Finite Group Databasebeta

← #30610736

Subgroups — #30610736

Order 15616.

1044 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462496C211yesmaximal, characteristic
2C4×D61⋊M16C211yesmaximal, characteristic
3#21461168C214yesmaximal, complemented
4#21461161C214yesmaximal, complemented
5#21461142C214yesmaximal, complemented
6C2×#16840317C211yesmaximal, characteristic
7C2×D61⋊M16—21no—
8#16840784—21no—
9#16840783C2212yes—
10#16840781C2212yes—
11#16840778—21no—
12#16840766C2212yes—
13#16840733—21no—
14C4×#11758213—21no—
15C4×#11758213—21no—
16C4×#11758213C2212yes—
17C4×#11758217—21no—
18C4×#11758217C2212yes—
19#16840721—21no—
20C4×D61⋊C8C2212yes—
21Q8⋊#92420C22116yescomplemented
22#16840317—21no—
23#16840317—21no—
24#16840317—21no—
25#16840317—21no—
26#16840317C412yes—
27#16840317C412yes—
28#16840317C2212yes—
29#16840317C2212yes—
30C2×#14519777C411yescharacteristic
31C2×#14519768—21no—
32C2×#14519757—21no—
33C2×#14519753C412yes—
34C2×Q8×Dic61C2212yes—
35C2×#14519742—21no—
36C2×#14519735—21no—
37C2×C42×D61C2211yescharacteristic
38C2×C4⋊Q8×C61C411yescharacteristic
39C2×#11758213—22no—
40C2×#11758213—22no—
41D61⋊M16—28no—
42C2×#11758217—22no—
43C2×D61⋊C8—22no—
44C4⋊#92420—22no—
45C4⋊#92420C2314yes—
46#14519834—22no—
47C22⋊#92420—22no—
48#14519832—22no—
49#14519832—22no—
50#14519832—22no—
51C4×#92420—22no—
52C4×#92420—22no—
53C4×#92420C2314yes—
54C22×Dic122—21no—
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58#14519777—21no—
59#14519777—21no—
60#14519777C2×C412yes—
61#14519777C2×C412yes—
62#14519768—22no—
63#14519768—22no—
64#14519768—22no—
65#14519768—22no—
66#14519757—22no—
67#14519757—22no—
68#14519757—22no—
69#14519757—22no—
70#14519753—22no—
71#14519753—22no—
72#14519753C2×C414yes—
73#14519753C2×C414yes—
74Q8×Dic61—22no—
75Q8×Dic61—22no—
76Q8×Dic61C2×C414yes—
77Q8×Dic61C2314yes—
78#14519742—22no—
79#14519742—22no—
80#14519742—22no—
81#14519742—22no—
82#14519735—22no—
83#14519735—22no—
84#14519735—22no—
85#14519735—22no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—22no—
88C2×C4⋊Dic61C2×C412yes—
89C2×#11758196—22no—
90C2×#11758198—22no—
91C2×C4×Dic61—22no—
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C42×D61C2×C411yescharacteristic
96C42×D61C2×C411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61D411yescharacteristic
100C42×D61D411yescharacteristic
101C42×D61C2311yescharacteristic
102C42×D61C2311yescharacteristic
103C22×Q8×C61C2×C412yes—
104C4⋊Q8×C61—21no—
105C4⋊Q8×C61—21no—
106C4⋊Q8×C61C2×C412yes—
107C4⋊Q8×C61C2×C412yes—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C42×C61C2×C411yescharacteristic
111C23×D61C22⋊C411yescharacteristic
112C2×Q8×C61—22no—
113C2×Q8×C61—44no—
114C2×Q8×C61C22×C414yes—
115C22×C4×C61—22no—
116C22×C4×C61C22⋊C411yescharacteristic
117C22×C4×C61C22⋊C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×Dic61—22no—
120C22×Dic61C22⋊C411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C2×C4×D61—24no—
124C2×C4×D61—24no—
125C2×C4×D61—42no—
126C2×C4×D61C22⋊C411yescharacteristic
127C2×C4×D61C22⋊C411yescharacteristic
128C2×C4×D61C22⋊C411yescharacteristic
129C2×C4×D61C22⋊C411yescharacteristic
130C2×C4×D61C22⋊C411yescharacteristic
131C2×C4×D61C22⋊C411yescharacteristic
132C2×C4×D61C22×C411yescharacteristic
133C2×C4×D61C22×C411yescharacteristic
134C2×C4×D61C2×D411yescharacteristic
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×D411yescharacteristic
137C2×C4×D61C2×D411yescharacteristic
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—44no—
141#11758217—28no—
142C2×#92420—24no—
143C2×#92420—24no—
144#11758213—28no—
145#11758213—44no—
146D61⋊C8—28no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C42×C61C22⋊C411yescharacteristic
152C42×C61C22⋊C411yescharacteristic
