Finite Group Databasebeta

← #21459895

Subgroups — #21459895

Order 7808.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16840494C213yesmaximal, complemented
2Q8×Dic122C211yesmaximal, characteristic, complemented
3#16840470C213yesmaximal, complemented
4#16840431C211yesmaximal, characteristic, complemented
5#16840423C211yesmaximal, characteristic, complemented
6#16840416C213yesmaximal, complemented
7#16840412C213yesmaximal, complemented
8#16840385C213yesmaximal, complemented
9#16840381C213yesmaximal, complemented
10#16840307C213yesmaximal, complemented
11#16840292C213yesmaximal, complemented
12#16840245C211yesmaximal, characteristic, complemented
13#16840244C211yesmaximal, characteristic, complemented
14C2×Q8×Dic61C211yesmaximal, characteristic, complemented
15C61×#548C211yesmaximal, characteristic, complemented
16C2×#11758226C2211yescharacteristic
17C2×#11758224C2216yes—
18#14519777C2213yes—
19#14519776C2213yes—
20#14519776C2213yes—
21#14519773C2213yes—
22#14519772C2213yes—
23#14519770C2213yes—
24#14519767C2213yes—
25#14519766C2213yes—
26#14519763C2213yes—
27#14519763C2216yes—
28#14519761C2216yes—
29#14519761C2213yes—
30#14519759C2216yes—
31#14519758C2213yes—
32#14519757C2216yes—
33#14519754C2216yes—
34#14519754C2211yescharacteristic
35#14519753C2213yes—
36#14519751C2213yes—
37#14519751C2213yes—
38#14519749C2213yes—
39#14519748C2213yes—
40Q8×Dic61C2213yes—
41Q8×Dic61C2211yescharacteristic
42Q8×Dic61C2211yescharacteristic
43Q8×Dic61C2213yes—
44Q8×Dic61C2211yescharacteristic
45C4×Dic122C2213yes—
46C4×D244C2213yes—
47D4×Dic61C2213yes—
48D4×Dic61C2213yes—
49#14519739C2213yes—
50C4×C22⋊D61C2213yes—
51C4×C22⋊D61C2213yes—
52#14519737C2216yes—
53#14519736C2213yes—
54#14519734C2213yes—
55C2×C4⋊Dic61C2213yes—
56C2×C4⋊Dic61C2213yes—
57C4⋊C4×D61C2213yes—
58C2×C4×Dic61C2213yes—
59C2×#41×C61C2211yescharacteristic
60C22×Q8×C61C2211yescharacteristic
61#124×C61C2213yes—
62C22⋊Q8×C61C2213yes—
63#121×C61C2213yes—
64C4×Q8×C61C2211yescharacteristic
65C4×D4×C61C2213yes—
66#41×C61—24no—
67C2×Q8×C61—24no—
68C2×Q8×C61C2313yes—
69C2×Q8×C61C2311yescharacteristic
70C2×D4×C61C2313yes—
71C2×D4×C61C2313yes—
72C22×C4×C61C2313yes—
73C22×C4×C61C2313yes—
74C2×C22⋊D61C2313yes—
75C2×C22⋊D61C2313yes—
76C2×C22⋊D61C2316yes—
77C22×Dic61C2313yes—
78C22×Dic61C2311yescharacteristic
79C22×Dic61C2316yes—
80#11758226—24no—
81#11758224—412no—
82C2×D244C2313yes—
83C2×C4×D61C2313yes—
84C2×C4×D61C2316yes—
85C2×Dic122C2311yescharacteristic
86C2×Dic122C2316yes—
87C4⋊C4×C61C2313yes—
88C4⋊C4×C61C2313yes—
89C22⋊C4×C61C2311yescharacteristic
90C22⋊C4×C61C2313yes—
91C22⋊C4×C61C2316yes—
92C42×C61C2313yes—
93C22⋊Dic61C2313yes—
94C22⋊Dic61C2313yes—
95C22⋊Dic61C2311yescharacteristic
96C22⋊Dic61C2313yes—
97#11758198C2313yes—
98#11758198C2313yes—
99#11758198C2316yes—
100#11758198C2313yes—
