Finite Group Databasebeta

← #46850474

Subgroups — #46850474

Order 34688.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774629C213yesmaximal, complemented
2#31774626C211yesmaximal, characteristic, complemented
3#31774612C213yesmaximal, complemented
4#31774594C213yesmaximal, complemented
5#31774581C213yesmaximal, complemented
6#31774566C211yesmaximal, characteristic, complemented
7#31774558C211yesmaximal, characteristic, complemented
8#31774516C213yesmaximal, complemented
9#31774500C213yesmaximal, complemented
10#31774433C213yesmaximal, complemented
11#31774427C213yesmaximal, complemented
12#31774379C211yesmaximal, characteristic, complemented
13#31774375C211yesmaximal, characteristic, complemented
14#41×Dic271C211yesmaximal, characteristic, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17#22576160C2213yes—
18#22576156C2213yes—
19#22576155C2213yes—
20#22576153C2216yes—
21#22576153C2211yescharacteristic
22#22576151C2213yes—
23#22576151C2213yes—
24#22576150C2213yes—
25#22576149C2213yes—
26#22576146C2216yes—
27#22576146C2213yes—
28#22576144C2216yes—
29#22576144C2213yes—
30#22576142C2216yes—
31#22576141C2213yes—
32#22576137C2213yes—
33#22576136C2213yes—
34#22576134C2213yes—
35#22576134C2213yes—
36#22576132C2213yes—
37#22576131C2216yes—
38#22576131C2213yes—
39Q8×Dic271C2213yes—
40Q8×Dic271C2211yescharacteristic
41Q8×Dic271C2211yescharacteristic
42Q8×Dic271C2213yes—
43Q8×Dic271C2211yescharacteristic
44C4×Dic542C2213yes—
45C4×D1084C2213yes—
46D4×Dic271C2213yes—
47D4×Dic271C2213yes—
48#22576122C2213yes—
49C4×C22⋊D271C2213yes—
50C4×C22⋊D271C2213yes—
51(C22⋊D271)⋊C4C2216yes—
52#22576119C2213yes—
53#22576119C2213yes—
54#22576119C2213yes—
55#22576118C2216yes—
56#22576117C2213yes—
57C4⋊C4×D271C2213yes—
58C2×C4×Dic271C2213yes—
59C2×#41×C271C2211yescharacteristic
60C22×Q8×C271C2211yescharacteristic
61#124×C271C2213yes—
62C22⋊Q8×C271C2213yes—
63#121×C271C2213yes—
64C4×Q8×C271C2211yescharacteristic
65C4×D4×C271C2213yes—
66#16864655—24no—
67C2×Dic542C2313yes—
68C2×D1084C2313yes—
69C2×C22⋊D271C2313yes—
70C2×C22⋊D271C2316yes—
71C2×C22⋊D271C2313yes—
72C22×Dic271C2313yes—
73C22×Dic271C2313yes—
74C2×C4×D271C2313yes—
75C2×C4×D271C2316yes—
76C4⋊Dic271C2313yes—
77C4⋊Dic271C2316yes—
78C4⋊Dic271C2313yes—
79C4⋊Dic271C2313yes—
80C4⋊Dic271C2313yes—
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2313yes—
83Dic271⋊C4C2313yes—
84Dic271⋊C4C2313yes—
85Dic271⋊C4C2311yescharacteristic
86Dic271⋊C4C2313yes—
87Dic271⋊C4C2313yes—
88Dic271⋊C4C2311yescharacteristic
89C22⋊Dic271C2313yes—
90C22⋊Dic271C2313yes—
91C22⋊Dic271C2316yes—
92C22⋊Dic271C2311yescharacteristic
93C22⋊Dic271C2311yescharacteristic
94(C2×D271)⋊C4C2313yes—
95(C2×D271)⋊C4C2313yes—
96(C2×D271)⋊C4C2316yes—
97(C2×D271)⋊C4C2313yes—
98(C2×D271)⋊C4C2313yes—
99C4×Dic271C2313yes—
100C4×Dic271C2316yes—
101C4×Dic271C2313yes—
102C4×Dic271C2313yes—
103C4×Dic271C2311yescharacteristic
