214454822
#214454762
#214454662
#214454642
#214454212
#214454202
#214454142
C8×#117581892
C61×#17072
#1962661