Finite Group Databasebeta

← #29595880

Subgroups — #29595880

Order 15616.

728 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 127 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#21447504C211yesmaximal, characteristic
2#21447473C212yesmaximal, complemented
3#21446372C212yesmaximal, complemented
4#21446203C211yesmaximal, characteristic
5C61×#3550C211yesmaximal, characteristic
6#16840529C2211yescharacteristic
7#16839779—21no—
8#16839746C2212yes—
9C2×#14519696—22no—
10#16839736C2211yescharacteristic
11#16839561—21no—
12#16839559—21no—
13#16839552—24no—
14#16839532—22no—
15#16839522—21no—
16C61×#563—21no—
17C61×#563C412yescomplemented
18C61×#559C412yescomplemented
19Q82×C61—21no—
20C61×#556—22no—
21D4×Q8×C61—22no—
22C61×#542—24no—
23C61×#536—21no—
24C61×#536C2212yes—
25C61×#533—21no—
26C61×#532C2211yescharacteristic
27#14519784—42no—
28C2×#11758194—22no—
29C2×#11758194—21no—
30C2×#11758194—22no—
31#14519736—21no—
32#14519704—42no—
33#14519696—22no—
34#14519696—44no—
35#14519694—22no—
36#14519692—22no—
37C2×#41×C61D411yescharacteristic
38C2×#41×C61C2311yescharacteristic
39C22×Q8×C61—21no—
40C22×Q8×C61D412yes—
41C4⋊Q8×C61—21no—
42C4⋊Q8×C61—21no—
43C4⋊Q8×C61—24no—
44C4⋊Q8×C61—22no—
45C4⋊Q8×C61—22no—
46C4⋊Q8×C61D411yescharacteristic
47#127×C61C2×C411yescharacteristic
48#126×C61—24no—
49#126×C61—24no—
50#125×C61—22no—
51#124×C61—22no—
52#124×C61—22no—
53#124×C61—24no—
54#124×C61C2×C411yescharacteristic
55#124×C61D411yescharacteristic
56#123×C61—22no—
57#123×C61—22no—
58#123×C61C2×C414yes—
59#123×C61D414yes—
60C22⋊Q8×C61—22no—
61C22⋊Q8×C61—22no—
62C22⋊Q8×C61—24no—
63C22⋊Q8×C61—24no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61—22no—
66#121×C61—22no—
67C4×Q8×C61—22no—
68C4×Q8×C61—24no—
69C4×Q8×C61—22no—
70C4×D4×C61—24no—
71C4×D4×C61—22no—
72#117×C61—22no—
73#117×C61—21no—
74#117×C61D412yes—
75C2×C42×C61D411yescharacteristic
76#41×C61—41no—
77#41×C61—41no—
78#41×C61—41no—
79#41×C61—41no—
80C2×Q8×C61—22no—
81C2×Q8×C61—22no—
82C2×Q8×C61—22no—
83C2×Q8×C61—22no—
84C2×Q8×C61—42no—
85C2×Q8×C61—42no—
86C2×Q8×C61—81no—
87C2×Q8×C61—81no—
88C2×Q8×C61C22⋊C411yescharacteristic
89C2×Q8×C61C2×D411yescharacteristic
90C2×D4×C61—22no—
91C2×D4×C61—21no—
92C2×D4×C61C22⋊C411yescharacteristic
93C2×D4×C61C22×C411yescharacteristic
94C2×D4×C61C2×D412yes—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—21no—
98C22×C4×C61C22⋊C414yes—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D412yes—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—24no—
105C4⋊C4×C61—24no—
106C4⋊C4×C61—24no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—24no—
110C22⋊C4×C61—22no—
111C22⋊C4×C61—24no—
112C22⋊C4×C61—22no—
113C22⋊C4×C61—24no—
114C22⋊C4×C61—24no—
115C22⋊C4×C61—22no—
116C22⋊C4×C61C22⋊C414yes—
117C42×C61—24no—
118C42×C61—22no—
119C42×C61—21no—
120C42×C61—22no—
121C42×C61C22⋊C411yescharacteristic
122C42×C61C22⋊C411yescharacteristic
123C22⋊Dic61—41no—
124C4⋊Dic61—41no—
125C4×Dic61—41no—
126#11758194—24no—
127#11758194—22no—
128#11758194—41no—
129#11758194—42no—
130#11758194—42no—
131C2×#92418—42no—
132C2×#92418—41no—
133C2×#92418—42no—
134C23×C61—21no—
135C23×C61C2×C22⋊C412yes—
136C23×C61#12011yescharacteristic
137Q8×C61—41no—
138Q8×C61—44no—
139Q8×C61—41no—
140Q8×C61—42no—
141Q8×C61—42no—
142Q8×C61—82no—
143Q8×C61—82no—
144D4×C61—41no—
145D4×C61—42no—
146D4×C61—41no—
147D4×C61—44no—
148D4×C61—81no—
149C2×C4×C61—21no—
150C2×C4×C61—22no—
151C2×C4×C61—21no—
152C2×C4×C61—22no—
153C2×C4×C61—21no—
154C2×C4×C61—24no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—24no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—44no—
162C2×C4×C61—82no—
163C2×C4×C61C2×C22⋊C411yescharacteristic
164C2×C4×C61#12012yes—
165C2×C4×C61#12011yescharacteristic
166C2×Dic61—81no—
167#92418—42no—
168#92418—82no—
169#12771—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—22no—
172C22×C61—41no—
173C22×C61—42no—
174C22×C61—42no—
175C22×C61—41no—
176C22×C61#37911yescharacteristic
177C4×C61—22no—
178C4×C61—22no—
179C4×C61—21no—
180C4×C61—41no—
