Finite Group Databasebeta

← #30598405

Subgroups — #30598405

Order 15616.

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

normal subgroups only

This group has 738 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
2#21461334C212yesmaximal
3#21461147C212yesmaximal
4#21461061C211yesmaximal, characteristic
5C2×Q8×Dic122C211yesmaximal, characteristic
6#16840779C2212yes—
7#16840765C2212yes—
8C2×#14519838—21no—
9#16840642—21no—
10#16840621—24no—
11Q8⋊#92420—24no—
12#16840608—21no—
13#16840607—21no—
14Q8×Dic122—24no—
15Q8×Dic122—24no—
16C2×#14519777—21no—
17C2×#14519777C411yescharacteristic
18C2×#14519776—21no—
19C2×#14519776—21no—
20C2×#14519776C411yescharacteristic
21C2×#14519776C2211yescharacteristic
22C2×Q8×Dic61C411yescharacteristic
23C2×Q8×Dic61C2211yescharacteristic
24C2×C4×Dic122—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×C4×Q8×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×Dic122—21no—
36C22×Dic122—21no—
37C22×Dic122D411yescharacteristic
38#14519777—28no—
39#14519777—22no—
40#14519777—22no—
41#14519776—28no—
42#14519776—28no—
43#14519776—22no—
44#14519776—22no—
45#14519776—22no—
46#14519776D414yes—
47Q8×Dic61—22no—
48Q8×Dic61—22no—
49Q8×Dic61—22no—
50Q8×Dic61D414yes—
51C4×Dic122—28no—
52C4×Dic122—22no—
53C4×Dic122—22no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61C2×C411yescharacteristic
58C2×C4⋊Dic61C2×C411yescharacteristic
59C2×C4⋊Dic61D411yescharacteristic
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×#11758196D411yescharacteristic
63C2×#11758196D411yescharacteristic
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61C2×C411yescharacteristic
67C2×C4×Dic61C2311yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C4×Q8×C61—22no—
70C4×Q8×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C42×C61—21no—
74C2×C42×C61C2×C411yescharacteristic
75C2×Q8×C61—24no—
76C2×Q8×C61—21no—
77C2×Q8×C61C22⋊C412yes—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C22⋊C411yescharacteristic
87C22×Dic61C2×D411yescharacteristic
88C22×Dic61C2×D411yescharacteristic
89C2×Dic122—22no—
90C2×Dic122—22no—
91C2×Dic122—24no—
92C2×Dic122—24no—
93C2×Dic122—24no—
94C2×Dic122—82no—
95C2×Dic122—82no—
96C2×Dic122—82no—
97C2×Dic122SD1612yes—
98C2×Dic122Q1612yes—
99#11758217—42no—
100#11758217—42no—
101C2×#92420—42no—
102C2×#92420—42no—
103C4⋊C4×C61—24no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61C22⋊C412yes—
106C42×C61—24no—
107C42×C61—21no—
108C42×C61—21no—
109C4⋊Dic61—24no—
110C4⋊Dic61—24no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113C4⋊Dic61—24no—
114C4⋊Dic61—22no—
115C4⋊Dic61C22⋊C412yes—
116C4⋊Dic61C22⋊C412yes—
117#11758196—24no—
118#11758196—24no—
119#11758196—22no—
120#11758196—22no—
121C4×Dic61—24no—
122C4×Dic61—21no—
123C4×Dic61—21no—
124C4×Dic61—24no—
125C4×Dic61C2×D412yes—
126C4×Dic61C2×D412yes—
127C23×C61C2×C22⋊C411yescharacteristic
128Q8×C61—24no—
129Q8×C61—22no—
130Q8×C61#10114yescomplemented
131C2×C4×C61—24no—
132C2×C4×C61—22no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—22no—
136C2×C4×C61—21no—
137C2×C4×C61—22no—
138C2×C4×C61—42no—
139C2×C4×C61—41no—
140C2×C4×C61#10312yes—
141C2×C4×C61C2×C22⋊C411yescharacteristic
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—41no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C2×Dic61—41no—
152C2×Dic61#10312yes—
153C2×Dic61#12012yes—
154C2×Dic61#12012yes—
155C2×Dic61C2×SD1611yescharacteristic
156C2×Dic61C2×Q1611yescharacteristic
157C2×Dic61#13712yes—
158Dic122—24no—
159Dic122—24no—
160Dic122—84no—
161Dic122—84no—
162Dic122—84no—
163#92420—82no—
164#92420—82no—
165#8396—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61#37911yescharacteristic
169C22×C61C2×#10311yescharacteristic
170C22×C61#41911yescharacteristic
171C4×C61—24no—
172C4×C61—21no—
173C4×C61—22no—
174C4×C61—41no—
175C4×C61—42no—
176C4×C61C2×#10112yes—
177Dic61—22no—
178Dic61—22no—
179Dic61—41no—
180Dic61—42no—
181Dic61—42no—
