Finite Group Databasebeta

← #30598389

Subgroups — #30598389

Order 15616.

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

normal subgroups only

This group has 966 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#21461685C211yesmaximal, characteristic, complemented
2#21461328C212yesmaximal, complemented
3#21461144C212yesmaximal
4#21461061C211yesmaximal, characteristic, complemented
5C2×D4×Dic122C211yesmaximal, characteristic
6#16840779C2212yes—
7#16840765C2212yes—
8C2×#14519838—21no—
9#16840642—21no—
10#16840621—24no—
11D4⋊#92420—24no—
12#16840608—21no—
13#16840607—21no—
14C23×Dic122—21no—
15D4×Dic122—24no—
16D4×Dic122—24no—
17C2×#14519777C411yescharacteristic
18C2×#14519776C411yescharacteristic
19C2×#14519776C2211yescharacteristic
20C2×#14519756—21no—
21C2×#14519754—21no—
22C2×#14519754—21no—
23C2×C4×Dic122C2211yescharacteristic
24C2×D4×Dic61C411yescharacteristic
25C2×D4×Dic61C2211yescharacteristic
26C2×C4×D4×C61C411yescharacteristic
27C2×#11758217—22no—
28#14519844—22no—
29#14519844—22no—
30C4⋊#92420—22no—
31#14519838—42no—
32C22⋊#92420—22no—
33C4×#92420—22no—
34C22×Dic122—28no—
35C22×Dic122—21no—
36C22×Dic122—21no—
37C22×Dic122—21no—
38C22×Dic122—44no—
39C22×Dic122—42no—
40C22×Dic122—42no—
41C22×Dic122D411yescharacteristic
42C23×Dic61—21no—
43C23×Dic61—21no—
44#14519777—22no—
45#14519777—22no—
46#14519776—22no—
47#14519776—22no—
48#14519776—22no—
49#14519776D414yes—
50#14519756—28no—
51#14519754—28no—
52#14519754—28no—
53C4×Dic122—22no—
54C4×Dic122—22no—
55D4×Dic61—22no—
56D4×Dic61—22no—
57D4×Dic61—22no—
58D4×Dic61D414yes—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2×C411yescharacteristic
61C2×C4⋊Dic61C2×C411yescharacteristic
62C2×C4⋊Dic61D411yescharacteristic
63C2×#11758196—21no—
64C2×#11758196—21no—
65C2×#11758196D411yescharacteristic
66C2×#11758196D411yescharacteristic
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—21no—
69C2×C4×Dic61C2×C411yescharacteristic
70C2×C4×Dic61C2311yescharacteristic
71C22×D4×C61C2×C411yescharacteristic
72C23×C4×C61—21no—
73C4×D4×C61—22no—
74C4×D4×C61—22no—
75C2×C4⋊C4×C61C2×C411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C2×C42×C61C2×C411yescharacteristic
78C24×C61—21no—
79C2×D4×C61—24no—
80C2×D4×C61—21no—
81C2×D4×C61C22⋊C412yes—
82C22×C4×C61—24no—
83C22×C4×C61—21no—
84C22×C4×C61—42no—
85C22×C4×C61—42no—
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22⋊C411yescharacteristic
88C22×C4×C61C22×C411yescharacteristic
89C22×Dic61—24no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—24no—
93C22×Dic61—42no—
94C22×Dic61—42no—
95C22×Dic61—41no—
96C22×Dic61—41no—
97C22×Dic61—42no—
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C2×D411yescharacteristic
101C22×Dic61C2×D411yescharacteristic
102C2×Dic122—22no—
103C2×Dic122—22no—
104C2×Dic122—24no—
105C2×Dic122—24no—
106C2×Dic122—24no—
107C2×Dic122—416no—
108C2×Dic122—416no—
109C2×Dic122—42no—
110C2×Dic122—42no—
111C2×Dic122—42no—
112C2×Dic122—42no—
113C2×Dic122—41no—
114C2×Dic122—41no—
115C2×Dic122—42no—
116C2×Dic122D812yes—
117C2×Dic122SD1612yes—
118#11758217—42no—
119#11758217—42no—
120C2×#92420—42no—
121C2×#92420—42no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61C22⋊C412yes—
