Finite Group Databasebeta

← #29652806

Subgroups — #29652806

Order 15616.

970 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 183 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#21455945C212yesmaximal, complemented
2#21449103C211yesmaximal, characteristic
3#21444810C214yesmaximal, complemented
4#21444776C214yesmaximal, complemented
5#21444331C212yesmaximal, complemented
6#21444160C211yesmaximal, characteristic
7C61×#3501C211yesmaximal, characteristic
8C2×C4×Dic122C2211yescharacteristic
9#16840090C2214yes—
10#16840044—22no—
11#16839969—22no—
12#16839917C2214yes—
13#16839867—24no—
14#16839819C2212yes—
15#16839700—21no—
16#16839687C2212yes—
17#16839676—21no—
18#16839674C2212yes—
19C4×C22⋊Dic61—22no—
20#16839608C2211yescharacteristic
21#16839499C2218yes—
22#16839486C2214yes—
23C61×#554—22no—
24C61×#554—24no—
25D4×Q8×C61—22no—
26C61×#544—21no—
27C61×#543C2214yes—
28C61×#533C2212yes—
29C2×C22⋊Q8×C61—22no—
30C61×#518—21no—
31C2×C4×D4×C61C2211yescharacteristic
32C22×Dic122C2311yescharacteristic
33C2×#11758202—24no—
34C4×Dic122—22no—
35C4×Dic122—22no—
36C2×C4⋊Dic61C2311yescharacteristic
37C2×#11758196—21no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61—21no—
42#14519708—24no—
43C4⋊#92418—22no—
44#14519701—24no—
45#14519700—24no—
46C22⋊#92418C2314yes—
47#14519688—22no—
48#14519684—22no—
49#14519684—21no—
50#14519684—22no—
51#14519684C2312yes—
52C22×Q8×C61—22no—
53C22×D4×C61D411yescharacteristic
54C23×C4×C61D412yes—
55C4⋊Q8×C61—24no—
56C4⋊Q8×C61D412yes—
57C4⋊Q8×C61D412yes—
58#125×C61—24no—
59#125×C61—22no—
60C22⋊Q8×C61—24no—
61C22⋊Q8×C61—24no—
62C22⋊Q8×C61—28no—
63C22⋊Q8×C61—22no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61—28no—
66C22⋊Q8×C61C2314yes—
67C4×Q8×C61—22no—
68C4×Q8×C61D414yes—
69C4×D4×C61—24no—
70C4×D4×C61—24no—
71C4×D4×C61—22no—
72C4×D4×C61—22no—
73#117×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—24no—
76C2×C4⋊C4×C61D411yescharacteristic
77C2×C22⋊C4×C61—24no—
78C2×C22⋊C4×C61C2312yes—
79C2×C42×C61C2311yescharacteristic
80C24×C61C2×D412yes—
81C2×Q8×C61—24no—
82C2×Q8×C61—22no—
83C2×Q8×C61—84no—
84C2×Q8×C61C2×D414yes—
85C2×D4×C61—24no—
86C2×D4×C61—22no—
87C22×C4×C61—24no—
88C22×C4×C61—24no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—44no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2411yescharacteristic
96C22×Dic61—21no—
97C22×Dic61—22no—
98C22×Dic61—21no—
99C2×Dic122—22no—
100C2×Dic122—42no—
101C4⋊C4×C61—24no—
102C4⋊C4×C61—28no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—24no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—24no—
109C4⋊C4×C61C2×D414yes—
110C22⋊C4×C61—28no—
111C22⋊C4×C61—24no—
112C22⋊C4×C61—24no—
113C42×C61—24no—
114C42×C61C2×D412yes—
115C42×C61C2×D412yes—
116C22⋊Dic61—44no—
117C22⋊Dic61—44no—
118#11758202—48no—
119C4⋊Dic61—22no—
120#11758196—42no—
121C4×Dic61—42no—
122C4×Dic61—41no—
123C4×Dic61—41no—
124C2×#92418—24no—
125C23×C61—22no—
126C23×C61—22no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61#13714yes—
130C23×C61C22×D411yescharacteristic
131Q8×C61—44no—
132Q8×C61—44no—
133Q8×C61—84no—
134D4×C61—28no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—24no—
138C2×C4×C61—24no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61—44no—
142C2×C4×C61—42no—
143C2×C4×C61—82no—
144C2×C4×C61—84no—
145C2×C4×C61—84no—
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12012yes—
148C2×C4×C61#12011yescharacteristic
149C2×C4×C61C22×D412yes—
150C2×Dic61—22no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—42no—
154C2×Dic61—42no—
155C2×Dic61—41no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158Dic122—24no—
159Dic122—42no—
160#92418—44no—
161#18181—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—24no—
165C22×C61—44no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—82no—
169C22×C61C2×#12011yescharacteristic
170C22×C61C2×#13712yes—
171C4×C61—22no—
172C4×C61—21no—
