Finite Group Databasebeta

← #30598366

Subgroups — #30598366

Order 15616.

1034 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461680C211yesmaximal, characteristic
2#21461658C212yesmaximal, complemented
3#21461258C212yesmaximal, complemented
4#21461056C211yesmaximal, characteristic
5C2×Q8×D244C211yesmaximal, characteristic
6C2×#14519840—24no—
7C2×#14519836—21no—
8D61⋊(C22⋊C8)C2212yes—
9#16840704C2212yes—
10#16840640—21no—
11#16840615—24no—
12#16840607—21no—
13#16840606—21no—
14C22×Q8×D61C411yescharacteristic, complemented
15C22×Q8×D61C2211yescharacteristic
16Q8×D244—24no—
17Q8×D244—24no—
18C2×#14519777—21no—
19C2×#14519777C411yescharacteristic, complemented
20C2×#14519757—21no—
21C2×#14519757—21no—
22C2×#14519757C411yescharacteristic, complemented
23C2×#14519757C2211yescharacteristic
24C2×C4×D244—21no—
25C2×C4×D244C2211yescharacteristic
26C2×C4×Q8×C61C411yescharacteristic, complemented
27C2×D61⋊C8—22no—
28C2×C4⋊#6923—22no—
29C2×C22⋊#6923—22no—
30#14519840—48no—
31#14519836—42no—
32#14519834—22no—
33C22⋊#92420—22no—
34(C2×#6923)⋊C4—22no—
35C22×Dic122—21no—
36C22×Dic122—21no—
37C22×Dic122C2×C411yescharacteristic
38C22×Dic122C2×C411yescharacteristic
39C2×Q8×D61—22no—
40C2×Q8×D61—22no—
41C2×Q8×D61—22no—
42C2×Q8×D61—44no—
43C2×Q8×D61—44no—
44C2×Q8×D61D414yes—
45C22×D244D411yescharacteristic
46C22×C4×D61—21no—
47C22×C4×D61—21no—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50#14519777—28no—
51#14519777—22no—
52#14519777—22no—
53#14519757—28no—
54#14519757—28no—
55#14519757—22no—
56#14519757—22no—
57#14519757—22no—
58#14519757D414yes—
59C4×D244—28no—
60C4×D244—22no—
61C4×D244—22no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×#11758198—21no—
68C2×#11758198—21no—
69C2×#11758198D411yescharacteristic
70C2×#11758198D411yescharacteristic
71C22×Q8×C61C2×C411yescharacteristic
72C4×Q8×C61—22no—
73C4×Q8×C61—22no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61C2×C411yescharacteristic
76C2×C42×C61—21no—
77C2×C42×C61C2×C411yescharacteristic
78C23×D61C22⋊C411yescharacteristic
79C23×D61C2×D411yescharacteristic
80C22×#6923—42no—
81C2×Q8×C61—24no—
82C2×Q8×C61—21no—
83C2×Q8×C61C22⋊C412yes—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22×C411yescharacteristic
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61C22⋊C411yescharacteristic
92C22×Dic61C2×D411yescharacteristic
93Q8×D61—28no—
94Q8×D61—44no—
95Q8×D61—48no—
96Q8×D61—44no—
97Q8×D61—48no—
98Q8×D61—44no—
99C2×D244—22no—
100C2×D244—22no—
101C2×D244SD1612yes—
102C2×D244Q1612yes—
103C2×C4×D61—24no—
104C2×C4×D61—24no—
105C2×C4×D61—21no—
106C2×C4×D61—21no—
107C2×C4×D61—42no—
108C2×C4×D61—44no—
109C2×C4×D61—44no—
110C2×C4×D61—42no—
111C2×C4×D61C2×D412yes—
112C2×C4×D61C2×D412yes—
113C2×Dic122—24no—
114C2×Dic122—21no—
115C2×Dic122—21no—
116C2×Dic122—24no—
117C2×Dic122—42no—
118C2×Dic122—44no—
119C2×Dic122—44no—
120C2×Dic122—42no—
121C2×Dic122C22⋊C412yes—
122C2×Dic122C22⋊C412yes—
123C22⋊#6923—42no—
124C22⋊#6923—42no—
125C2×#92420—42no—
126C4⋊#6923—44no—
127D61⋊C8—44no—
128C4⋊C4×C61—24no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61C22⋊C412yes—
131C42×C61—24no—
132C42×C61—21no—
133C42×C61—21no—
134#11758198—24no—
135#11758198—24no—
136#11758198—22no—
137#11758198—22no—
138C4⋊Dic61—24no—
139C4⋊Dic61—24no—
140C4⋊Dic61—24no—
141C4⋊Dic61—24no—
142C4⋊Dic61—22no—