124C22⋊C4×C61—24no—
125C42×C61—21no—
126C42×C61—21no—
127C22⋊Dic61—24no—
128C22⋊Dic61—24no—
129C4⋊Dic61—24no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61—22no—
133C4⋊Dic61C22⋊C412yes—
134C4⋊Dic61C22⋊C412yes—
135#11758196—24no—
136#11758196—24no—
137#11758196—22no—
138#11758196—22no—
139C4×Dic61—21no—
140C4×Dic61—21no—
141C4×Dic61C2×D412yes—
142C4×Dic61C2×D412yes—
143C23×C61—24no—
144C23×C61—22no—
145C23×C61—42no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61C2×C22⋊C411yescharacteristic
149D4×C61—24no—
150D4×C61—22no—
151D4×C61#10114yescomplemented
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—22no—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—48no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61#10212yes—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—21no—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—41no—
170C2×Dic61—48no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—44no—
176C2×Dic61—44no—
177C2×Dic61—41no—
178C2×Dic61#10212yes—
179C2×Dic61#12012yes—
180C2×Dic61#12012yes—
181C2×Dic61C2×D811yescharacteristic
182C2×Dic61C2×SD1611yescharacteristic
183C2×Dic61AGL(1,ℤ/8)12yes—
184Dic122—24no—
185Dic122—24no—
186Dic122—48no—
187Dic122—48no—
188Dic122—44no—
189Dic122—44no—
190#92420—82no—
191#92420—82no—
192#8388—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—24no—
196C22×C61—44no—
197C22×C61—44no—
198C22×C61—42no—
199C22×C61—42no—
200C22×C61#37911yescharacteristic
201C22×C61C2×#10211yescharacteristic
202C22×C61#41811yescharacteristic
203C4×C61—21no—
204C4×C61—22no—
205C4×C61—41no—
206C4×C61—42no—
207C4×C61C2×#10112yes—
208Dic61—22no—
209Dic61—22no—
210Dic61—41no—
211Dic61—42no—
212Dic61—42no—
213Dic61#44712yes—
214Dic61#45012yes—
215C2×D4×Q8—611no—
216#1801—611no—
217#1546—612no—
218#1486—612no—
219#1461—611no—
220C2×C61—21no—
221C2×C61—21no—
222C2×C61—44no—
223C2×C61#178411yescharacteristic
224C2×C61#178611yescharacteristic
225C2×C61#179611yescharacteristic
226C23×Q8—1221no—
227D4×Q8—1224no—
228D4×Q8—1224no—
229C2×C4⋊Q8—611no—
230C2×C4⋊Q8—611no—
231C2×C4⋊Q8—611no—
232C2×C22⋊Q8—1221no—
233C2×C22⋊Q8—1221no—
234C2×C22⋊Q8—1221no—
235C2×C4×Q8—611no—
236C2×C4×D4—611no—
237C2×C4×D4—611no—
238C2×C4×D4#692311yescharacteristic
239#431—612no—
240#423—612no—
241C2×#101—1221no—
242#343—1221no—
243#332—1224no—
244D4⋊C8—1224no—
245#324—1221no—
246#324—1221no—
247C61#838811yescharacteristic, complemented, Sylow
248C22×Q8—611no—
249C22×Q8—1221no—
250C22×Q8—1221no—
251C22×Q8—1228no—
252C22×Q8—1221no—
253C22×Q8—2442no—
254C22×Q8—2442no—
255C22×Q8—2444no—
256C22×D4C2×#692311yescharacteristic
257C23×C4—21no—
258C23×C4—1221no—
259C23×C4—1221no—
260C2×M16—1222no—
261C4⋊Q8—614no—
262C4⋊Q8—1222no—
263C4⋊Q8—1222no—
264C4⋊Q8—1222no—
265C4⋊Q8—1222no—
266C4⋊Q8—1222no—
267C22⋊Q8—1228no—
268C22⋊Q8—1228no—
269C22⋊Q8—1228no—
270C4×Q8—1222no—
271C4×Q8—1222no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—614no—
275C4×D4—1222no—
276C4×D4—1222no—
277C4×D4—1222no—
278C2×C4⋊C4—611no—
279C2×C4⋊C4—611no—
280C2×C4⋊C4—611no—
281C2×C4⋊C4—611no—
282C2×C4⋊C4—611no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4—1221no—