153C42×C61C22×C411yescharacteristic
154C42×C61C22×C411yescharacteristic
155#11758198—24no—
156#11758198—24no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—24no—
161C4⋊Dic61—24no—
162C4⋊Dic61C22×C414yes—
163#11758196—24no—
164#11758196—24no—
165C4×Dic61—24no—
166C4×Dic61—24no—
167C4×Dic61C22⋊C411yescharacteristic
168C4×Dic61C22⋊C411yescharacteristic
169C4×Dic61C22⋊C411yescharacteristic
170C4×Dic61C22⋊C411yescharacteristic
171C4×Dic61C22×C411yescharacteristic
172C4×Dic61C2×D411yescharacteristic
173C4×Dic61C2×D411yescharacteristic
174C4×Dic61C2×D411yescharacteristic
175C4×Dic61C2×D411yescharacteristic
176C4×Dic61C2×D411yescharacteristic
177C4×Dic61C2×D411yescharacteristic
178C4×Dic61C2411yescharacteristic
179C23×C61C2×C22⋊C411yescharacteristic
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—41no—
183C22×D61C2×C22⋊C411yescharacteristic
184C22×D61C2×C22⋊C411yescharacteristic
185Q8×C61—28no—
186Q8×C61—44no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—24no—
190C2×C4×C61—44no—
191C2×C4×C61#10414yes—
192C2×C4×C61C2×C22⋊C411yescharacteristic
193C2×C4×C61C2×C22⋊C411yescharacteristic
194C2×C4×C61C2×C22⋊C411yescharacteristic
195C2×C4×C61C2×C22⋊C411yescharacteristic
196C2×C4×C61C2×C22⋊C411yescharacteristic
197C2×C4×C61C23×C411yescharacteristic
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—24no—
201C2×Dic61—41no—
202C2×Dic61—44no—
203C2×Dic61C2×C22⋊C411yescharacteristic
204C2×Dic61C2×C22⋊C411yescharacteristic
205C2×Dic61C2×C22⋊C411yescharacteristic
206C2×Dic61C2×C22⋊C411yescharacteristic
207C2×Dic61C22×D411yescharacteristic
208C2×Dic61C22×D411yescharacteristic
209C4×D61—22no—
210C4×D61—22no—
211C4×D61—22no—
212C4×D61—22no—
213C4×D61—22no—
214C4×D61—22no—
215C4×D61—22no—
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—41no—
219C4×D61—41no—
220C4×D61#13714yes—
221Dic122—44no—
222Dic122—44no—
223#92420—28no—
224#92420—44no—
225#17501—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61C2×#10412yes—
229C22×C61C22×C22⋊C411yescharacteristic
230C2×D61—21no—
231C2×D61—21no—
232C2×D61—21no—
233C2×D61—21no—
234C2×D61—21no—
235C2×D61—44no—
236C2×D61#41912yes—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—44no—
240C4×C61#41912yes—
241C4×C61C2×#10412yes—
242C4×C61C2×#10412yes—
243C4×C61#56812yes—
244Dic61—21no—
245Dic61—21no—
246Dic61—21no—
247Dic61—21no—
248Dic61—21no—
249Dic61—44no—
250Dic61C2×#13712yes—
251C2×#533—611no—
252#2920—611no—
253C2×C4×M16—611no—
254#1493—614no—
255#1491—614no—
256#1485—614no—
257C2×C61—22no—
258C2×C61#287911yescharacteristic
259C2×C61C2×#41911yescharacteristic
260C2×C61C22×#10411yescharacteristic
261D61—22no—
262D61—41no—
263C22×M16—1221no—
264#533—612no—
265#533—612no—
266#533—612no—
267#533—612no—
268#533—1221no—
269#533—1221no—
270#533—1221no—
271#533—1221no—
272C2×C4⋊Q8—611no—
273C2×C4⋊Q8#692311yescharacteristic
274C2×#125—1221no—
275C2×C22⋊Q8—612no—
276C2×C22⋊Q8—1221no—
277C2×C4×Q8—612no—
278C2×C4×Q8—1221no—
279C2×#117—1221no—
280C22×C42—611no—
281#432—612no—
282#432—1221no—
283#431—612no—
284#431—1221no—
285#423—612no—
286#407—1221no—
287C4×M16—612no—
288C4×M16—612no—
289C4×M16—1221no—
290C4×M16—1221no—
291C4×M16—1221no—
292C2×#97—1221no—
293C2×C4×C8—612no—
294Q8⋊C8—6116no—
295C61#1750111yescharacteristic, complemented, Sylow
296C22×Q8—1221no—
297C22×Q8C2×#692312yes—
298C23×C4—611no—
299C23×C4—611no—
300C23×C4—611no—
301C2×M16—1222no—
302C2×M16—1222no—
303C2×M16—1228no—
304C2×M16—1222no—
305C22×C8—1222no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—612no—
309C4⋊Q8—612no—
310C4⋊Q8—1221no—
311C4⋊Q8—1221no—
312C4⋊Q8C2×#692312yes—
313C4⋊Q8C2×#692312yes—
314#125—1222no—
315#125—1222no—
316#125—1222no—
317#125—1222no—
318C22⋊Q8—614no—
319C22⋊Q8—614no—