101#11758198C2313yes—
102C4⋊Dic61C2313yes—
103C4⋊Dic61C2313yes—
104C4⋊Dic61C2313yes—
105C4⋊Dic61C2313yes—
106C4⋊Dic61C2316yes—
107C4⋊Dic61C2311yescharacteristic
108C4⋊Dic61C2313yes—
109#11758196C2313yes—
110#11758196C2311yescharacteristic
111#11758196C2313yes—
112#11758196C2311yescharacteristic
113#11758196C2313yes—
114#11758196C2313yes—
115C4×Dic61C2313yes—
116C4×Dic61C2313yes—
117C4×Dic61C2313yes—
118C4×Dic61C2313yes—
119C4×Dic61C2313yes—
120C23×C61C2413yes—
121C23×C61C2411yescharacteristic
122C22×D61C2413yes—
123Q8×C61—26no—
124Q8×C61#4114yescomplemented
125D4×C61—26no—
126D4×C61—26no—
127C2×C4×C61—26no—
128C2×C4×C61—26no—
129C2×C4×C61C2413yes—
130C2×C4×C61C2413yes—
131C2×C4×C61C2411yescharacteristic
132C2×C4×C61C2411yescharacteristic
133C2×C4×C61C2413yes—
134C22⋊D61—26no—
135C22⋊D61—26no—
136C22⋊D61—46no—
137C2×Dic61—26no—
138C2×Dic61—22no—
139C2×Dic61—46no—
140C2×Dic61C2411yescharacteristic
141C2×Dic61C2411yescharacteristic
142C2×Dic61C2413yes—
143C2×Dic61C2413yes—
144C2×Dic61C2413yes—
145C2×Dic61C2411yescharacteristic
146C2×Dic61C2411yescharacteristic
147D244—26no—
148C4×D61—26no—
149C4×D61—46no—
150Dic122—22no—
151Dic122—46no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—23no—
155C22×C61—23no—
156C22×C61—23no—
157C22×C61H(5,𝔽2)12yes—
158C22×C61C2511yescharacteristic
159C2×D61—23no—
160C2×D61—23no—
161C2×D61—23no—
162C4×C61—21no—
163C4×C61—23no—
164C4×C61—23no—
165C4×C61—21no—
166C4×C61C2×#4116yes—
167Dic61—21no—
168Dic61—23no—
169Dic61—23no—
170Dic61—23no—
171Dic61—21no—
172Dic61—21no—
173Dic612−1+412yes—
174#3522—611nomaximal, Sylow
175C2×C61—22no—
176C2×C61—43no—
177C2×C61C22×#4111yescharacteristic
178C2×C61C2×H(5,𝔽2)11yescharacteristic
179C2×C61C2×2−1+411yescharacteristic
180D61—43no—
181#559—613no—
182Q82—611no—
183#554—613no—
184#550—611no—
185#548—611no—
186#548D6111yescharacteristic, complemented
187#547—613no—
188#547—613no—
189#542—613no—
190#542—613no—
191#532—613no—
192#529—613no—
193#520—611no—
194#520—611no—
195C2×C4×Q8—611no—
196C61#352211yescharacteristic, complemented, Sylow
197C2×#41—611no—
198C2×#41—616no—
199C2×#41C2×D6111yescharacteristic
200C22×Q8C2×D6111yescharacteristic
201C4⋊Q8—613no—
202C4⋊Q8—613no—
203C4⋊Q8—613no—
204#127—613no—
205#126—613no—
206#126—613no—
207#125—613no—
208#124—613no—
209#124—613no—
210#124—616no—
211#124C2×D6113yes—
212#123—613no—
213#123—613no—
214#123—616no—
215#123—616no—
216C22⋊Q8—613no—
217C22⋊Q8—616no—
218C22⋊Q8—611no—
219C22⋊Q8—616no—
220C22⋊Q8C2×D6113yes—
221#121—613no—
222#121—613no—
223#121—613no—
224#121—613no—
225#121C2×D6113yes—
226C4×Q8—611no—
227C4×Q8—613no—
228C4×Q8—611no—
229C4×Q8—613no—
230C4×Q8—611no—
231C4×Q8—613no—
232C4×Q8C2×D6111yescharacteristic
233C4×D4—613no—
234C4×D4—613no—
235C4×D4—613no—
236C4×D4—613no—
237C4×D4—613no—
238C4×D4—616no—
239C4×D4—613no—