286C2×C4⋊C4C2×#692311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1221no—
290C2×C42—611no—
291C2×C42—611no—
292C2×C42C2×#692311yescharacteristic
293C4⋊C8—1222no—
294C4⋊C8—1222no—
295C4⋊C8—1222no—
296#101—2442no—
297C22⋊C8—1222no—
298C4×C8—1222no—
299C24—21no—
300C2×Q8—612no—
301C2×Q8—612no—
302C2×Q8—1222no—
303C2×Q8—1222no—
304C2×Q8—1224no—
305C2×Q8—1224no—
306C2×Q8—1224no—
307C2×Q8—2442no—
308C2×Q8—2441no—
309C2×Q8—2441no—
310C2×Q8—2442no—
311C2×Q8—2442no—
312C2×Q8—2442no—
313C2×Q8—24416no—
314C2×Q8—24416no—
315C2×Q8—2442no—
316C2×D4—24no—
317C2×D4—21no—
318C2×D4C22⋊#692312yes—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—611no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—1224no—
328C22×C4—1224no—
329C22×C4—1221no—
330C22×C4—1221no—
331C22×C4—2441no—
332C22×C4—2441no—
333C22×C4—2442no—
334C22×C4—2442no—
335C22×C4—2442no—
336C22×C4C22⋊#692311yescharacteristic
337C22×C4C22⋊#692311yescharacteristic
338C22×C4C22×#692311yescharacteristic
339M16—2442no—
340M16—2442no—
341C2×C8—2442no—
342C2×C8—2442no—
343C4⋊C4—21no—
344C4⋊C4—612no—
345C4⋊C4—612no—
346C4⋊C4—1222no—
347C4⋊C4—1222no—
348C4⋊C4—1222no—
349C4⋊C4—1224no—
350C4⋊C4—1224no—
351C4⋊C4—1224no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4C22⋊#692312yes—
355C22⋊C4—24no—
356C22⋊C4—1224no—
357C22⋊C4—1224no—
358C42—21no—
359C42—21no—
360C42—612no—
361C42—612no—
362C42—1221no—
363C42—1221no—
364C23—24no—
365C23—22no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23C2×C22⋊#692311yescharacteristic
370Q8—1224no—
371Q8—1224no—
372Q8—2448no—
373Q8—2444no—
374Q8—2444no—
375Q8—2448no—
376D4—24no—
377D4—22no—
378D4#1451983814yescomplemented
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—48no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—611no—
389C2×C4—611no—
390C2×C4—612no—
391C2×C4—612no—
392C2×C4—612no—
393C2×C4—612no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—1221no—
397C2×C4—1222no—
398C2×C4—1222no—
399C2×C4—2444no—
400C2×C4—2444no—
401C2×C4—2441no—
402C2×C4—2448no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4#1451983912yes—
409C2×C4C2×C22⋊#692311yescharacteristic
410C2×C4C2×C22⋊#692311yescharacteristic
411C8—4882no—
412C8—4882no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—44no—
417C22—44no—
418C22—42no—
419C22—42no—
420C22#1684070311yescharacteristic
421C22C2×#1451983911yescharacteristic
422C22#1684075411yescharacteristic
423C4—21no—
424C4—22no—
425C4—42no—
426C4—41no—
427C4—612no—
428C4—612no—
429C4—1222no—
430C4—1222no—
431C4—2442no—
432C4—2442no—
433C4—2441no—
434C4C2×#1451983812yes—
435C2—21no—
436C2—21no—
437C2—44no—
438C2#2146165311yescharacteristic
439C2#2146165711yescharacteristic
440C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC22×C61#379
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×D4#6923
hypocenterC61#8388
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #21461685C2semidirect
2. #21461328C2semidirect
4. #21461061C2semidirect
151. D4×C61#101semidirect
247. C61#8388semidirect
378. D4#14519838semidirect

Maximal subgroups

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

subgroupindex
#214616852
#214613282
#214611442
#214610612
C2×D4×Dic1222
#838861