Finite Group Databasebeta

← #30606837

Subgroups — #30606837

Order 15616.

1018 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C2×#16840673C212yesmaximal, complemented
2C2×#16840672C212yesmaximal, complemented
3#21461654C212yesmaximal
4C2×C4⋊Q8×D61C211yesmaximal, characteristic
5C2×#14519838—22no—
6C2×#14519838C2214yes—
7#16840734—22no—
8#16840673—24no—
9#16840672—24no—
10C22×Q8×D61C411yescharacteristic
11C22×Q8×D61C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13C2×#14519776—22no—
14C2×#14519775C411yescharacteristic
15C2×#14519775C2211yescharacteristic
16C4⋊Q8×D61—24no—
17C4⋊Q8×D61—24no—
18C2×C4⋊C4×D61—22no—
19C2×C42×D61C411yescharacteristic
20C2×C4⋊Q8×C61C411yescharacteristic
21C2×#11758217—24no—
22#14519838—24no—
23#14519838—24no—
24#14519838—44no—
25#14519834—44no—
26C22×Dic122—21no—
27C22×Dic122C2×C411yescharacteristic
28C22×Dic122D412yes—
29C22×Dic122C2311yescharacteristic
30C2×Q8×D61—22no—
31C2×Q8×D61—22no—
32C2×Q8×D61—22no—
33C2×Q8×D61—22no—
34C2×Q8×D61—42no—
35C2×Q8×D61—42no—
36C2×Q8×D61—42no—
37C2×Q8×D61—42no—
38C22×C4×D61—21no—
39C22×C4×D61—21no—
40C22×C4×D61C2×C411yescharacteristic
41C22×C4×D61D412yes—
42#14519777—22no—
43#14519777D414yes—
44#14519776—216no—
45#14519775—22no—
46#14519775—22no—
47#14519775—22no—
48#14519775D414yes—
49C2×C4⋊Dic61—22no—
50C2×#11758196—22no—
51C2×#11758196—22no—
52C4⋊C4×D61—216no—
53C2×C4×Dic61—21no—
54C2×C4×Dic61C2×C411yescharacteristic
55C42×D61—22no—
56C42×D61—22no—
57C22×Q8×C61C2×C411yescharacteristic
58C22×Q8×C61C2×C411yescharacteristic
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61—22no—
61C2×C4⋊C4×C61—22no—
62C2×C42×C61C2×C411yescharacteristic
63C23×D61C22⋊C411yescharacteristic
64C2×Q8×C61—21no—
65C2×Q8×C61—21no—
66C2×Q8×C61—21no—
67C2×Q8×C61—41no—
68C2×Q8×C61—41no—
69C2×Q8×C61—41no—
70C2×Q8×C61—41no—
71C2×Q8×C61C22⋊C412yes—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C22⋊C412yes—
75C22×C4×C61C22×C411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C2×D412yes—
80Q8×D61—48no—
81Q8×D61—48no—
82Q8×D61—48no—
83Q8×D61—48no—
84C2×C4×D61—28no—
85C2×C4×D61—24no—
86C2×C4×D61—22no—
87C2×C4×D61—21no—
88C2×C4×D61—21no—
89C2×C4×D61—22no—
90C2×C4×D61—24no—
91C2×C4×D61—24no—
92C2×C4×D61—24no—
93C2×C4×D61—42no—
94C2×C4×D61—42no—
95C2×C4×D61—42no—
96C2×Dic122—24no—
97C2×Dic122—21no—
98C2×Dic122—24no—
99C2×Dic122—24no—
100C2×Dic122—42no—
101C2×Dic122—41no—
102C2×Dic122—41no—
103C2×Dic122—42no—
104C2×Dic122—42no—
105C2×Dic122—41no—
106C2×Dic122—41no—
107C2×Dic122C22⋊C412yes—
108C2×Dic122C2×D412yes—
109C2×Dic122C2×D412yes—
110#11758217—44no—
111#11758217—44no—
112C2×#92420—44no—
113C4⋊C4×C61—28no—
114C42×C61—21no—
115C42×C61C22⋊C412yes—
116C4⋊Dic61—28no—
117#11758196—28no—
118#11758196—28no—
119C4×Dic61—24no—
120C4×Dic61—21no—
121C4×Dic61C22⋊C412yes—
122C23×C61C2×C22⋊C411yescharacteristic
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—24no—
126C22×D61#9912yes—
127Q8×C61—42no—
128Q8×C61—42no—
129Q8×C61—42no—
130Q8×C61—42no—
131C2×C4×C61—24no—
132C2×C4×C61—22no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61C2×C22⋊C411yescharacteristic
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×Dic61—24no—
144C2×Dic61—21no—
145C2×Dic61—22no—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—21no—
149C2×Dic61—22no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61#9912yes—
153C2×Dic61#12014yes—
154C4×D61—216no—
155C4×D61—24no—
156C4×D61—24no—
157C4×D61—48no—
158C4×D61—48no—
159Dic122—42no—
160Dic122—42no—
161Dic122—48no—
162Dic122—48no—
163Dic122—42no—
164Dic122—42no—
165#92420—84no—
166#12700—611nomaximal, Sylow
167C22×C61—22no—
168C22×C61#37911yescharacteristic
169C22×C61C2×#9911yescharacteristic
170C22×C61C2×#9911yescharacteristic
171C2×D61—24no—
172C2×D61—24no—
173C2×D61—22no—
174C2×D61—22no—
175C2×D61#35614yescomplemented
176C4×C61—24no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—42no—
180C4×C61—42no—
