Finite Group Databasebeta

← #29657163

Subgroups — #29657163

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459631C211yesmaximal, characteristic
2C2×#16840141C211yesmaximal, characteristic, complemented
3#21455628C211yesmaximal, characteristic
4#21444958C214yesmaximal, complemented
5#21444952C214yesmaximal, complemented
6#21444499C211yesmaximal, characteristic, complemented
7#21444274C211yesmaximal, characteristic, complemented
8#21444199C211yesmaximal, characteristic
9C61×#3230C211yesmaximal, characteristic, complemented
10C22×#11758202—22no—
11Q8×Dic122C2212yes—
12#16840490—21no—
13#16840472C2212yes—
14#16840470—22no—
15Q8×D244—21no—
16#16840425—22no—
17#16840392—22no—
18#16840390—21no—
19#16840317—21no—
20C2×#14519777C2211yescharacteristic
21C4⋊Q8×D61—21no—
22#16840242—21no—
23C4×#11758222C2211yescharacteristic
24#16840141—22no—
25#16840141C2214yes—
26#16840054C2212yes—
27#16840049C2212yes—
28#16840044—21no—
29#16840039—21no—
30#16839802—21no—
31#16839786C2211yescharacteristic
32#16839761—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839510C2214yes—
35Q8⋊#92418C2214yes—
36#16839492C2212yes—
37#16839489C2212yes—
38C61×#533C2211yescharacteristic
39C2×C4⋊Q8×C61C2211yescharacteristic
40C61×#500—21no—
41C61×#487—21no—
42C61×#484—22no—
43C61×#477C2214yes—
44C61×#431C2211yescharacteristic
45C2×#11758222D411yescharacteristic
46C22×Dic122D412yes—
47C2×Q8×D61—22no—
48C2×#11758202—28no—
49C2×#11758202—48no—
50C2×#11758194—21no—
51C22×#92418—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—22no—
55#14519777D412yes—
56#14519777C2312yes—
57#14519776—22no—
58#14519776—24no—
59#14519776—22no—
60#14519775—22no—
61#14519769—22no—
62#14519768—22no—
63#14519767—24no—
64#14519757—24no—
65#14519757—22no—
66#14519755—24no—
67#14519754—24no—
68#14519753—22no—
69Q8×Dic61—22no—
70C4×Dic122—22no—
71C4×Dic122D411yescharacteristic
72C4×Dic122C2311yescharacteristic
73#14519742—22no—
74C4×D244—22no—
75C4×D244D411yescharacteristic
76C4×D244C2311yescharacteristic
77C4×C22⋊D61—21no—
78C4×C22⋊D61—21no—
79#14519737—24no—
80#14519736D411yescharacteristic
81#14519735—22no—
82#14519734—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61—22no—
87C42×D61—21no—
88#14519710—22no—
89#14519708—22no—
90C4⋊#92418—21no—
91C4⋊#92418C2312yes—
92#14519700—22no—
93#14519697—22no—
94C4×#92418—21no—
95C4×#92418C2312yes—
96C22×Q8×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61D411yescharacteristic
101C4⋊Q8×C61D411yescharacteristic
102C4⋊Q8×C61C2312yes—
103#125×C61—21no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61D412yes—
106C4×Q8×C61—21no—
107C4×Q8×C61C2312yes—
108#117×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C42×C61C2311yescharacteristic
112#107×C61—24no—
113C4⋊C8×C61C2312yes—
114#103×C61—24no—
115C22⋊C8×C61—21no—
116C4×C8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—22no—
119C2×Q8×C61—44no—
120C2×Q8×C61—41no—
121C2×Q8×C61C2×D412yes—
122C22×C4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C2×C22⋊D61—21no—
127C22×Dic61—22no—
128Q8×D61—84no—
129#11758222—22no—
130#11758222Q1614yescomplemented
131C2×D244C2×D411yescharacteristic
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—24no—
136C2×Dic122—22no—
137C2×Dic122—24no—
138C2×Dic122—44no—
139C2×Dic122C2×D411yescharacteristic
140C2×Dic122C2×D414yes—
141C2×C8×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—22no—
153C42×C61—22no—
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2411yescharacteristic
