Finite Group Databasebeta

← #29648016

Subgroups — #29648016

Order 15616.

960 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454839C211yesmaximal, characteristic, complemented
2#21454247C211yesmaximal, characteristic, complemented
3#21444434C213yesmaximal, complemented
4#21444426C213yesmaximal, complemented
5#21444414C213yesmaximal, complemented
6#21444407C213yesmaximal, complemented
7C61×#3518C211yesmaximal, characteristic
8C2×#14519784—21no—
9Q8×#92418—21no—
10#16839840—23no—
11#16839819C2213yes—
12#16839718—23no—
13C4×#11758194C2213yes—
14C4×#11758194C2213yes—
15C2×#14519688C2213yes—
16#16839504C2213yes—
17#16839504C2213yes—
18#16839504C2216yes—
19#16839502C2211yescharacteristic
20#16839502C2213yes—
21Q82×C61C411yescharacteristic
22C61×#556—23no—
23C61×#556C413yes—
24C61×#555C413yes—
25D4×Q8×C61C411yescharacteristic
26C61×#548—21no—
27C61×#542—23no—
28C61×#533C2213yes—
29C61×#529C2213yes—
30C61×#519—21no—
31C4×#41×C61C2211yescharacteristic
32#14519784—28no—
33C2×#11758194—23no—
34C2×#11758194—23no—
35C22×#92418—23no—
36C4⋊#92418—23no—
37C22⋊#92418—23no—
38#14519688—23no—
39#14519688C2311yescharacteristic
40#14519688C2313yes—
41#14519688C2311yescharacteristic
42#14519688C2313yes—
43C4×#92418—23no—
44C4×#92418—23no—
45C2×#41×C61D411yescharacteristic
46C22×Q8×C61—21no—
47C4⋊Q8×C61—23no—
48C4⋊Q8×C61C2×C413yes—
49C4⋊Q8×C61C2×C413yes—
50#126×C61—26no—
51#126×C61—23no—
52#125×C61C2313yes—
53#124×C61—23no—
54#124×C61—26no—
55#124×C61C2311yescharacteristic
56#123×C61—23no—
57C22⋊Q8×C61—23no—
58C22⋊Q8×C61—23no—
59C22⋊Q8×C61—23no—
60C22⋊Q8×C61C2×C413yes—
61#121×C61—23no—
62#121×C61C2×C413yes—
63C4×Q8×C61—21no—
64C4×Q8×C61—23no—
65C4×Q8×C61—23no—
66C4×Q8×C61—23no—
67C4×Q8×C61C2×C413yes—
68C4×Q8×C61C2×C411yescharacteristic
69C4×Q8×C61D411yescharacteristic
70C4×D4×C61—23no—
71C4×D4×C61C2×C413yes—
72C4×D4×C61C2×C413yes—
73C4×D4×C61D413yes—
74#117×C61—23no—
75#117×C61—23no—
76#117×C61C2×C416yes—
77#117×C61D413yes—
78C2×C42×C61C2313yes—
79#41×C61—24no—
80C2×Q8×C61—23no—
81C2×Q8×C61—23no—
82C2×Q8×C61—44no—
83C2×Q8×C61C22⋊C411yescharacteristic
84C2×Q8×C61C22×C411yescharacteristic
85C2×D4×C61—23no—
86C2×D4×C61C22⋊C413yes—
87C2×D4×C61C22×C411yescharacteristic
88C22×C4×C61—23no—
89C22×C4×C61C22×C413yes—
90C22×C4×C61C2×D413yes—
91C22×C4×C61C2×D413yes—
92C4⋊C4×C61—23no—
93C4⋊C4×C61—23no—
94C4⋊C4×C61—23no—
95C4⋊C4×C61—23no—
96C4⋊C4×C61—23no—
97C4⋊C4×C61—23no—
98C4⋊C4×C61C22⋊C413yes—
99C4⋊C4×C61C22⋊C413yes—
100C4⋊C4×C61C22×C413yes—
101C4⋊C4×C61C22×C413yes—
102C22⋊C4×C61—26no—
103C22⋊C4×C61—23no—
104C22⋊C4×C61—23no—
105C22⋊C4×C61—23no—
106C22⋊C4×C61—21no—
107C22⋊C4×C61—21no—
