Finite Group Databasebeta

← #29592571

Subgroups — #29592571

Order 15616.

422 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 63 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#21447070C211yesmaximal, characteristic
2#21444838C211yesmaximal, characteristic
3#21444770C211yesmaximal, characteristic
4#21443843C211yesmaximal, characteristic, complemented
5#21443840C211yesmaximal, characteristic, complemented
6#21443767C211yesmaximal, characteristic, complemented
7C61×#2742C211yesmaximal, characteristic, complemented
8#16840324C2211yescharacteristic
9#16840080—21no—
10#16839901—21no—
11C2×#14519691—21no—
12#16839641—21no—
13#16839529C2211yescharacteristic
14#16839495—21no—
15#16839495—21no—
16#16839486C2211yescharacteristic
17#16839486C2211yescharacteristic
18C61×#400—21no—
19C61×#399C2211yescharacteristic
20C61×#395—21no—
21C61×#393C2211yescharacteristic
22C61×#389—21no—
23C61×#388—21no—
24C61×#387—21no—
25C61×#385C2211yescharacteristic
26C61×#380—21no—
27C22×C22⋊D61—21no—
28C2×#11758201—41no—
29#14519761—41no—
30#14519751—21no—
31#14519751—21no—
32#14519747—41no—
33C2×C22⋊Dic61—21no—
34C2×#11758198—41no—
35#14519708—41no—
36#14519701—41no—
37#14519697—41no—
38#14519691—42no—
39C22⋊#92418—21no—
40C22⋊#92418—21no—
41#14519684—41no—
42C22×Q8×C61D411yescharacteristic
43C23×C4×C61D411yescharacteristic
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61D411yescharacteristic
53C2×C22⋊C4×C61C2311yescharacteristic
54C2×C42×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61D411yescharacteristic
62#95×C61D411yescharacteristic
63C24×C61C2×D411yescharacteristic
64C23×D61—21no—
65C2×Q8×C61—41no—
66C2×Q8×C61—81no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—42no—
74C22×C4×C61—41no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C2×C22⋊D61—41no—
78C2×C22⋊D61—41no—
79C2×C22⋊D61—41no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61—41no—
82C2×C22⋊D61—41no—
83C22×Dic61—21no—
84C22×Dic61—41no—
85C2×D244—21no—
86C2×D244—21no—
87C4⋊C4×C61—42no—
88C4⋊C4×C61—42no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—81no—
91C4⋊C4×C61—81no—
92C4⋊C4×C61—81no—
93C22⋊C4×C61—21no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—81no—
96C22⋊C4×C61—81no—
97C22⋊C4×C61SD1612yescomplemented
98C42×C61—81no—
99C22⋊Dic61—41no—
100C22⋊Dic61—41no—
101#11758201—82no—
102#11758198—41no—
103#11758198—41no—
104#11758198—82no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C2×#92418—41no—
108C2×#92418—41no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61#12011yescharacteristic
114C23×C61C2×SD1611yescharacteristic
115C23×C61#13711yescharacteristic
116C22×D61—41no—
117C22×D61—41no—
118C22×D61—41no—
119C22×D61—81no—
120C22×D61—81no—
121Q8×C61—42no—
122Q8×C61—81no—
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—81no—
147C22⋊D61—81no—
148C22⋊D61—81no—
149C22⋊D61—81no—
150C22⋊D61—81no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—81no—
155C2×Dic61—81no—
156C2×Dic61—81no—
157D244—81no—
158D244—81no—
159#92418—81no—
160#92418—81no—
161#12178—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—81no—
171C22×C61#44811yescharacteristic
172C22×C61#45411yescharacteristic
173C22×C61B(4,𝔽2)11yescharacteristic
174C2×D61—82no—
175C2×D61—81no—
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—161no—
179C4×C61—42no—
180C4×C61—41no—
181C4×C61—42no—
182C4×C61—41no—
183C4×C61—81no—
184C4×C61—81no—
185C4×C61—81no—
186C4×C61—81no—
187Dic61—81no—
188Dic61—161no—
189#2742D6111yescharacteristic, complemented
190#2044—611no—
191#1613—611no—
192#1601—611no—
193#1352—611no—
194#1351—611no—