158C22⋊Dic61—21no—
159#11758202—416no—
160#11758198—24no—
161#11758198—21no—
162#11758198—21no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61C2×D411yescharacteristic
169C4⋊Dic61C2×D411yescharacteristic
170#11758196—24no—
171#11758196—24no—
172#11758196—24no—
173#11758196—21no—
174#11758196—21no—
175C4×Dic61—24no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179#11758194—42no—
180C2×#92418—24no—
181C2×#92418—22no—
182C2×#92418—22no—
183C23×C61#12011yescharacteristic
184C22×D61—21no—
185Q8×C61—24no—
186Q8×C61—44no—
187Q8×C61—42no—
188Q8×C61—82no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—42no—
194C2×C4×C61—82no—
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61C2×Q1612yes—
199C2×C4×C61#13712yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C22⋊D61—24no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—24no—
208C2×Dic61—82no—
209D244—21no—
210D244C2×Q1612yescomplemented
211C4×D61—24no—
212C4×D61—42no—
213C4×D61—42no—
214C4×D61—84no—
215Dic122—21no—
216Dic122—44no—
217Dic122—44no—
218Dic122—84no—
219Dic122C2×Q1612yescomplemented
220C8×C61—42no—
221#92418—24no—
222#92418—42no—
223#18623—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61#45112yes—
227C22×C61C2×#12011yescharacteristic
228C2×D61—22no—
229C2×D61—41no—
230C2×D61—41no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—42no—
234C4×C61—42no—
235C4×C61—82no—
236C4×C61#45112yes—
237C4×C61#45312yes—
238C4×C61C22×Q1611yescharacteristic
239C4×C61C2×#13711yescharacteristic
240C4×C61#57812yes—
241Dic61—22no—
242Dic61—41no—
243Dic61—44no—
244Dic61—41no—
245#3503—611no—
246#3230D6111yescharacteristic, complemented
247C2×#494—611no—
248#3079—611no—
249#1633—614no—
250#1632—614no—
251#1548—611no—
252#1493—611no—
253#1476—611no—
254C2×C61—22no—
255C2×C61C2×#45111yescharacteristic
256C2×C61#301111yescharacteristic
257C2×C61C2×#45311yescharacteristic
258D61—42no—
259C22×Q16—1222no—
260Q82—612no—
261#557—1221no—
262#554—612no—
263#554—1222no—
264D4×Q8—1221no—
265D4×Q8—1222no—
266#544—1221no—
267#544—1222no—
268#533—1221no—
269#533C2×D6111yescharacteristic
270C2×C4⋊Q8—611no—
271C2×C4⋊Q8—1221no—
272C2×C4⋊Q8C2×D6111yescharacteristic
273#520—1221no—
274C4×#41—611no—
275#500—21no—
276#494—614no—
277#494—1222no—
278#487—21no—
279#484—22no—
280#477—612no—
281#477C2×D6114yes—
282#476—612no—
283#475—1221no—
284#474—1221no—
285#431C2×D6111yescharacteristic
286#428—1221no—
287#423—611no—
288#417—1221no—
289C2×C4×C8—611no—
290#333—614no—
291Q8⋊C8—614no—
292Q8⋊C8—612no—
293D4⋊C8—612no—
294C61#1862311yescharacteristic, complemented, Sylow
295C2×#41—611no—
296C22×Q8—21no—
297C22×Q8—612no—
298C22×Q8—1222no—
299C2×Q16—1228no—
300C2×Q16—2448no—
301C2×M16—1221no—
302C22×C8—1221no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—612no—
307C4⋊Q8—612no—
308C4⋊Q8—1222no—
309C4⋊Q8—1224no—
310C4⋊Q8—1222no—
311C4⋊Q8—1222no—
312C4⋊Q8—1222no—
313C4⋊Q8—1221no—
314C4⋊Q8—1221no—
315C4⋊Q8C22⋊D6111yescharacteristic
316C4⋊Q8C22⋊D6111yescharacteristic
317C4⋊Q8C22×D6112yes—
318#125—21no—
319#125—1222no—
320#125—1222no—
321#125—1224no—
322C22⋊Q8—21no—
323C22⋊Q8—1224no—
324C22⋊Q8—1224no—
325C22⋊Q8—1222no—
326C22⋊Q8—1224no—
327C22⋊Q8—1222no—
328C22⋊Q8C22⋊D6112yes—
329C4×Q8—21no—
330C4×Q8—611no—
331C4×Q8—611no—
332C4×Q8—1222no—
333C4×Q8—1222no—
334C4×Q8—1222no—
335C4×Q8C22×D6112yes—
336C4×D4—611no—
337C4×D4—611no—
338C4×D4—1222no—
339C4×D4—1224no—
340C4×D4—1221no—
341C4×D4—1221no—
342#117—21no—
343#117—611no—
344#117—1222no—