108C22⋊C4×C61C22⋊C416yes—
109C42×C61—26no—
110C42×C61—23no—
111C42×C61C22×C413yes—
112C42×C61C2×D413yes—
113C42×C61C2×D413yes—
114C42×C61C2411yescharacteristic
115#11758194—212no—
116#11758194—46no—
117C2×#92418—212no—
118C2×#92418—21no—
119C2×#92418—23no—
120C2×#92418—21no—
121C2×#92418—23no—
122C23×C61—21no—
123C23×C61C2×C22⋊C413yes—
124Q8×C61—41no—
125Q8×C61—46no—
126Q8×C61—83no—
127Q8×C61#10414yes—
128D4×C61—26no—
129D4×C61—41no—
130D4×C61—83no—
131C2×C4×C61—26no—
132C2×C4×C61—21no—
133C2×C4×C61—23no—
134C2×C4×C61—23no—
135C2×C4×C61—43no—
136C2×C4×C61—43no—
137C2×C4×C61—46no—
138C2×C4×C61C2×C22⋊C413yes—
139C2×C4×C61C2×C22⋊C413yes—
140C2×C4×C61C2×C22⋊C413yes—
141C2×C4×C61C23×C411yescharacteristic
142C2×C4×C61C22×D411yescharacteristic
143C2×C4×C61C22×D411yescharacteristic
144#92418—22no—
145#92418—26no—
146#92418—22no—
147#92418—43no—
148#17654—611nomaximal, Sylow
149C22×C61—23no—
150C22×C61—23no—
151C22×C61—23no—
152C22×C61—41no—
153C22×C61—41no—
154C22×C61—41no—
155C22×C61C22×C22⋊C411yescharacteristic
156C4×C61—21no—
157C4×C61—23no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—23no—
161C4×C61—41no—
162C4×C61—43no—
163C4×C61—43no—
164C4×C61C2×#10416yes—
165#3518D6111yescharacteristic
166#2971—611no—
167#2875—611no—
168#1531—613no—
169#1529—613no—
170#1526—613no—
171#1524—613no—
172C2×C61—43no—
173C2×C61—81no—
174C2×C61#288911yescharacteristic
175C2×C61C22×#10411yescharacteristic
176C2×C61#290111yescharacteristic
177C2×#131—1221no—
178Q82Dic6111yescharacteristic
179#556—23no—
180#556Dic6113yes—
181#555Dic6113yes—
182D4×Q8Dic6111yescharacteristic
183#548—21no—
184#542—23no—
185#533C2×D6113yes—
186#529C2×D6113yes—
187#519—21no—
188C4×#41C2×D6111yescharacteristic
189C8×Q8—1221no—
190#436—1223no—
191#432—613no—
192#405—1223no—
193C4×M16—613no—
194C4×M16—613no—
195C2×#97—613no—
196#330—613no—
197#330—613no—
198#330—616no—
199#329—611no—
200#329—613no—
201C61#1765411yescharacteristic, complemented, Sylow
202C2×#41C22⋊D6111yescharacteristic
203C22×Q8—21no—
204#131—1228no—
205C2×M16—1223no—
206C2×M16—1223no—
207C22×C8—1223no—
208C4⋊Q8—23no—
209C4⋊Q8C2×Dic6113yes—
210C4⋊Q8C2×Dic6113yes—
211#126—26no—
212#126—23no—
213#125C22×D6113yes—
214#124—23no—
215#124—26no—
216#124C22×D6111yescharacteristic
217#123—23no—
218C22⋊Q8—23no—
219C22⋊Q8—23no—
220C22⋊Q8—23no—
221C22⋊Q8C2×Dic6113yes—
222#121—23no—
223#121C2×Dic6113yes—
224C4×Q8—21no—
225C4×Q8—23no—
226C4×Q8—23no—
227C4×Q8—23no—
228C4×Q8C2×Dic6113yes—
229C4×Q8C2×Dic6111yescharacteristic
230C4×Q8C22⋊D6111yescharacteristic
231C4×D4—23no—
232C4×D4C2×Dic6113yes—
233C4×D4C2×Dic6113yes—
234C4×D4C22⋊D6113yes—
