Finite Group Databasebeta

← #30598733

Subgroups — #30598733

Order 15616.

656 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461352C212yesmaximal, complemented
2#21461301C212yesmaximal, complemented
3#21461072C211yesmaximal, characteristic
4#21461072C211yesmaximal, characteristic
5C2×#16840370C211yesmaximal, characteristic
6#16840782C2212yes—
7#16840782C2212yes—
8#16840622—24no—
9#16840619—24no—
10#16840608—21no—
11#16840608—21no—
12#16840606—21no—
13#16840606—21no—
14#16840370—24no—
15#16840370—24no—
16C2×#14519775C2211yescharacteristic
17C2×#14519772—21no—
18C2×#14519771—21no—
19C2×#14519769—21no—
20C2×#14519768C2211yescharacteristic
21C2×#14519767—21no—
22C2×#14519757C411yescharacteristic
23C2×#14519753C411yescharacteristic
24C2×C4×Dic122C411yescharacteristic
25C2×#14519734C2211yescharacteristic
26C2×C4×Q8×C61C411yescharacteristic
27#14519844—22no—
28#14519844—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31#14519832—22no—
32#14519832—22no—
33C22×Dic122C2×C411yescharacteristic
34C22×C4×D61C2×C411yescharacteristic
35#14519775—22no—
36#14519775D414yes—
37#14519772—28no—
38#14519771—28no—
39#14519769—28no—
40#14519768—22no—
41#14519768D414yes—
42#14519767—28no—
43#14519757—22no—
44#14519757—22no—
45#14519753—22no—
46#14519753—22no—
47C4×Dic122—22no—
48C4×Dic122—22no—
49#14519734—22no—
50#14519734—22no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2×C411yescharacteristic
54C2×#11758196—21no—
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196D411yescharacteristic
60C2×#11758196D411yescharacteristic
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×#11758198D411yescharacteristic
64C2×#11758198D411yescharacteristic
65C2×C4×Dic61—21no—
66C2×C4×Dic61C2311yescharacteristic
67C22×Q8×C61C2×C411yescharacteristic
68C4×Q8×C61—22no—
69C4×Q8×C61—22no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61C2×C411yescharacteristic
72C2×C42×C61—21no—
73C2×C42×C61C2×C411yescharacteristic
74C23×D61C22⋊C411yescharacteristic
75C2×Q8×C61—21no—
76C2×Q8×C61—42no—
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×Dic61C2×D411yescharacteristic
87C22×Dic61C2×D411yescharacteristic
88C2×C4×D61—21no—
89C2×C4×D61—21no—
90C2×C4×D61—42no—
91C2×Dic122—21no—
92C2×Dic122—42no—
93C2×Dic122C22⋊C412yes—
94C2×#92420—42no—
95C2×#92420—42no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61C22⋊C412yes—
99C42×C61—24no—
100C42×C61—21no—
101C42×C61—21no—
102#11758198—24no—
103#11758198—24no—
104#11758198—22no—
105#11758198—22no—
106C4⋊Dic61—24no—
107C4⋊Dic61—21no—
108C4⋊Dic61—24no—
109C4⋊Dic61C22⋊C412yes—
110#11758196—24no—
111#11758196—24no—
112#11758196—24no—
113#11758196—24no—
114#11758196—24no—
115#11758196—22no—
116#11758196—22no—
117C4×Dic61—24no—
118C4×Dic61C2×D412yes—
119C4×Dic61C2×D412yes—
120C23×C61C2×C22⋊C411yescharacteristic
121C22×D61—21no—
122C22×D61—22no—
123C22×D61—41no—
124C22×D61—41no—
125Q8×C61—42no—
126Q8×C61—42no—
127C2×C4×C61—22no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61C2×C22⋊C411yescharacteristic
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×Dic61—22no—
139C2×Dic61—22no—
140C2×Dic61—21no—
141C2×Dic61—22no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61#12012yes—
147C2×Dic61#12012yes—
148C4×D61—42no—
149C4×D61—42no—
150Dic122—42no—
151Dic122—42no—
152#92420—82no—
153#92420—82no—
154#8527—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61#37911yescharacteristic
158C22×C61#41312yes—
159C2×D61—41no—
160C2×D61—41no—
161C2×D61—41no—
162C2×D61—82no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—41no—
166C4×C61—41no—
167C4×C61—42no—
168C4×C61—42no—
169Dic61—22no—
170Dic61—22no—
171Dic61—42no—
172Dic61—42no—
