Finite Group Databasebeta

← #29592724

Subgroups — #29592724

Order 15616.

380 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 59 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#21446865C211yesmaximal, characteristic
2#21444855C211yesmaximal, characteristic
3#21444821C211yesmaximal, characteristic
4#21443846C211yesmaximal, characteristic, complemented
5#21443825C211yesmaximal, characteristic, complemented
6#21443753C211yesmaximal, characteristic, complemented
7C61×#2833C211yesmaximal, characteristic, complemented
8C2×#14519808—21no—
9#16840343C2211yescharacteristic
10#16840080—21no—
11#16840075—21no—
12C2×#14519705—21no—
13#16839531C2211yescharacteristic
14#16839495—21no—
15#16839495—21no—
16#16839488C2211yescharacteristic
17#16839488C2211yescharacteristic
18C61×#401—21no—
19C61×#400—21no—
20C61×#400—21no—
21C61×#400C2211yescharacteristic
22C61×#398—21no—
23C61×#398C2211yescharacteristic
24C61×#395—21no—
25C61×#391C2211yescharacteristic
26C61×#387—21no—
27C2×#11758222—21no—
28#14519808—42no—
29#14519808—42no—
30C2×#11758202—41no—
31C2×#11758201—41no—
32C2×#11758194—21no—
33#14519757—41no—
34#14519752—41no—
35#14519751—21no—
36#14519751—21no—
37#14519710—41no—
38#14519708—41no—
39#14519705—42no—
40#14519697—41no—
41#14519697—41no—
42C22⋊#92418—21no—
43C22⋊#92418—21no—
44C22×Q8×C61D411yescharacteristic
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×C4⋊C4×C61D411yescharacteristic
51C2×C4⋊C4×C61C2311yescharacteristic
52C2×C42×C61D411yescharacteristic
53#95×C61—21no—
54#95×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×C61—21no—
62#95×C61—21no—
63#95×C61D411yescharacteristic
64#95×C61D411yescharacteristic
65C2×Q8×C61—21no—
66C2×Q8×C61—21no—
67C2×Q8×C61—41no—
68C2×Q8×C61—41no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C2×C22⋊D61—41no—
79#11758222—41no—
80#11758222—41no—
81#11758222—41no—
82#11758222—41no—
83C2×D244—21no—
84C2×D244—21no—
85C2×D244—21no—
86C2×C4×D61—41no—
87C2×Dic122—21no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—81no—
94C4⋊C4×C61—81no—
95C4⋊C4×C61—81no—
96C42×C61—41no—
97#11758202—44no—
98#11758201—44no—
99#11758198—41no—
100#11758198—41no—
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103#11758194—22no—
104#11758194—41no—
105C2×#92418—41no—
106C2×#92418—41no—
107C2×#92418—41no—
108C23×C61#12011yescharacteristic
109C22×D61—41no—
110Q8×C61—42no—
111Q8×C61—81no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—42no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
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—41no—
134C2×C4×C61—41no—
135C2×C4×C61#13511yescharacteristic
136C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
137C22⋊D61—81no—
138C22⋊D61—81no—
139C2×Dic61—41no—
140D244—41no—
141D244—41no—
142D244—81no—
143D244—81no—
144C4×D61—81no—
145C4×D61—81no—
146Dic122—41no—
147Dic122—41no—
148#92418—42no—
149#92418—81no—
150#92418—81no—
151#12201—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61#44911yescharacteristic
155C22×C61#45311yescharacteristic
156C22×C61B(4,𝔽2)11yescharacteristic
157C2×D61—81no—
158C2×D61—81no—
159C2×D61—81no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61—81no—
165C4×C61—81no—
166C4×C61—81no—
167C4×C61—81no—
168C4×C61—81no—
169C4×C61—81no—
170Dic61—81no—
171#2833D6111yescharacteristic, complemented
172#2007—611no—
173#1616—611no—
174#1610—611no—
175#1353—611no—
176#1346—611no—
177#1319—611no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61#160011yescharacteristic
181C2×C61#220111yescharacteristic
182C2×C61#220311yescharacteristic
183D61—161no—
184C2×#137—1221no—
185#537—611no—
186#481—1221no—
187#480—1221no—
188C2×#104—1221no—
189#401—21no—
190#400—21no—
191#400—21no—
192#400C2×D6111yescharacteristic
193#398—21no—
194#398C2×D6111yescharacteristic
195#395—21no—
196#391C2×D6111yescharacteristic
197#387—21no—
198#343—611no—
199#327—1221no—
200#327—1221no—
201#324—611no—
202#324—611no—
203C61#1220111yescharacteristic, complemented, Sylow
204C2×#41—1221no—
205C22×Q8C22⋊D6111yescharacteristic
206#137—2442no—
207#137—2442no—
208C2×Q16—2441no—
209C2×SD16—2441no—
210C2×M16—1221no—
211C22⋊Q8—2441no—
212#121—1221no—
213#121—1221no—
214#121—2441no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22⋊D6111yescharacteristic
220C2×C4⋊C4C22⋊D6111yescharacteristic
221C2×C4⋊C4C22×D6111yescharacteristic
222C2×C42C22⋊D6111yescharacteristic
223#107—2441no—
224#106—2441no—
225#104—2442no—
226#102—2441no—
227#102—2441no—
228C22⋊C8—1221no—
229C22⋊C8—1221no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C22⋊D6111yescharacteristic
241#95C22⋊D6111yescharacteristic
242#41—2441no—
243#41—2441no—
244#41—2441no—
245#41—2441no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—1221no—
251C2×D4—1221no—
252C2×D4—1221no—
253C2×D4—1221no—
254C2×D4—2441no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—2441no—
262C22×C4C2×C22⋊D6111yescharacteristic
263C22×C4C2×C22⋊D6111yescharacteristic
264C22×C4C2×C22⋊D6111yescharacteristic
265Q16—2444no—
266SD16—2444no—
267M16—1222no—
268M16—2441no—
269C2×C8—2441no—
270C2×C8—2441no—
271C2×C8—2441no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C4⋊C4—1221no—
281C4⋊C4—1221no—
282C22⋊C4—2441no—
283C22⋊C4—2441no—
284C42—41no—
285C23—2441no—
286C23#1451974611yescharacteristic
287Q8—42no—
288Q8—81no—
289Q8—2441no—
290Q8—2441no—
291D4—4881no—
292D4—4881no—
293D4—4881no—
294D4—4881no—
295D4—2441no—
296D4—2441no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—4881no—
321C2×C4—4881no—
322C2×C4—2441no—
323C2×C4#1451979711yescharacteristic
324C2×C4#1451980211yescharacteristic
325C8—4881no—
326C8—4881no—
327C8—2442no—
328C22—21no—
329C22—21no—
330C22—4881no—
331C22—4881no—
332C22—4881no—
333C22#1683990711yescharacteristic
334C22#1683993011yescharacteristic
335C22#1683994911yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—4881no—
347C2—21no—
348C2—21no—
349C2—9761no—
350C2#2144476511yescharacteristic
351C2#2144783211yescharacteristic
352C2#2144784011yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2833C2
socleC22×C61B(4,𝔽2)
prime radical#16840343C22
hypercenter#2833D61
hypocenterC61#12201
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2833

Complements

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

NH ≅ G/Ntype
4. #21443846C2semidirect
5. #21443825C2semidirect
6. #21443753C2semidirect
7. C61×#2833C2semidirect
171. #2833D61semidirect
203. C61#12201semidirect

Maximal subgroups

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

subgroupindex
#214468652
#214448552
#214448212
#214438462
#214438252
#214437532
C61×#28332
#1220161