104C4×Dic271C2313yes—
105#41×C271—24no—
106C2×Q8×C271—24no—
107C2×Q8×C271C2313yes—
108C2×Q8×C271C2311yescharacteristic
109C2×D4×C271C2313yes—
110C2×D4×C271C2313yes—
111C22×C4×C271C2313yes—
112C22×C4×C271C2313yes—
113C4⋊C4×C271C2313yes—
114C4⋊C4×C271C2313yes—
115C22⋊C4×C271C2313yes—
116C22⋊C4×C271C2316yes—
117C22⋊C4×C271C2311yescharacteristic
118C42×C271C2313yes—
119C22×D271C2413yes—
120Dic542—43no—
121D1084—26no—
122C22⋊D271—43no—
123C22⋊D271—43no—
124C22⋊D271—46no—
125C2×Dic271—43no—
126C2×Dic271—43no—
127C2×Dic271C2411yescharacteristic
128C2×Dic271C2411yescharacteristic
129C2×Dic271C2413yes—
130C2×Dic271C2413yes—
131C2×Dic271C2411yescharacteristic
132C2×Dic271C2413yes—
133C2×Dic271C2411yescharacteristic
134C4×D271—26no—
135C4×D271—46no—
136C23×C271C2413yes—
137C23×C271C2411yescharacteristic
138Q8×C271—26no—
139Q8×C271#4114yes—
140D4×C271—26no—
141D4×C271—43no—
142C2×C4×C271—26no—
143C2×C4×C271—26no—
144C2×C4×C271C2411yescharacteristic
145C2×C4×C271C2413yes—
146C2×C4×C271C2413yes—
147C2×C4×C271C2411yescharacteristic
148C2×C4×C271C2413yes—
149C22×C271—21no—
150C22×C271—23no—
151C22×C271—21no—
152C22×C271—23no—
153C22×C271—23no—
154C22×C271—21no—
155C22×C271C2511yescharacteristic
156C2×D271—23no—
157C2×D271—23no—
158C2×D271—23no—
159C4×C271—21no—
160C4×C271—23no—
161C4×C271—23no—
162C4×C271—21no—
163C4×C271C2×#4116yes—
164Dic271—21no—
165Dic271—23no—
166Dic271—21no—
167Dic271—21no—
168Dic271—23no—
169Dic271—21no—
170Dic271—23no—
171C2×C271—41no—
172C2×C271—43no—
173C2×C271C22×#4111yescharacteristic
174C2×C271#58511yescharacteristic
175C2×C271#58511yescharacteristic
176D271—43no—
177C271#352411yescharacteristic, complemented, Sylow
178#3524—2711nomaximal, Sylow
179#559—2713no—
180#557—2711no—
181#555—2713no—
182#553—2713no—
183#552—2713no—
184#550—2711no—
185#548—2711no—
186#548D27111yescharacteristic, complemented
187#542—2713no—
188#540—2713no—
189#529—2713no—
190#529—2713no—
191#520—2711no—
192#519—2711no—
193C4×#41—2711no—
194C2×#41—2711no—
195C2×#41C2×D27111yescharacteristic
196C22×Q8C2×D27111yescharacteristic
197C4⋊Q8—2713no—
198#127—2713no—
199#126—2713no—
200#126—2711no—
201#126—2716no—
202#125—2713no—
203#125—2713no—
204#125—2713no—
205#124—2713no—
206#124—2713no—
207#124—2716no—
208#124C2×D27113yes—
209#123—2713no—
210#123—2716no—
211#123—2716no—
212#123—2713no—
213C22⋊Q8—2713no—
214C22⋊Q8—2713no—
215C22⋊Q8C2×D27113yes—
216#121—2716no—
217#121—2713no—
218#121—2713no—
219#121—2713no—
220#121—2713no—
221#121C2×D27113yes—
222C4×Q8—2711no—
223C4×Q8—2711no—
224C4×Q8—2713no—
225C4×Q8—2711no—
226C4×Q8—2713no—
227C4×Q8—2713no—
228C4×Q8C2×D27111yescharacteristic
229C4×D4—2713no—
230C4×D4—2713no—
231C4×D4—2713no—
232C4×D4—2713no—
233C4×D4—2713no—
234C4×D4—2716no—
235C4×D4—2713no—
236C4×D4C2×D27113yes—
237#117—2713no—
238#117—2713no—
239#117—2713no—
240#117—2716no—
241#117—2713no—
242C2×C4⋊C4—2713no—
