Finite Group Databasebeta

← #29654310

Subgroups — #29654310

Order 15616.

928 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C212yesmaximal, complemented
2#21451024C211yesmaximal, characteristic, complemented
3#21444793C212yesmaximal, complemented
4#21444788C212yesmaximal
5#21444782C214yesmaximal, complemented
6#21444344C212yesmaximal, complemented
7#21444163C211yesmaximal, characteristic, complemented
8C61×#3461C211yesmaximal, characteristic
9C2×#14519777C2211yescharacteristic
10#16840157—22no—
11#16840132—22no—
12#16839972—22no—
13#16839967—22no—
14#16839917C2214yes—
15#16839900C2214yes—
16#16839819C2212yes—
17#16839682C2212yes—
18#16839682C2212yes—
19#16839669—21no—
20#16839660—21no—
21#16839660—21no—
22#16839631—21no—
23C4×C4⋊Dic61C2211yescharacteristic
24#16839499C2214yes—
25#16839496C2214yes—
26#16839488C2212yes—
27#16839488C2212yes—
28C61×#544C2211yescharacteristic
29C61×#542—22no—
30C61×#541—22no—
31C61×#539—21no—
32C61×#539C2212yes—
33C61×#537C2211yescharacteristic
34C61×#536—21no—
35C61×#533—22no—
36C61×#529—22no—
37C61×#529—21no—
38C61×#529C2212yes—
39C4×#41×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C22×Dic122C2312yes—
42C2×#11758202—24no—
43C2×#11758200—24no—
44#14519777—22no—
45#14519777—22no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61—21no—
49C2×#11758196—22no—
50C2×#11758196—22no—
51C2×C4×Dic61—21no—
52C4⋊#92418—22no—
53#14519700—24no—
54#14519700—24no—
55C22⋊#92418C2314yes—
56#14519688—22no—
57#14519684—22no—
58#14519684—22no—
59#14519684C2312yes—
60C2×#41×C61—21no—
61C22×Q8×C61D411yescharacteristic
62C4⋊Q8×C61—22no—
63C4⋊Q8×C61—22no—
64#126×C61—22no—
65#126×C61—22no—
66#126×C61—22no—
67#126×C61—22no—
68#125×C61—22no—
69#125×C61—21no—
70#125×C61—22no—
71#125×C61—22no—
72#125×C61D412yes—
73#124×C61—22no—
74#124×C61—21no—
75#124×C61D412yes—
76#123×C61—22no—
77#123×C61—22no—
78#123×C61—22no—
79#123×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—21no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61C2312yes—
87#121×C61—22no—
88#121×C61—21no—
89#121×C61C2312yes—
90C4×Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61D412yes—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61D412yes—
102#117×C61—22no—
103#117×C61—22no—
104#117×C61—21no—
105#117×C61—22no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C42×C61—22no—
110C2×C42×C61D411yescharacteristic
111C2×C42×C61D411yescharacteristic
112C2×C42×C61C2311yescharacteristic
113#41×C61—42no—
114#41×C61—42no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—41no—
120C2×Q8×C61C2×D412yes—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—21no—
124C2×D4×C61C2×D412yes—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—22no—
136C22×Dic61—22no—
137C2×Dic122—24no—
138C2×Dic122—82no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C2×D412yes—
155C4⋊C4×C61C2×D412yes—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C42×C61—22no—
166C42×C61—22no—
167C42×C61—21no—
168C42×C61—22no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61C2×D412yes—
173C42×C61C2×D412yes—
174#11758202—84no—
175#11758200—84no—
176C4⋊Dic61—24no—
177C4⋊Dic61—41no—
178C4⋊Dic61—42no—
179C4⋊Dic61—41no—
180#11758196—82no—
181#11758196—82no—
182C4×Dic61—42no—
183C2×#92418—24no—
184C23×C61—22no—
185C23×C61C22×D411yescharacteristic
186Q8×C61—44no—
187Q8×C61—42no—
188Q8×C61—42no—
189Q8×C61—81no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—81no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—22no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—41no—
209C2×C4×C61—44no—
210C2×C4×C61—44no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—82no—
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12012yes—
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61#13712yes—
218C2×C4×C61C22×D412yes—
219C2×Dic61—24no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225Dic122—44no—
226Dic122—44no—
227#92418—44no—
228#18338—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61C2×#12011yescharacteristic
235C22×C61C2×#13711yescharacteristic
236C22×C61#57511yescharacteristic
237C4×C61—22no—
238C4×C61—22no—
239C4×C61—21no—
240C4×C61—22no—
241C4×C61—41no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61—41no—
245C4×C61—41no—
246C4×C61—42no—
247C4×C61#45612yes—
248Dic61—44no—
249Dic61—82no—
250#3461D6111yescharacteristic
251#3218—612no—
252#2567—611no—
253#1605—612no—
254#1604—612no—
255#1603—614no—
256#1509—612no—
257#1467—611no—
258C2×C61—21no—