143C23×C61C2×C22⋊C411yescharacteristic
144C22×D61—21no—
145C22×D61—21no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150C22×D61#10312yescomplemented
151C22×D61#12012yes—
152C22×D61C2×SD1611yescharacteristic
153C22×D61C2×Q1611yescharacteristic
154C22×D61#13712yes—
155C2×#6923—82no—
156C2×#6923—82no—
157C2×#6923—82no—
158Q8×C61—24no—
159Q8×C61—22no—
160Q8×C61#9914yescomplemented
161C2×C4×C61—24no—
162C2×C4×C61—21no—
163C2×C4×C61—22no—
164C2×C4×C61—21no—
165C2×C4×C61—22no—
166C2×C4×C61—21no—
167C2×C4×C61—22no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61#10312yescomplemented
171C2×C4×C61C2×C22⋊C411yescharacteristic
172C2×C4×C61C2×C22⋊C411yescharacteristic
173C2×Dic61—21no—
174C2×Dic61—22no—
175C2×Dic61—22no—
176C2×Dic61—41no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—41no—
180C2×Dic61#12012yes—
181D244—24no—
182D244—24no—
183C4×D61—24no—
184C4×D61—42no—
185C4×D61—48no—
186C4×D61—42no—
187C4×D61—48no—
188C4×D61—42no—
189Dic122—24no—
190Dic122—48no—
191Dic122—42no—
192Dic122—48no—
193Dic122—42no—
194Dic122—42no—
195#92420—82no—
196#8379—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61#37911yescharacteristic
200C22×C61C2×#10311yescharacteristic
201C22×C61#41911yescharacteristic
202C2×D61—22no—
203C2×D61—22no—
204C2×D61—22no—
205C2×D61—41no—
206C2×D61—41no—
207C2×D61—42no—
208C2×D61—42no—
209C2×D61—41no—
210C2×D61—41no—
211C2×D61—42no—
212C2×D61#44812yes—
213C2×D61#45112yes—
214#6923—162no—
215C4×C61—24no—
216C4×C61—21no—
217C4×C61—22no—
218C4×C61—41no—
219C4×C61—42no—
220C4×C61C2×#9912yes—
221Dic61—22no—
222Dic61—42no—
223Dic61—41no—
224Dic61—42no—
225C2×D4×Q8—611no—
226#1799—611no—
227#1787—612no—
228#1520—612no—
229#1459—611no—
230C2×C61—21no—
231C2×C61—21no—
232C2×C61#178111yescharacteristic
233C2×C61#178711yescharacteristic
234C2×C61#179611yescharacteristic
235D61—22no—
236D61—22no—
237D61—42no—
238C23×Q8—611no—
239C23×Q8—611no—
240D4×Q8—1224no—
241D4×Q8—1224no—
242C2×C4⋊Q8—611no—
243C2×C4⋊Q8—1221no—
244C2×C22⋊Q8—611no—
245C2×C22⋊Q8—611no—
246C2×C22⋊Q8—1221no—
247C2×C22⋊Q8—1221no—
248C2×C4×Q8#692311yescharacteristic, complemented
249C2×C4×D4—611no—
250C2×C4×D4—1221no—
251C2×#103—1224no—
252C2×#99—1221no—
253C2×C22⋊C8—612no—
254#380—612no—
255#342—1221no—
256#328—1224no—
257#324—1221no—
258#323—1221no—
259C61#837911yescharacteristic, complemented, Sylow
260C22×Q8—614no—
261C22×Q8—611no—
262C22×Q8—611no—
263C22×Q8—1221no—
264C22×Q8—1221no—
265C22×Q8—1222no—
266C22×Q8—1222no—
267C22×Q8—1222no—
268C22×Q8—2444no—
269C22×Q8—2444no—
270C22×Q8C2×#692311yescharacteristic
271C22×D4—611no—
272C23×C4—611no—
273C23×C4—611no—
274C23×C4—1221no—
275C23×C4—1221no—
276C22×C8—1222no—
277C4⋊Q8—1222no—
278C4⋊Q8—1222no—
279C4⋊Q8—1228no—
280C22⋊Q8—614no—
281C22⋊Q8—1222no—
282C22⋊Q8—1222no—
283C22⋊Q8—1228no—
284C22⋊Q8—1228no—
285C22⋊Q8—1222no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×D4—1222no—
289C4×D4—1222no—
290C4×D4—1228no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C4⋊C4—1222no—
298C2×C4⋊C4C2×#692311yescharacteristic
299C2×C22⋊C4—611no—
300C2×C22⋊C4—611no—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4—1221no—
303C2×C22⋊C4—1222no—
304C2×C42—21no—
305C2×C42C2×#692311yescharacteristic
306#103—2448no—
307#99—2442no—
308C22⋊C8—1222no—
309C22⋊C8—1222no—
310#95—1222no—
311C24—611no—
312C24—611no—