320C22⋊Q8—1222no—
321C22⋊Q8—1222no—
322C22⋊Q8—1222no—
323C22⋊Q8—1222no—
324C22⋊Q8—1222no—
325C22⋊Q8—1222no—
326C4×Q8—614no—
327C4×Q8—614no—
328C4×Q8—1222no—
329C4×Q8—1222no—
330C4×Q8—1222no—
331C4×Q8—1222no—
332C4×Q8—1222no—
333C4×Q8—1222no—
334#117—1222no—
335#117—1222no—
336#117—1222no—
337#117—1222no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—1222no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4—1221no—
344C2×C22⋊C4—1222no—
345C2×C42—611no—
346C2×C42—611no—
347C2×C42—611no—
348C2×C42—611no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—611no—
355C2×C42—611no—
356C2×C42—1222no—
357C2×C42C2×#692311yescharacteristic
358C4⋊C8—614no—
359C4⋊C8—1222no—
360C22⋊C8—1222no—
361C22⋊C8—1222no—
362#97—1222no—
363#97—1222no—
364#97—1222no—
365C4×C8—614no—
366C4×C8—1222no—
367C4×C8—1222no—
368C24—611no—
369C2×Q8—22no—
370C2×Q8—44no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×Q8—2444no—
374C2×Q8C22×#692314yes—
375C22×C4—22no—
376C22×C4—611no—
377C22×C4—611no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—1224no—
392C22×C4—1224no—
393C22×C4—1222no—
394C22×C4—2442no—
395C22×C4C22⋊#692311yescharacteristic
396C22×C4C22⋊#692311yescharacteristic
397C22×C4C22×#692311yescharacteristic
398M16—1228no—
399M16—1228no—
400M16—2444no—
401C2×C8—1224no—
402C2×C8—1228no—
403C2×C8—1224no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—614no—
409C4⋊C4—1224no—
410C4⋊C4—1224no—
411C4⋊C4—1224no—
412C4⋊C4—1224no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C22⋊C4—1224no—
417C22⋊C4—1224no—
418C42—611no—
419C42—611no—
420C42—611no—
421C42—611no—
422C42—611no—
423C42—611no—
424C42—611no—
425C42—611no—
426C42—611no—
427C42—611no—
428C42—611no—
429C42—611no—
430C42—1224no—
431C42—1224no—
432C42C22⋊#692311yescharacteristic
433C42C22⋊#692311yescharacteristic
434C42C22×#692311yescharacteristic
435C42C22×#692311yescharacteristic
436C23—611no—
437C23—611no—
438C23—1222no—
439C23—1222no—
440C23—2441no—
441C23C2×C22⋊#692311yescharacteristic
442Q8—28no—
443Q8—44no—
444Q8—2444no—
445Q8—2444no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—44no—
450C2×C4—611no—
451C2×C4—614no—
452C2×C4—611no—
453C2×C4—611no—
454C2×C4—611no—
455C2×C4—611no—
456C2×C4—611no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—1222no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—1224no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2444no—
473C2×C4#1451984214yes—
474C2×C4C2×C22⋊#692311yescharacteristic
475C2×C4C2×C22⋊#692311yescharacteristic
476C2×C4C2×C22⋊#692311yescharacteristic
477C2×C4C2×C22⋊#692311yescharacteristic
478C2×C4C2×C22⋊#692311yescharacteristic
479C2×C4C23×#692311yescharacteristic
480C8—1228no—
481C8—2444no—
482C22—21no—
483C22—21no—
484C22—612no—
485C22—1221no—
486C22—1221no—
487C22—1221no—
488C22—1221no—
489C22—1221no—
490C22—2444no—
491C22C2×#1451984212yes—
492C22C22×C22⋊#692311yescharacteristic
493C4—21no—
494C4—21no—
495C4—44no—
496C4—612no—
497C4—1221no—
498C4—1221no—
499C4—1221no—
500C4—1221no—
501C4—1221no—
502C4—2444no—
503C4#1684075512yes—
504C4C2×#1451984212yes—
505C4C2×#1451984212yes—
506C4#1684084012yes—
507C2—22no—
508C2—1222no—
509C2—2441no—
510C2#2146240511yescharacteristic
511C2C2×#1684075511yescharacteristic
512C2C22×#1451984211yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC4×Dic61C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61#17501
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21461168C2semidirect
4. #21461161C2semidirect
5. #21461142C2semidirect
21. Q8⋊#92420C22semidirect
295. C61#17501semidirect

Maximal subgroups

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

subgroupindex
#214624962
C4×D61⋊M162
#214611682
#214611612
#214611422
C2×#168403172
#1750161