259C2×C61—21no—
260C2×C61—82no—
261C2×C61#300711yescharacteristic
262C2×C61C2×#45611yescharacteristic
263C2×C61#302711yescharacteristic
264#544C2×D6111yescharacteristic
265#542—22no—
266#541—22no—
267#539—21no—
268#539C2×D6112yes—
269#537C2×D6111yescharacteristic
270#536—21no—
271#533—22no—
272C2×C4⋊Q8—611no—
273#529—22no—
274#529—21no—
275#529C2×D6112yes—
276C4×#41—21no—
277C2×C4×Q8C2×D6111yescharacteristic
278#497—1222no—
279#491—1222no—
280#462—1222no—
281#461—1222no—
282#451—614no—
283#448—614no—
284#432—612no—
285#395—612no—
286#395—612no—
287#391—1221no—
288#389—1221no—
289#389—1221no—
290#384—1221no—
291C4×C4⋊C4—611no—
292#328—614no—
293#327—614no—
294#324—612no—
295#324—612no—
296C61#1833811yescharacteristic, complemented, Sylow
297C2×#41—21no—
298C22×Q8—612no—
299C22×Q8C22⋊D6111yescharacteristic
300C2×Q16—1224no—
301C2×SD16—1224no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—1222no—
305C4⋊Q8—1222no—
306#126—22no—
307#126—22no—
308#126—22no—
309#126—22no—
310#125—21no—
311#125—22no—
312#125—22no—
313#125—22no—
314#125C22⋊D6112yes—
315#124—22no—
316#124—21no—
317#124C22⋊D6112yes—
318#123—22no—
319#123—22no—
320#123—22no—
321#123—22no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—21no—
327C22⋊Q8—22no—
328C22⋊Q8C22×D6112yes—
329#121—22no—
330#121—21no—
331#121C22×D6112yes—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8C22⋊D6112yes—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4—21no—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4C22⋊D6112yes—
344#117—22no—
345#117—22no—
346#117—21no—
347#117—22no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4C22⋊D6111yescharacteristic
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C4⋊C4C22×D6111yescharacteristic
356C2×C42—22no—
357C2×C42—1221no—
358C2×C42C22⋊D6111yescharacteristic
359C2×C42C22⋊D6111yescharacteristic
360C2×C42C22×D6111yescharacteristic
361C4⋊C8—1222no—
362#103—1224no—
363#103—1224no—
364C22⋊C8—614no—
365#97—1222no—
366#95—612no—
367#95—1222no—
368#95—1222no—
369#41—42no—
370#41—42no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—41no—
376C2×Q8—1224no—
377C2×Q8—4882no—
378C2×Q8C2×C22⋊D6112yes—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4C2×C22⋊D6112yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4C2×C22⋊D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390C22×C4C2×C22⋊D6111yescharacteristic
391C22×C4C2×C22⋊D6111yescharacteristic
392C22×C4C2×C22⋊D6111yescharacteristic
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C23×D6111yescharacteristic
395Q16—4884no—
396SD16—4884no—
397C2×C8—1224no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—1224no—
414C4⋊C4—4882no—
415C4⋊C4—4882no—
416C4⋊C4—2441no—
417C4⋊C4—2442no—
418C4⋊C4—2441no—
419C4⋊C4C2×C22⋊D6112yes—
420C4⋊C4C2×C22⋊D6112yes—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C42—22no—
431C42—22no—
432C42—21no—
433C42—22no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—2442no—
438C42C2×C22⋊D6112yes—
439C42C2×C22⋊D6112yes—
440C23—22no—
441C23C22×C22⋊D6111yescharacteristic
442Q8—44no—
443Q8—42no—
444Q8—42no—
445Q8—81no—
446Q8—2444no—
447Q8—2444no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—81no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—82no—
472C2×C4—1224no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4#1451974611yescharacteristic
479C2×C4#1451974612yes—
480C2×C4#1451974611yescharacteristic
481C2×C4#1451980712yes—
482C2×C4C22×C22⋊D6112yes—
483C8—2444no—
484C22—21no—
485C22—21no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22C2×#1451974611yescharacteristic
490C22C2×#1451980711yescharacteristic
491C22#1684056011yescharacteristic
492C4—21no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—42no—
497C4—41no—
498C4—41no—
499C4—42no—
500C4—41no—
501C4—42no—
502C4—4882no—
503C4—2444no—
504C4#1683994512yes—
505C2—21no—
506C2—21no—
507C2—82no—
508C2#2145507911yescharacteristic
509C2C2×#1683994511yescharacteristic
510C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3461C2
socleC22×C61C2×#120
prime radicalC61×#537C22
hypercenter#3461D61
hypocenterC61#18338
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
1. #21456843C2semidirect
2. #21451024C2semidirect
3. #21444793C2semidirect
5. #21444782C2semidirect
6. #21444344C2semidirect
7. #21444163C2semidirect
296. C61#18338semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214510242
#214447932
#214447882
#214447822
#214443442
#214441632
C61×#34612
#1833861