181C4×C61—44no—
182C4×C61—41no—
183C4×C61—44no—
184C4×C61B(4,𝔽2)11yescharacteristic
185C4×C61#45811yescharacteristic
186Dic61—161no—
187#3550D6111yescharacteristic
188#2132—611no—
189#2123—612no—
190#1915—612no—
191#1884—611no—
192C2×C61—21no—
193C2×C61—41no—
194C2×C61—42no—
195C2×C61—81no—
196C2×C61#179011yescharacteristic
197#568—611no—
198#563—21no—
199#563Dic6112yescomplemented
200#559Dic6112yescomplemented
201Q82—21no—
202#556—22no—
203D4×Q8—22no—
204#542—24no—
205#536—21no—
206#536C2×D6112yes—
207#533—21no—
208#532C2×D6111yescharacteristic
209#421—1221no—
210#413—612no—
211C2×#101—1222no—
212#410—611no—
213#356—1221no—
214#355—1221no—
215#352—1224no—
216#344—1222no—
217#339—1221no—
218C61#1277111yescharacteristic, complemented, Sylow
219C2×#41C22⋊D6111yescharacteristic
220C2×#41C22×D6111yescharacteristic
221C22×Q8—21no—
222C22×Q8C22⋊D6112yes—
223#131—2442no—
224C2×M16—1221no—
225C2×M16—1222no—
226C2×M16—1222no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—24no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8C22⋊D6111yescharacteristic
233#127C2×Dic6111yescharacteristic
234#126—24no—
235#126—24no—
236#125—22no—
237#124—22no—
238#124—22no—
239#124—24no—
240#124C2×Dic6111yescharacteristic
241#124C22⋊D6111yescharacteristic
242#123—22no—
243#123—22no—
244#123C2×Dic6114yes—
245#123C22⋊D6114yes—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252#121—22no—
253C4×Q8—22no—
254C4×Q8—24no—
255C4×Q8—22no—
256C4×D4—24no—
257C4×D4—22no—
258#117—22no—
259#117—21no—
260#117—1221no—
261#117C22⋊D6112yes—
262C2×C42C22⋊D6111yescharacteristic
263#104—2442no—
264#101—1222no—
265#101—2444no—
266#100—1222no—
267#99—1222no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—81no—
279C2×Q8—81no—
280C2×Q8C22⋊Dic6111yescharacteristic
281C2×Q8C2×C22⋊D6111yescharacteristic
282C2×D4—22no—
283C2×D4—21no—
284C2×D4C22⋊Dic6111yescharacteristic
285C2×D4C22×Dic6111yescharacteristic
286C2×D4C2×C22⋊D6112yes—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4C22⋊Dic6114yes—
291C22×C4C2×C22⋊D6111yescharacteristic
292C22×C4C2×C22⋊D6112yes—
293M16—1224no—
294M16—1222no—
295M16—2442no—
296M16—2442no—
297M16—2441no—
298C2×C8—2442no—
299C2×C8—2441no—
300C2×C8—2442no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—2441no—
311C22⋊C4—22no—
312C22⋊C4—24no—
313C22⋊C4—22no—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—22no—
317C22⋊C4—2441no—
318C22⋊C4C22⋊Dic6114yes—
319C42—24no—
320C42—22no—
321C42—22no—
322C42—21no—
323C42—2441no—
324C42C22⋊Dic6111yescharacteristic
325C42C22⋊Dic6111yescharacteristic
326C23—21no—
327C23C2×C22⋊Dic6112yes—
328C23#1451974611yescharacteristic
329Q8—41no—
330Q8—41no—
331Q8—44no—
332Q8—42no—
333Q8—42no—
334Q8—82no—
335Q8—82no—
336D4—41no—
337D4—42no—
338D4—41no—
339D4—44no—
340D4—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—82no—
355C2×C4—4881no—
356C2×C4C2×C22⋊Dic6111yescharacteristic
357C2×C4#1451974612yes—
358C2×C4#1451974611yescharacteristic
359C8—4882no—
360C8—2442no—
361C22—22no—
362C22—21no—
363C22—41no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22#1683961811yescharacteristic
368C4—22no—
369C4—22no—
370C4—21no—
371C4—41no—
372C4—44no—
373C4—41no—
374C4—44no—
375C4—9761no—
376C4#1683994911yescharacteristic
377C4#1683995211yescharacteristic
378C2—21no—
379C2—41no—
380C2—42no—
381C2—81no—
382C2#2144576711yescharacteristic
subgroupG/H
centerC2#21445767
commutatorC2×D4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#3550C2
socleC2×C61#1790
prime radicalC61×#532C22
cosocleC2×#41×C61C23
hypercenter#3550D61
hypocenterC61#12771
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3550

Complements

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

NH ≅ G/Ntype
2. #21447473C2semidirect
3. #21446372C2semidirect
17. C61×#563C4semidirect
18. C61×#559C4semidirect
199. #563Dic61semidirect
200. #559Dic61semidirect
218. C61#12771semidirect

Maximal subgroups

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

subgroupindex
#214475042
#214474732
#214463722
#214462032
C61×#35502
#1277161