173C4×C61—22no—
174C4×C61—42no—
175C4×C61—44no—
176C4×C61—44no—
177C4×C61#45512yes—
178Dic61—42no—
179Dic61—82no—
180Dic61—81no—
181#3501D6111yescharacteristic
182#3113—612no—
183#2398—611no—
184#1608—614no—
185#1602—614no—
186#1506—612no—
187#1466—611no—
188C2×C61—22no—
189C2×C61—44no—
190C2×C61#300411yescharacteristic
191C2×C61C2×#45511yescharacteristic
192C2×C61#302511yescharacteristic
193#554—22no—
194#554—24no—
195D4×Q8—22no—
196#544—21no—
197#543C2×D6114yes—
198#533C2×D6112yes—
199C2×C22⋊Q8—22no—
200#518—21no—
201C2×C4×Q8—611no—
202C2×C4×D4C2×D6111yescharacteristic
203#483—614no—
204#475—1222no—
205#462—1222no—
206#451—614no—
207#441—1224no—
208#432—612no—
209#400—1221no—
210#397—612no—
211#394—1221no—
212#393—612no—
213C4×C22⋊C4—1222no—
214#376—611no—
215#328—618no—
216#323—614no—
217C61#1818111yescharacteristic, complemented, Sylow
218C22×Q8—22no—
219C22×Q8—611no—
220C22×D4C22⋊D6111yescharacteristic
221C23×C4C22⋊D6112yes—
222C2×Q16—1224no—
223C4⋊Q8—24no—
224C4⋊Q8C22⋊D6112yes—
225C4⋊Q8C22⋊D6112yes—
226#125—24no—
227#125—22no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—28no—
231C22⋊Q8—22no—
232C22⋊Q8—24no—
233C22⋊Q8—28no—
234C22⋊Q8C22×D6114yes—
235C4×Q8—22no—
236C4×Q8—1222no—
237C4×Q8—1222no—
238C4×Q8C22⋊D6114yes—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—22no—
242C4×D4—22no—
243#117—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—611no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4C22⋊D6111yescharacteristic
249C2×C22⋊C4—24no—
250C2×C22⋊C4—1222no—
251C2×C22⋊C4—1222no—
252C2×C22⋊C4C22×D6112yes—
253C2×C42—1221no—
254C2×C42—1221no—
255C2×C42C22×D6111yescharacteristic
256#106—1224no—
257C4⋊C8—1222no—
258#103—1224no—
259#103—1224no—
260C22⋊C8—614no—
261#97—1222no—
262#95—612no—
263#95—1221no—
264#95—1222no—
265#95—1222no—
266C24C2×C22⋊D6112yes—
267C2×Q8—24no—
268C2×Q8—22no—
269C2×Q8—84no—
270C2×Q8—1222no—
271C2×Q8—2442no—
272C2×Q8C2×C22⋊D6114yes—
273C2×D4—24no—
274C2×D4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—44no—
280C22×C4—1221no—
281C22×C4—1222no—
282C22×C4—1221no—
283C22×C4C2×C22⋊D6111yescharacteristic
284C22×C4C2×C22⋊D6111yescharacteristic
285C22×C4C2×C22⋊D6112yes—
286C22×C4C23×D6111yescharacteristic
287Q16—2448no—
288C2×C8—1224no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—1222no—
298C4⋊C4—2442no—
299C4⋊C4C2×C22⋊D6114yes—
300C22⋊C4—28no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—2444no—
304C22⋊C4—2444no—
305C42—24no—
306C42—2442no—
307C42—2441no—
308C42—2441no—
309C42C2×C22⋊D6112yes—
310C42C2×C22⋊D6112yes—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23#1451980814yes—
316C23C22×C22⋊D6111yescharacteristic
317Q8—44no—
318Q8—44no—
319Q8—84no—
320Q8—1224no—
321Q8—2442no—
322D4—28no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—1222no—
335C2×C4—2442no—
336C2×C4—2442no—
337C2×C4—2441no—
338C2×C4—2442no—
339C2×C4—2442no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4#1451974612yes—
343C2×C4#1451974611yescharacteristic
344C2×C4#1451974611yescharacteristic
345C2×C4C22×C22⋊D6112yes—
346C8—2444no—
347C22—21no—
348C22—21no—
349C22—24no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22C2×#1451974611yescharacteristic
355C22C2×#1451980812yes—
356C4—21no—
357C4—22no—
358C4—22no—
359C4—42no—
360C4—44no—
361C4—44no—
362C4—4882no—
363C4—4881no—
364C4—2442no—
365C4#1683994012yes—
366C2—22no—
367C2—44no—
368C2#2145505311yescharacteristic
369C2C2×#1683994011yescharacteristic
370C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3501D61
hypocenterC61#18181
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21455945C2semidirect
3. #21444810C2semidirect
4. #21444776C2semidirect
5. #21444331C2semidirect
217. C61#18181semidirect

Maximal subgroups

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

subgroupindex
#214559452
#214491032
#214448102
#214447762
#214443312
#214441602
C61×#35012
#1818161