Finite Group Databasebeta

← #30598409

Subgroups — #30598409

Order 15616.

782 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 112 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#21461334C212yesmaximal, complemented
3#21461144C212yesmaximal
4#21461061C211yesmaximal, characteristic, complemented
5C2×#16840471C211yesmaximal, characteristic
6#16840779C2212yes—
7#16840765C2212yes—
8C2×#14519838—21no—
9#16840642—21no—
10#16840621—24no—
11D4⋊#92420—24no—
12#16840608—21no—
13#16840607—21no—
14#16840471—24no—
15#16840471—24no—
16C2×#14519776C2211yescharacteristic
17C2×#14519769C411yescharacteristic
18C2×#14519768C411yescharacteristic
19C2×#14519756—21no—
20C2×#14519754—21no—
21C2×#14519754—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×D4×Dic61C411yescharacteristic
24C2×D4×Dic61C2211yescharacteristic
25C22×#11758196—21no—
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—21no—
35C22×Dic122D411yescharacteristic
36C23×Dic61—21no—
37C23×Dic61—21no—
38#14519776—22no—
39#14519776D414yes—
40#14519769—22no—
41#14519769—22no—
42#14519768—22no—
43#14519768—22no—
44#14519756—28no—
45#14519754—28no—
46#14519754—28no—
47C4×Dic122—22no—
48C4×Dic122—22no—
49D4×Dic61—22no—
50D4×Dic61—22no—
51D4×Dic61—22no—
52D4×Dic61D414yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×C4⋊Dic61D411yescharacteristic
57C2×#11758196—28no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×#11758196D411yescharacteristic
63C2×#11758196D411yescharacteristic
64C2×C22⋊Dic61—21no—
65C2×C22⋊Dic61—21no—
66C2×C4×Dic61C2×C411yescharacteristic
67C2×C4×Dic61C2311yescharacteristic
68C22×D4×C61C2×C411yescharacteristic
69C23×C4×C61—21no—
70C4×D4×C61—22no—
71C4×D4×C61—22no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C2×C42×C61C2×C411yescharacteristic
75C24×C61—21no—
76C2×D4×C61—24no—
77C2×D4×C61—21no—
78C2×D4×C61C22⋊C412yes—
79C22×C4×C61—21no—
80C22×C4×C61—24no—
81C22×C4×C61—82no—
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—24no—
88C22×Dic61—24no—
89C22×Dic61—42no—
90C22×Dic61—41no—
91C22×Dic61—41no—
92C22×Dic61—82no—
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C2×D411yescharacteristic
96C22×Dic61C2×D411yescharacteristic
97C2×Dic122—22no—
98C2×Dic122—22no—
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—24no—
102C2×Dic122—82no—
103#11758217—42no—
104#11758217—42no—
105C2×#92420—42no—
106C2×#92420—42no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61C22⋊C412yes—
109C22⋊C4×C61—24no—
110C42×C61—21no—
111C42×C61—21no—
112C22⋊Dic61—24no—
113C22⋊Dic61—24no—
114C4⋊Dic61—24no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—22no—
118C4⋊Dic61C22⋊C412yes—
119C4⋊Dic61C22⋊C412yes—
120#11758196—24no—
121#11758196—24no—
122#11758196—24no—
123#11758196—22no—
124#11758196—24no—
125#11758196—22no—
126C4×Dic61—21no—
127C4×Dic61—21no—
128C4×Dic61C2×D412yes—
129C4×Dic61C2×D412yes—
130C23×C61—24no—
131C23×C61—22no—
132C23×C61—42no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61C2×C22⋊C411yescharacteristic
136D4×C61—24no—
137D4×C61—22no—
138D4×C61#10114yescomplemented
139C2×C4×C61—22no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—22no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—42no—
146C2×C4×C61—41no—
147C2×C4×C61—84no—
148C2×C4×C61C2×C22⋊C411yescharacteristic
149C2×C4×C61C2×C22⋊C411yescharacteristic
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—22no—
153C2×Dic61—22no—
154C2×Dic61—21no—
155C2×Dic61—21no—
156C2×Dic61—41no—
157C2×Dic61—42no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—44no—
161C2×Dic61—44no—
162C2×Dic61—41no—
163C2×Dic61—84no—
164C2×Dic61#12012yes—
165C2×Dic61#12012yes—
166C2×Dic61#13512yes—
167C2×Dic61AGL(1,ℤ/8)11yescharacteristic
168C2×Dic61#13711yescharacteristic
169Dic122—42no—
170Dic122—42no—
171Dic122—84no—
172#92420—82no—
173#92420—82no—
174#8398—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—24no—
178C22×C61—44no—
179C22×C61—44no—
180C22×C61—42no—
181C22×C61—42no—
182C22×C61#37911yescharacteristic
183C22×C61#41611yescharacteristic
184C22×C61#41711yescharacteristic
185C4×C61—21no—
186C4×C61—42no—
187C4×C61—41no—
188C4×C61—41no—
189C4×C61C2×#10112yes—
190Dic61—22no—
191Dic61—42no—
192Dic61—41no—
193Dic61—41no—
194Dic61—42no—
195Dic61#44912yes—