195#1326—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61—41no—
199C2×C61—41no—
200C2×C61#159911yescharacteristic
201C2×C61#219711yescharacteristic
202C2×C61#220011yescharacteristic
203D61—161no—
204#534—611no—
205#481—1221no—
206#448—1221no—
207C2×#99—1221no—
208#400—21no—
209#399C2×D6111yescharacteristic
210#395—21no—
211#393C2×D6111yescharacteristic
212#389—21no—
213#388—21no—
214#387—21no—
215#386—1221no—
216#385C2×D6111yescharacteristic
217#380—21no—
218#342—611no—
219#327—1221no—
220#327—1221no—
221#323—611no—
222#323—611no—
223C61#1217811yescharacteristic, complemented, Sylow
224C22×Q8C22⋊D6111yescharacteristic
225C22×D4—1221no—
226C23×C4C22⋊D6111yescharacteristic
227C2×SD16—2441no—
228#123—2441no—
229#121—1221no—
230#121—1221no—
231#120—2441no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C22⋊D6111yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1221no—
241C2×C22⋊C4—2441no—
242C2×C22⋊C4C22⋊D6111yescharacteristic
243C2×C22⋊C4C22×D6111yescharacteristic
244C2×C42—21no—
245#106—2441no—
246#103—2441no—
247#102—2441no—
248#99—2442no—
249C22⋊C8—1221no—
250C22⋊C8—1221no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—2441no—
258#95C22⋊D6111yescharacteristic
259#95C22⋊D6111yescharacteristic
260C24—1221no—
261C24C2×C22⋊D6111yescharacteristic
262C2×Q8—41no—
263C2×Q8—81no—
264C2×D4—1221no—
265C2×D4—1221no—
266C2×D4—2441no—
267C2×D4—2441no—
268C2×D4—2441no—
269C2×D4—2441no—
270C2×D4—2441no—
271C2×D4—2441no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—1221no—
281C22×C4—2441no—
282C22×C4C2×C22⋊D6111yescharacteristic
283C22×C4C2×C22⋊D6111yescharacteristic
284SD16—4882no—
285C2×C8—2441no—
286C2×C8—2441no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—1221no—
294C4⋊C4—1221no—
295C22⋊C4—21no—
296C22⋊C4—41no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C22⋊C4—4882no—
300C22⋊C4—2441no—
301C22⋊C4—2441no—
302C22⋊C4—2441no—
303C22⋊C4—2441no—
304C22⋊C4#1175820112yescomplemented
305C42—81no—
306C23—21no—
307C23—21no—
308C23—41no—
309C23—41no—
310C23—4881no—
311C23—4881no—
312C23—2441no—
313C23—2441no—
314C23—2441no—
315C23#1451974611yescharacteristic
316C23C2×#1175820111yescharacteristic
317C23#1451980811yescharacteristic
318Q8—42no—
319Q8—81no—
320D4—4881no—
321D4—4881no—
322D4—4881no—
323D4—4881no—
324D4—4881no—
325D4—4881no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—4881no—
351C2×C4—4881no—
352C2×C4—4881no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C8—4881no—
357C8—4881no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22—9761no—
368C22—4882no—
369C22—4881no—
370C22—4881no—
371C22—4881no—
372C22#1683990111yescharacteristic
373C22#1683993511yescharacteristic
374C22#1683994911yescharacteristic
375C4—42no—
376C4—41no—
377C4—42no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—9761no—
384C4—4881no—
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—9761no—
390C2#2144475911yescharacteristic
391C2#2144780911yescharacteristic
392C2#2144782611yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2742C2
socleC22×C61B(4,𝔽2)
prime radical#16840324C22
hypercenter#2742D61
hypocenterC61#12178
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2742

Complements

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

NH ≅ G/Ntype
4. #21443843C2semidirect
5. #21443840C2semidirect
6. #21443767C2semidirect
7. C61×#2742C2semidirect
97. C22⋊C4×C61SD16semidirect
189. #2742D61semidirect
223. C61#12178semidirect
304. C22⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214470702
#214448382
#214447702
#214438432
#214438402
#214437672
C61×#27422
#1217861