240C4×D4C2×D6113yes—
241#117—613no—
242#117—613no—
243C2×C4⋊C4—613no—
244C2×C4⋊C4—613no—
245C2×C4⋊C4—613no—
246C2×C42—613no—
247#41—24no—
248#41—1224no—
249#41—24412no—
250C2×Q8—24no—
251C2×Q8—611no—
252C2×Q8—616no—
253C2×Q8C22×D6113yes—
254C2×Q8C22×D6111yescharacteristic
255C2×D4—613no—
256C2×D4—613no—
257C2×D4—616no—
258C2×D4—613no—
259C2×D4C22×D6113yes—
260C2×D4C22×D6113yes—
261C22×C4—611no—
262C22×C4—613no—
263C22×C4—616no—
264C22×C4—613no—
265C22×C4—616no—
266C22×C4C22×D6113yes—
267C22×C4C22×D6113yes—
268C4⋊C4—613no—
269C4⋊C4—613no—
270C4⋊C4—613no—
271C4⋊C4—613no—
272C4⋊C4—613no—
273C4⋊C4—616no—
274C4⋊C4—611no—
275C4⋊C4—611no—
276C4⋊C4—613no—
277C4⋊C4—613no—
278C4⋊C4—613no—
279C4⋊C4—611no—
280C4⋊C4—613no—
281C4⋊C4C22×D6113yes—
282C4⋊C4C22×D6113yes—
283C22⋊C4—613no—
284C22⋊C4—613no—
285C22⋊C4—611no—
286C22⋊C4—613no—
287C22⋊C4—613no—
288C22⋊C4—613no—
289C22⋊C4—613no—
290C22⋊C4—616no—
291C22⋊C4—613no—
292C22⋊C4C22×D6116yes—
293C22⋊C4C22×D6111yescharacteristic
294C22⋊C4C22×D6113yes—
295C42—613no—
296C42—613no—
297C42—613no—
298C42—613no—
299C42—613no—
300C42C22×D6113yes—
301C23—613no—
302C23C23×D6113yes—
303C23C23×D6111yescharacteristic
304Q8—26no—
305Q8—1222no—
306Q8—2446no—
307Q8#1175822414yescomplemented
308D4—26no—
309D4—26no—
310D4—1226no—
311D4—1226no—
312D4—1226no—
313D4—2446no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—611no—
317C2×C4—611no—
318C2×C4—611no—
319C2×C4—613no—
320C2×C4—613no—
321C2×C4—613no—
322C2×C4—611no—
323C2×C4—1222no—
324C2×C4—1226no—
325C2×C4—1226no—
326C2×C4—2446no—
327C2×C4—2446no—
328C2×C4C23×D6113yes—
329C2×C4C23×D6111yescharacteristic
330C2×C4C23×D6113yes—
331C2×C4C23×D6111yescharacteristic
332C2×C4C23×D6113yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—1223no—
339C22—1223no—
340C22—1223no—
341C22#1451982312yes—
342C22C24×D6111yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—612no—
348C4—1221no—
349C4—1223no—
350C4—1223no—
351C4—1223no—
352C4—1221no—
353C4—1221no—
354C4C2×#1175822416yes—
355C2—22no—
356C2—43no—
357C2—2443no—
358C2C22×#1175822411yescharacteristic
359C2C2×#1451982311yescharacteristic
360C22−1+4×D6111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D61
commutator, socleC22×C61C25
FittingC61×#548C2
hypercenter#548D61
hypocenterC61#3522
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16840494C2semidirect
2. Q8×Dic122C2semidirect
3. #16840470C2semidirect
4. #16840431C2semidirect
5. #16840423C2semidirect
6. #16840416C2semidirect
7. #16840412C2semidirect
8. #16840385C2semidirect
9. #16840381C2semidirect
10. #16840307C2semidirect
11. #16840292C2semidirect
12. #16840245C2semidirect
13. #16840244C2semidirect
14. C2×Q8×Dic61C2semidirect
15. C61×#548C2semidirect
124. Q8×C61#41semidirect
186. #548D61semidirect
196. C61#3522semidirect
307. Q8#11758224semidirect