313C2×Q8—24no—
314C2×Q8—21no—
315C2×Q8—612no—
316C2×Q8—612no—
317C2×Q8—1224no—
318C2×Q8—1224no—
319C2×Q8—1221no—
320C2×Q8—1221no—
321C2×Q8—1228no—
322C2×Q8—2442no—
323C2×Q8—2442no—
324C2×Q8—2444no—
325C2×Q8—2444no—
326C2×Q8—2444no—
327C2×Q8—2448no—
328C2×Q8—2448no—
329C2×Q8—2444no—
330C2×Q8—2444no—
331C2×Q8C22⋊#692312yes—
332C2×D4—612no—
333C2×D4—612no—
334C2×D4—1222no—
335C2×D4—1222no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—611no—
339C22×C4—612no—
340C22×C4—612no—
341C22×C4—611no—
342C22×C4—1224no—
343C22×C4—1224no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1221no—
348C22×C4—2442no—
349C22×C4—2444no—
350C22×C4—2444no—
351C22×C4—2442no—
352C22×C4—2442no—
353C22×C4C22⋊#692311yescharacteristic
354C22×C4C22⋊#692311yescharacteristic
355C22×C4C22×#692311yescharacteristic
356C2×C8—2442no—
357C2×C8—2444no—
358C4⋊C4—24no—
359C4⋊C4—21no—
360C4⋊C4—1222no—
361C4⋊C4—1224no—
362C4⋊C4—1224no—
363C4⋊C4—1224no—
364C4⋊C4—1224no—
365C4⋊C4—2444no—
366C4⋊C4C22⋊#692312yes—
367C22⋊C4—1222no—
368C22⋊C4—1222no—
369C22⋊C4—1224no—
370C22⋊C4—1224no—
371C22⋊C4—2442no—
372C22⋊C4—2442no—
373C42—24no—
374C42—21no—
375C42—21no—
376C23—612no—
377C23—611no—
378C23—611no—
379C23—612no—
380C23—612no—
381C23—1221no—
382C23—1221no—
383C23—2441no—
384C23—2441no—
385C23—2441no—
386C23—2441no—
387C23C2×C22⋊#692311yescharacteristic
388Q8—24no—
389Q8—22no—
390Q8—1224no—
391Q8—2448no—
392Q8—2442no—
393Q8—2448no—
394Q8—2442no—
395Q8—2442no—
396Q8#1451983614yescomplemented
397D4—1224no—
398D4—1224no—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—612no—
409C2×C4—1224no—
410C2×C4—1221no—
411C2×C4—1222no—
412C2×C4—1222no—
413C2×C4—4882no—
414C2×C4—4882no—
415C2×C4—4882no—
416C2×C4—2442no—
417C2×C4—2448no—
418C2×C4—2448no—
419C2×C4—2442no—
420C2×C4—2441no—
421C2×C4—2442no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2441no—
425C2×C4#1451984012yescomplemented
426C2×C4C2×C22⋊#692311yescharacteristic
427C2×C4C2×C22⋊#692311yescharacteristic
428C8—4882no—
429C22—21no—
430C22—21no—
431C22—612no—
432C22—612no—
433C22—1222no—
434C22—1222no—
435C22—1222no—
436C22—2441no—
437C22—2441no—
438C22—2442no—
439C22—2442no—
440C22—2442no—
441C22—2441no—
442C22—2441no—
443C22#1684070311yescharacteristic
444C22C2×#1451984011yescharacteristic
445C22#1684075511yescharacteristic
446C4—24no—
447C4—21no—
448C4—22no—
449C4—41no—
450C4—42no—
451C4—1222no—
452C4—9762no—
453C4—2442no—
454C4—2442no—
455C4—2441no—
456C4C2×#1451983612yes—
457C2—21no—
458C2—21no—
459C2—1222no—
460C2—1222no—
461C2—2442no—
462C2#2146164711yescharacteristic
463C2#2146165811yescharacteristic
464C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61#379
prime radicalC22×D244D4
cosocleC22×C4×D61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8379
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #21461658C2semidirect
3. #21461258C2semidirect
14. C22×Q8×D61C4semidirect
19. C2×#14519777C4semidirect
22. C2×#14519757C4semidirect
26. C2×C4×Q8×C61C4semidirect
150. C22×D61#103semidirect
160. Q8×C61#99semidirect
170. C2×C4×C61#103semidirect
248. C2×C4×Q8#6923semidirect
259. C61#8379semidirect
396. Q8#14519836semidirect
425. C2×C4#14519840semidirect

Maximal subgroups

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

subgroupindex
#214616802
#214616582
#214612582
#214610562
C2×Q8×D2442
#837961