196Dic61#45212yes—
197C2×#554—611no—
198#1801—611no—
199#1549—612no—
200#1486—612no—
201#1461—611no—
202C2×C61—21no—
203C2×C61—21no—
204C2×C61—44no—
205C2×C61#178411yescharacteristic
206C2×C61#178811yescharacteristic
207C2×C61#179511yescharacteristic
208#554—1224no—
209#554—1224no—
210C2×C4⋊Q8—611no—
211C2×#125—611no—
212C2×#125—611no—
213C2×C22⋊Q8—1221no—
214C2×C22⋊Q8—1221no—
215C2×C22⋊Q8—1221no—
216C2×C4×Q8—611no—
217C2×C4×D4—611no—
218C2×C4×D4—611no—
219C2×C4×D4#692311yescharacteristic
220C22×C4⋊C4—1221no—
221#431—612no—
222#423—612no—
223C2×#101—1221no—
224#343—1221no—
225#332—1224no—
226D4⋊C8—1224no—
227#324—1221no—
228#324—1221no—
229C61#839811yescharacteristic, complemented, Sylow
230C22×Q8—611no—
231C22×Q8—1221no—
232C22×D4C2×#692311yescharacteristic
233C23×C4—21no—
234C23×C4—1221no—
235C23×C4—1221no—
236C2×M16—1222no—
237C4⋊Q8—614no—
238C4⋊Q8—1222no—
239#125—1222no—
240#125—1222no—
241#125—1222no—
242#125—1222no—
243C22⋊Q8—1228no—
244C22⋊Q8—1228no—
245C22⋊Q8—1228no—
246C4×Q8—1222no—
247C4×Q8—1222no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—614no—
251C4×D4—1222no—
252C4×D4—1222no—
253C4×D4—1222no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—611no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—1228no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1221no—
265C2×C4⋊C4C2×#692311yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C42—611no—
270C2×C42—611no—
271C2×C42C2×#692311yescharacteristic
272C4⋊C8—1222no—
273C4⋊C8—1222no—
274C4⋊C8—1222no—
275#101—2442no—
276C22⋊C8—1222no—
277C4×C8—1222no—
278C24—21no—
279C2×Q8—1224no—
280C2×Q8—1221no—
281C2×Q8—1222no—
282C2×Q8—1222no—
283C2×Q8—1221no—
284C2×Q8—4882no—
285C2×D4—24no—
286C2×D4—21no—
287C2×D4C22⋊#692312yes—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—82no—
291C22×C4—611no—
292C22×C4—611no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—1224no—
296C22×C4—1224no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—4882no—
300C22×C4—2442no—
301C22×C4—2441no—
302C22×C4—2441no—
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22⋊#692311yescharacteristic
305C22×C4C22×#692311yescharacteristic
306M16—2442no—
307M16—2442no—
308C2×C8—2442no—
309C2×C8—2442no—
310C4⋊C4—21no—
311C4⋊C4—612no—
312C4⋊C4—612no—
313C4⋊C4—1222no—
314C4⋊C4—1222no—
315C4⋊C4—1222no—
316C4⋊C4—1224no—
317C4⋊C4—1224no—
318C4⋊C4—1224no—
319C4⋊C4—1224no—
320C4⋊C4—1224no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C4⋊C4C22⋊#692312yes—
324C22⋊C4—24no—
325C22⋊C4—1224no—
326C22⋊C4—1224no—
327C42—21no—
328C42—21no—
329C42—612no—
330C42—612no—
331C42—1221no—
332C42—1221no—
333C23—24no—
334C23—22no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23C2×C22⋊#692311yescharacteristic
339Q8—4884no—
340Q8—2442no—
341Q8—2442no—
342D4—24no—
343D4—22no—
344D4#1451983814yescomplemented
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—84no—
354C2×C4—612no—
355C2×C4—611no—
356C2×C4—611no—
357C2×C4—612no—
358C2×C4—612no—
359C2×C4—1221no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—1221no—
363C2×C4—1222no—
364C2×C4—1222no—
365C2×C4—4884no—
366C2×C4—2444no—
367C2×C4—2444no—
368C2×C4—2441no—
369C2×C4—2442no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4C2×C22⋊#692311yescharacteristic
374C2×C4C2×C22⋊#692311yescharacteristic
375C8—4882no—
376C8—4882no—
377C22—21no—
378C22—21no—
379C22—24no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22#1684070311yescharacteristic
385C22#1684075211yescharacteristic
386C22#1684075311yescharacteristic
387C4—21no—
388C4—41no—
389C4—42no—
390C4—41no—
391C4—612no—
392C4—612no—
393C4—1222no—
394C4—2441no—
395C4—2442no—
396C4—2441no—
397C4—2442no—
398C4C2×#1451983812yes—
399C2—21no—
400C2—21no—
401C2—44no—
402C2#2146165311yescharacteristic
403C2#2146165911yescharacteristic
404C2#2146166911yescharacteristic
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#8398
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. #21461334C2semidirect
4. #21461061C2semidirect
138. D4×C61#101semidirect
229. C61#8398semidirect
344. D4#14519838semidirect

Maximal subgroups

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

subgroupindex
#214616852
#214613342
#214611442
#214610612
C2×#168404712
#839861