345#117—1221no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1222no—
352C2×C42—1221no—
353C2×C42C22×D6111yescharacteristic
354#107—24no—
355#107—1222no—
356#106—1222no—
357C4⋊C8—612no—
358C4⋊C8—1221no—
359C4⋊C8C22×D6112yes—
360#103—24no—
361#103—1222no—
362#102—1222no—
363C22⋊C8—21no—
364C4×C8—21no—
365C4×C8—612no—
366C4×C8—1221no—
367#41—614no—
368#41—1222no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—44no—
372C2×Q8—41no—
373C2×Q8—611no—
374C2×Q8—614no—
375C2×Q8—1224no—
376C2×Q8—1222no—
377C2×Q8—4884no—
378C2×Q8—2444no—
379C2×Q8C2×C22⋊D6112yes—
380C2×D4—611no—
381C2×D4—1221no—
382C22×C4—41no—
383C22×C4—1224no—
384C22×C4—1221no—
385C22×C4—1222no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4D4×D6111yescharacteristic
389C22×C4D4×D6111yescharacteristic
390C22×C4C2×C22⋊D6111yescharacteristic
391Q16—24416no—
392M16—2442no—
393C2×C8—22no—
394C2×C8—1224no—
395C2×C8—1222no—
396C2×C8—1222no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—611no—
407C4⋊C4—611no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1224no—
411C4⋊C4—1224no—
412C4⋊C4—1222no—
413C4⋊C4—1221no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—1221no—
418C4⋊C4C2×C22⋊D6112yes—
419C22⋊C4—22no—
420C22⋊C4—1224no—
421C22⋊C4—1221no—
422C22⋊C4—1221no—
423C22⋊C4—1221no—
424C42—22no—
425C42—1224no—
426C42—1221no—
427C42—1221no—
428C42—1221no—
429C42D4×D6111yescharacteristic
430C42D4×D6111yescharacteristic
431C42C2×C22⋊D6111yescharacteristic
432C42C23×D6111yescharacteristic
433C23—1221no—
434C23#1451974511yescharacteristic
435Q8—24no—
436Q8—44no—
437Q8—42no—
438Q8—82no—
439Q8—612no—
440Q8—1221no—
441Q8—4884no—
442Q8—2444no—
443Q8—2444no—
444D4—612no—
445D4—1224no—
446D4—1221no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—42no—
452C2×C4—82no—
453C2×C4—1221no—
454C2×C4—1224no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—1224no—
458C2×C4—4884no—
459C2×C4—4882no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4#1451974511yescharacteristic
463C2×C4#1451974511yescharacteristic
464C2×C4#1451974511yescharacteristic
465C2×C4Q16×D6112yes—
466C2×C4#1451980912yes—
467C2×C4C2×D4×D6111yescharacteristic
468C2×C4C2×D4×D6111yescharacteristic
469C2×C4C22×C22⋊D6111yescharacteristic
470C8—42no—
471C8—1224no—
472C8—2442no—
473C22—21no—
474C22—21no—
475C22—1222no—
476C22—2441no—
477C22—2441no—
478C22#1683991612yes—
479C22C2×#1451974511yescharacteristic
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—82no—
485C4—1222no—
486C4—2441no—
487C4—2444no—
488C4—2441no—
489C4#1683991612yes—
490C4#1683992812yes—
491C4C2×Q16×D6111yescharacteristic
492C4C2×#1451980911yescharacteristic
493C4#1684057712yes—
494C2—22no—
495C2—2442no—
496C2C2×#1683991611yescharacteristic
497C2#2145511211yescharacteristic
498C2C2×#1683992811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3230C2
socleC22×C61C2×#120
prime radicalC2×#11758222D4
hypercenter#3230D61
hypocenterC61#18623
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
2. C2×#16840141C2semidirect
4. #21444958C2semidirect
5. #21444952C2semidirect
6. #21444499C2semidirect
7. #21444274C2semidirect
9. C61×#3230C2semidirect
130. #11758222Q16semidirect
210. D244C2×Q16semidirect
219. Dic122C2×Q16semidirect
246. #3230D61semidirect
294. C61#18623semidirect

Maximal subgroups

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

subgroupindex
#214596312
C2×#168401412
#214556282
#214449582
#214449522
#214444992
#214442742
#214441992
C61×#32302
#1862361