173Dic61—41no—
174C2×#541—611no—
175#1558—612no—
176#1534—612no—
177#1464—611no—
178#1464—611no—
179C2×C61—22no—
180C2×C61#178511yescharacteristic
181C2×C61#179611yescharacteristic
182C2×C61#179611yescharacteristic
183D61—81no—
184#541—1224no—
185#541—1224no—
186C2×C4⋊Q8—611no—
187C2×#126—1221no—
188C2×#126—1221no—
189C2×#125—611no—
190C2×#125—1221no—
191C2×#125—1221no—
192C2×C22⋊Q8—611no—
193C2×C22⋊Q8—611no—
194C2×C4×Q8—611no—
195C2×C4×Q8#692311yescharacteristic
196C2×#117—611no—
197#432—612no—
198#432—612no—
199#333—1224no—
200#330—1224no—
201#324—1221no—
202#324—1221no—
203#323—1221no—
204#323—1221no—
205C61#852711yescharacteristic, complemented, Sylow
206C22×Q8—611no—
207C22×Q8C2×#692311yescharacteristic
208C23×C4—611no—
209C4⋊Q8—614no—
210C4⋊Q8—1222no—
211#126—1228no—
212#126—1228no—
213#125—614no—
214#125—1228no—
215#125—1222no—
216#125—1228no—
217C22⋊Q8—1222no—
218C22⋊Q8—1222no—
219C22⋊Q8—1222no—
220C22⋊Q8—1222no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—1222no—
224C4×Q8—1222no—
225#117—1222no—
226#117—1222no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—611no—
229C2×C4⋊C4—611no—
230C2×C4⋊C4—611no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1221no—
236C2×C4⋊C4—1221no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4C2×#692311yescharacteristic
239C2×C22⋊C4—611no—
240C2×C22⋊C4—611no—
241C2×C22⋊C4—1221no—
242C2×C22⋊C4—1221no—
243C2×C42—21no—
244C2×C42—611no—
245C2×C42—1221no—
246C2×C42C2×#692311yescharacteristic
247C4⋊C8—1222no—
248C4⋊C8—1222no—
249C22⋊C8—1222no—
250C22⋊C8—1222no—
251#97—1222no—
252#97—1222no—
253C24—611no—
254C2×Q8—21no—
255C2×Q8—42no—
256C2×Q8—612no—
257C2×Q8—1221no—
258C2×Q8—2442no—
259C2×Q8C22⋊#692312yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—611no—
263C22×C4—611no—
264C22×C4—611no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—2442no—
270C22×C4C22⋊#692311yescharacteristic
271C22×C4C22⋊#692311yescharacteristic
272C22×C4C22×#692311yescharacteristic
273C2×C8—2442no—
274C2×C8—2442no—
275C4⋊C4—24no—
276C4⋊C4—21no—
277C4⋊C4—612no—
278C4⋊C4—1222no—
279C4⋊C4—1222no—
280C4⋊C4—1224no—
281C4⋊C4—1224no—
282C4⋊C4—1224no—
283C4⋊C4—1224no—
284C4⋊C4—1224no—
285C4⋊C4—1224no—
286C4⋊C4—1224no—
287C4⋊C4—1221no—
288C4⋊C4C22⋊#692312yes—
289C22⋊C4—1222no—
290C22⋊C4—1222no—
291C22⋊C4—1224no—
292C22⋊C4—1224no—
293C42—24no—
294C42—21no—
295C42—21no—
296C42—612no—
297C42—612no—
298C42—1224no—
299C23—1222no—
300C23—1221no—
301C23—2441no—
302C23—2441no—
303C23C2×C22⋊#692311yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—2442no—
307Q8—2442no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—612no—
318C2×C4—612no—
319C2×C4—1222no—
320C2×C4—1222no—
321C2×C4—1222no—
322C2×C4—1221no—
323C2×C4—2442no—
324C2×C4—2442no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4C2×C22⋊#692311yescharacteristic
330C2×C4C2×C22⋊#692311yescharacteristic
331C8—4882no—
332C8—4882no—
333C22—21no—
334C22—21no—
335C22—4882no—
336C22—2441no—
337C22—2441no—
338C22—2441no—
339C22#1684070311yescharacteristic
340C22#1684074812yes—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—42no—
347C4—1222no—
348C4—1222no—
349C4—2441no—
350C4—2442no—
351C4—2442no—
352C2—22no—
353C2—4881no—
354C2#2146165511yescharacteristic
355C2#2146167111yescharacteristic
356C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61#379
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8527
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
1. #21461352C2semidirect
2. #21461301C2semidirect
205. C61#8527semidirect

Maximal subgroups

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

subgroupindex
#214613522
#214613012
#214610722
C2×#168403702
#852761