243C2×C42—2713no—
244#41—24no—
245#41—5424no—
246C2×Q8—24no—
247C2×Q8—2713no—
248C2×Q8C22×D27113yes—
249C2×Q8C22×D27111yescharacteristic
250C2×D4—2713no—
251C2×D4—2716no—
252C2×D4—2713no—
253C2×D4—2713no—
254C2×D4C22×D27113yes—
255C2×D4C22×D27113yes—
256C22×C4—2713no—
257C22×C4—2713no—
258C22×C4—2713no—
259C22×C4—2716no—
260C22×C4C22×D27113yes—
261C22×C4C22×D27113yes—
262C4⋊C4—2713no—
263C4⋊C4—2713no—
264C4⋊C4—2711no—
265C4⋊C4—2713no—
266C4⋊C4—2713no—
267C4⋊C4—2713no—
268C4⋊C4—2713no—
269C4⋊C4—2716no—
270C4⋊C4—2711no—
271C4⋊C4—2713no—
272C4⋊C4—2713no—
273C4⋊C4—2711no—
274C4⋊C4—2713no—
275C4⋊C4C22×D27113yes—
276C4⋊C4C22×D27113yes—
277C22⋊C4—2711no—
278C22⋊C4—2713no—
279C22⋊C4—2716no—
280C22⋊C4—2711no—
281C22⋊C4—2713no—
282C22⋊C4—2713no—
283C22⋊C4—2713no—
284C22⋊C4—2713no—
285C22⋊C4—2716no—
286C22⋊C4—2713no—
287C22⋊C4C22×D27116yes—
288C22⋊C4C22×D27111yescharacteristic
289C22⋊C4C22×D27113yes—
290C42—2711no—
291C42—2713no—
292C42—2713no—
293C42—2713no—
294C42—2716no—
295C42—2713no—
296C42C22×D27113yes—
297C23—2713no—
298C23C23×D27111yescharacteristic
299C23C23×D27113yes—
300Q8—26no—
301Q8—10843no—
302Q8#1686465314yes—
303D4—26no—
304D4—43no—
305D4—5426no—
306D4—10843no—
307D4—10843no—
308D4—10846no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—2711no—
312C2×C4—2713no—
313C2×C4—2711no—
314C2×C4—2713no—
315C2×C4—2711no—
316C2×C4—2713no—
317C2×C4—2711no—
318C2×C4—5426no—
319C2×C4—10846no—
320C2×C4—10843no—
321C2×C4—10843no—
322C2×C4C23×D27111yescharacteristic
323C2×C4C23×D27113yes—
324C2×C4C23×D27111yescharacteristic
325C2×C4C23×D27113yes—
326C2×C4C23×D27113yes—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—23no—
332C22—21no—
333C22—5423no—
334C22—5423no—
335C22—5423no—
336C22C24×D27111yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—5421no—
342C4—5423no—
343C4—5421no—
344C4—5421no—
345C4—5421no—
346C4—5423no—
347C4—5423no—
348C4C2×#1686465316yes—
349C2—41no—
350C2—43no—
351C2—10843no—
352C2C22×#1686465311yescharacteristic
353C2#3177473311yescharacteristic
354C2#3177473411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#548×C271C2
hypercenter#548D271
hypocenterC271#3524
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31774629C2semidirect
2. #31774626C2semidirect
3. #31774612C2semidirect
4. #31774594C2semidirect
5. #31774581C2semidirect
6. #31774566C2semidirect
7. #31774558C2semidirect
8. #31774516C2semidirect
9. #31774500C2semidirect
10. #31774433C2semidirect
11. #31774427C2semidirect
12. #31774379C2semidirect
13. #31774375C2semidirect
14. #41×Dic271C2semidirect
15. #548×C271C2semidirect
177. C271#3524semidirect
186. #548D271semidirect

Maximal subgroups

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

subgroupindex
#317746292
#317746262
#317746122
#317745942
#317745812
#317745662
#317745582
#317745162
#317745002
#317744332
#317744272
#317743792
#317743752
#41×Dic2712
#548×C2712
#3524271