235#117—23no—
236#117—23no—
237#117C2×Dic6116yes—
238#117C22⋊D6113yes—
239C2×C42C22×D6113yes—
240C4⋊C8—1223no—
241C22⋊C8—1223no—
242#97—611no—
243#97—613no—
244#97—611no—
245#97—613no—
246#97—1223no—
247C4×C8—1223no—
248C4×C8—1223no—
249#41—24no—
250C2×Q8—23no—
251C2×Q8—23no—
252C2×Q8—44no—
253C2×Q8C22⋊Dic6111yescharacteristic
254C2×Q8C22×Dic6111yescharacteristic
255C2×D4—23no—
256C2×D4C22⋊Dic6113yes—
257C2×D4C22×Dic6111yescharacteristic
258C22×C4—23no—
259C22×C4C22×Dic6113yes—
260C22×C4C2×C22⋊D6113yes—
261C22×C4C2×C22⋊D6113yes—
262M16—12212no—
263M16—2446no—
264C2×C8—12212no—
265C2×C8—1221no—
266C2×C8—1223no—
267C2×C8—1221no—
268C2×C8—1223no—
269C4⋊C4—23no—
270C4⋊C4—23no—
271C4⋊C4—23no—
272C4⋊C4—23no—
273C4⋊C4—23no—
274C4⋊C4—23no—
275C4⋊C4C22⋊Dic6113yes—
276C4⋊C4C22⋊Dic6113yes—
277C4⋊C4C22×Dic6113yes—
278C4⋊C4C22×Dic6113yes—
279C22⋊C4—26no—
280C22⋊C4—23no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—23no—
284C22⋊C4—23no—
285C22⋊C4C22⋊Dic6116yes—
286C42—26no—
287C42—23no—
288C42C22×Dic6113yes—
289C42C2×C22⋊D6113yes—
290C42C2×C22⋊D6113yes—
291C42C23×D6111yescharacteristic
292C23—21no—
293C23C2×C22⋊Dic6113yes—
294Q8—41no—
295Q8—46no—
296Q8—83no—
297Q8#1451970414yes—
298D4—26no—
299D4—41no—
300D4—83no—
301C2×C4—26no—
302C2×C4—21no—
303C2×C4—23no—
304C2×C4—23no—
305C2×C4—43no—
306C2×C4—43no—
307C2×C4—46no—
308C2×C4C2×C22⋊Dic6113yes—
309C2×C4C2×C22⋊Dic6113yes—
310C2×C4C2×C22⋊Dic6113yes—
311C2×C4C23×Dic6111yescharacteristic
312C2×C4C22×C22⋊D6111yescharacteristic
313C2×C4C22×C22⋊D6111yescharacteristic
314C8—1222no—
315C8—1226no—
316C8—1222no—
317C8—2443no—
318C22—23no—
319C22—23no—
320C22—23no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22C22×C22⋊Dic6111yescharacteristic
325C4—21no—
326C4—21no—
327C4—23no—
328C4—21no—
329C4—23no—
330C4—43no—
331C4—43no—
332C4—41no—
333C4C2×#1451970416yes—
334C2—43no—
335C2—81no—
336C2#2145430211yescharacteristic
337C2C22×#1451970411yescharacteristic
338C2#2145436611yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
Fitting, prime radicalC61×#3518C2
socleC22×C61C22×C22⋊C4
cosocleC42×C61C24
hypercenter#3518D61
hypocenterC61#17654
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3518

Complements

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

NH ≅ G/Ntype
1. #21454839C2semidirect
2. #21454247C2semidirect
3. #21444434C2semidirect
4. #21444426C2semidirect
5. #21444414C2semidirect
6. #21444407C2semidirect
201. C61#17654semidirect

Maximal subgroups

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

subgroupindex
#214548392
#214542472
#214444342
#214444262
#214444142
#214444072
C61×#35182
#1765461