181Dic61—24no—
182Dic61—24no—
183Dic61—22no—
184Dic61—42no—
185C22×C4⋊Q8—611no—
186C2×#356—612no—
187C2×#356—612no—
188#1784—612no—
189C2×C61—21no—
190C2×C61—21no—
191C2×C61#178111yescharacteristic
192C2×C61C2×#35612yes—
193D61—24no—
194C23×Q8—611no—
195C23×Q8—611no—
196C2×C4⋊Q8—611no—
197C2×C4⋊Q8—611no—
198C2×C4⋊Q8—611no—
199C2×C4⋊Q8—1222no—
200C2×C4⋊Q8—1224no—
201C2×C4⋊Q8—1224no—
202C2×C4⋊Q8#692311yescharacteristic
203C22×C4⋊C4—1222no—
204C22×C42—611no—
205C2×#101—614no—
206C2×#101—1222no—
207#407—1222no—
208#356—1224no—
209#356—1224no—
210C61#1270011yescharacteristic, complemented, Sylow
211C22×Q8—612no—
212C22×Q8—611no—
213C22×Q8—611no—
214C22×Q8—1221no—
215C22×Q8—1222no—
216C22×Q8—1222no—
217C22×Q8—1222no—
218C22×Q8—1222no—
219C22×Q8—2442no—
220C22×Q8—2442no—
221C22×Q8—2442no—
222C22×Q8—2442no—
223C22×Q8C2×#692311yescharacteristic
224C22×Q8C2×#692311yescharacteristic
225C23×C4—611no—
226C23×C4—612no—
227C23×C4—1221no—
228C23×C4—1221no—
229C2×M16—1224no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—614no—
233C4⋊Q8—614no—
234C4⋊Q8—1222no—
235C4⋊Q8—1222no—
236C4⋊Q8—12216no—
237C4⋊Q8—1222no—
238C4⋊Q8—1222no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—12216no—
241C2×C4⋊C4—1222no—
242C2×C4⋊C4—1222no—
243C2×C4⋊C4—1222no—
244C2×C42—611no—
245C2×C42—1222no—
246C2×C42—1221no—
247C2×C42—1222no—
248C2×C42C2×#692311yescharacteristic
249#101—1224no—
250#101—1224no—
251#101—2444no—
252C22⋊C8—2444no—
253C24—611no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—612no—
262C2×Q8—612no—
263C2×Q8—612no—
264C2×Q8—1224no—
265C2×Q8—1224no—
266C2×Q8—1224no—
267C2×Q8—1221no—
268C2×Q8—2441no—
269C2×Q8—2441no—
270C2×Q8—2442no—
271C2×Q8—2441no—
272C2×Q8—2441no—
273C2×Q8—2442no—
274C2×Q8—2442no—
275C2×Q8—2448no—
276C2×Q8—2448no—
277C2×Q8—2448no—
278C2×Q8—2448no—
279C2×Q8C22⋊#692312yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—611no—
283C22×C4—612no—
284C22×C4—1222no—
285C22×C4—1224no—
286C22×C4—1222no—
287C22×C4—1224no—
288C22×C4—1228no—
289C22×C4—1224no—
290C22×C4—1224no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—2442no—
296C22×C4—2442no—
297C22×C4—2442no—
298C22×C4C22⋊#692312yes—
299C22×C4C22×#692311yescharacteristic
300M16—2444no—
301M16—2444no—
302C2×C8—2444no—
303C4⋊C4—28no—
304C4⋊C4—1228no—
305C4⋊C4—1228no—
306C4⋊C4—1228no—
307C42—21no—
308C42—612no—
309C42—1224no—
310C42—1221no—
311C42C22⋊#692312yes—
312C23—612no—
313C23—1224no—
314C23—1221no—
315C23—1221no—
316C23C2×C22⋊#692311yescharacteristic
317Q8—42no—
318Q8—42no—
319Q8—42no—
320Q8—42no—
321Q8—2442no—
322Q8—2442no—
323Q8—2442no—
324Q8—2442no—
325Q8—2448no—
326Q8—2448no—
327C2×C4—24no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—612no—
338C2×C4—614no—
339C2×C4—12216no—
340C2×C4—1224no—
341C2×C4—1224no—
342C2×C4—1224no—
343C2×C4—1222no—
344C2×C4—1222no—
345C2×C4—1221no—
346C2×C4—1222no—
347C2×C4—1221no—
348C2×C4—1222no—
349C2×C4—2448no—
350C2×C4—2448no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4C2×C22⋊#692311yescharacteristic
354C2×C4C2×C22⋊#692311yescharacteristic
355C8—4884no—
356C22—22no—
357C22—614no—
358C22—1224no—
359C22—1224no—
360C22—1222no—
361C22—1222no—
362C22#1684070311yescharacteristic
363C22D61⋊#9911yescharacteristic
364C22D61⋊#9911yescharacteristic
365C4—24no—
366C4—21no—
367C4—21no—
368C4—42no—
369C4—42no—
370C4—1224no—
371C4—1224no—
372C4—1222no—
373C4—2442no—
374C2—21no—
375C2—21no—
376C2—1224no—
377C2#2146164611yescharacteristic
378C2D61⋊#35612yes—
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC22×Dic122C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#12700
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840673C2semidirect
2. C2×#16840672C2semidirect
175. C2×D61#356semidirect
210. C61#12700semidirect

Maximal subgroups

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

subgroupindex
C2×#168406732
C2×#168406722
#214616542
C2×C4⋊Q8×D612
#1270061