182Dic61#44812yes—
183Dic61#45112yes—
184C2×Q82—611no—
185#1801—611no—
186#1549—612no—
187#1487—612no—
188#1461—611no—
189C2×C61—21no—
190C2×C61—21no—
191C2×C61#178411yescharacteristic
192C2×C61#178711yescharacteristic
193C2×C61#179511yescharacteristic
194Q82—1224no—
195Q82—1224no—
196C2×C4⋊Q8—611no—
197C2×C4⋊Q8—611no—
198C2×C4⋊Q8—611no—
199C2×C4⋊Q8—1221no—
200C2×C4⋊Q8—1221no—
201C2×C4⋊Q8—1221no—
202C2×C4×Q8—611no—
203C2×C4×Q8—611no—
204C2×C4×Q8—611no—
205C2×C4×Q8—1221no—
206C2×C4×Q8#692311yescharacteristic
207#431—612no—
208#423—612no—
209C2×#101—1221no—
210#343—1221no—
211#332—1224no—
212Q8⋊C8—1224no—
213#324—1221no—
214#324—1221no—
215C61#839611yescharacteristic, complemented, Sylow
216C22×Q8—611no—
217C22×Q8—1221no—
218C22×Q8—1221no—
219C22×Q8—1221no—
220C22×Q8C2×#692311yescharacteristic
221C2×M16—1222no—
222C4⋊Q8—614no—
223C4⋊Q8—1222no—
224C4⋊Q8—1222no—
225C4⋊Q8—1228no—
226C4⋊Q8—1222no—
227C4⋊Q8—1222no—
228C4⋊Q8—1228no—
229C4⋊Q8—1228no—
230C4⋊Q8—1222no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—614no—
234C4×Q8—1222no—
235C4×Q8—1222no—
236C4×Q8—1228no—
237C4×Q8—1222no—
238C4×Q8—1222no—
239C4×Q8—1222no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—611no—
242C2×C4⋊C4—611no—
243C2×C4⋊C4—611no—
244C2×C4⋊C4—611no—
245C2×C4⋊C4—611no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4C2×#692311yescharacteristic
252C2×C42—21no—
253C2×C42—611no—
254C2×C42—611no—
255C2×C42—1221no—
256C2×C42—1221no—
257C2×C42C2×#692311yescharacteristic
258C4⋊C8—1222no—
259C4⋊C8—1222no—
260C4⋊C8—1222no—
261#101—2442no—
262C22⋊C8—1222no—
263C4×C8—1222no—
264C2×Q8—24no—
265C2×Q8—21no—
266C2×Q8—612no—
267C2×Q8—612no—
268C2×Q8—1222no—
269C2×Q8—1222no—
270C2×Q8—1224no—
271C2×Q8—1224no—
272C2×Q8—1224no—
273C2×Q8—4882no—
274C2×Q8—4882no—
275C2×Q8—4882no—
276C2×Q8C22⋊#692312yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—611no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4C22⋊#692311yescharacteristic
286C22×C4C22⋊#692311yescharacteristic
287C22×C4C22×#692311yescharacteristic
288M16—2442no—
289M16—2442no—
290C2×C8—2442no—
291C2×C8—2442no—
292C4⋊C4—24no—
293C4⋊C4—21no—
294C4⋊C4—612no—
295C4⋊C4—612no—
296C4⋊C4—1222no—
297C4⋊C4—1222no—
298C4⋊C4—1222no—
299C4⋊C4—1224no—
300C4⋊C4—1224no—
301C4⋊C4—1224no—
302C4⋊C4—1224no—
303C4⋊C4—1224no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4C22⋊#692312yes—
307C42—24no—
308C42—21no—
309C42—21no—
310C42—612no—
311C42—612no—
312C42—1224no—
313C42—1224no—
314C42—1221no—
315C42—1221no—
316C23C2×C22⋊#692311yescharacteristic
317Q8—24no—
318Q8—22no—
319Q8—1224no—
320Q8—1224no—
321Q8—4884no—
322Q8—4884no—
323Q8—4884no—
324Q8#1451983814yescomplemented
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—611no—
335C2×C4—611no—
336C2×C4—612no—
337C2×C4—612no—
338C2×C4—612no—
339C2×C4—612no—
340C2×C4—1221no—
341C2×C4—1221no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—1221no—
345C2×C4—2441no—
346C2×C4—2442no—
347C2×C4—2442no—
348C2×C4—2441no—
349C2×C4#1451984012yes—
350C2×C4C2×C22⋊#692311yescharacteristic
351C2×C4C2×C22⋊#692311yescharacteristic
352C8—4882no—
353C8—4882no—
354C22—21no—
355C22—21no—
356C22#1684070311yescharacteristic
357C22C2×#1451984011yescharacteristic
358C22#1684075511yescharacteristic
359C4—24no—
360C4—21no—
361C4—22no—
362C4—42no—
363C4—41no—
364C4—612no—
365C4—612no—
366C4—1222no—
367C4—1222no—
368C4—2442no—
369C4—2442no—
370C4—2441no—
371C4C2×#1451983812yes—
372C2—21no—
373C2—21no—
374C2#2146165311yescharacteristic
375C2#2146165811yescharacteristic
376C2#2146167011yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61#379
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8396
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
130. Q8×C61#101semidirect
215. C61#8396semidirect
324. Q8#14519838semidirect

Maximal subgroups

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

subgroupindex
#214616852
#214613342
#214611472
#214610612
C2×Q8×Dic1222
#839661