Finite Group Databasebeta

← #46839288

Subgroups — #46839288

Order 34688.

660 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576122C211yesmaximal, characteristic
2C2×#22576117C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D271C211yesmaximal, characteristic, complemented
4#31773789C211yesmaximal, characteristic, complemented
5#31773787C211yesmaximal, characteristic, complemented
6#31773783C211yesmaximal, characteristic, complemented
7#31773775C211yesmaximal, characteristic
8#31773775C211yesmaximal, characteristic
9#31773761C211yesmaximal, characteristic, complemented
10#31773756C211yesmaximal, characteristic, complemented
11#31773756C211yesmaximal, characteristic, complemented
12#31773748C211yesmaximal, characteristic, complemented
13C4×(C2×D271)⋊C4C211yesmaximal, characteristic, complemented
14#31773744C211yesmaximal, characteristic, complemented
15#387×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C22×C4×D271C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21#22576122C418yescomplemented
22#22576117—24no—
23C2×C4⋊Dic271C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C4⋊C4×D271—24no—
29C2×(C2×D271)⋊C4C2211yescharacteristic
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×C4×Dic271C2211yescharacteristic
38C2×C4×Dic271C2211yescharacteristic
39C2×C4×Dic271C2211yescharacteristic
40C2×C4×Dic271C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C2×C4⋊C4×C271C2211yescharacteristic
48C2×C4⋊C4×C271C2211yescharacteristic
49C2×C42×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52#95×C271C2211yescharacteristic
53#95×C271C2211yescharacteristic
54C23×D271C2311yescharacteristic
55C23×D271C2311yescharacteristic
56C2×D1084—22no—
57C2×D1084—22no—
58C2×D1084C2×C414yes—
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C22×Dic271C2311yescharacteristic
64C22×Dic271C2311yescharacteristic
65C2×C4×D271—22no—
66C2×C4×D271—22no—
67C2×C4×D271—22no—
68C2×C4×D271—22no—
69C2×C4×D271—22no—
70C2×C4×D271—22no—
71C2×C4×D271—22no—
72C2×C4×D271—22no—
73C2×C4×D271—41no—
74C2×C4×D271—41no—
75C2×C4×D271C2×C414yes—
76C2×C4×D271C2×C414yescomplemented
77C4⋊Dic271—22no—
78Dic271⋊C4—22no—
79Dic271⋊C4—22no—
80Dic271⋊C4—22no—
81Dic271⋊C4D414yescomplemented
82(C2×D271)⋊C4—22no—
83(C2×D271)⋊C4—22no—
84(C2×D271)⋊C4—22no—
85(C2×D271)⋊C4—22no—
86(C2×D271)⋊C4—41no—
87(C2×D271)⋊C4—41no—
88(C2×D271)⋊C4C2×C414yes—
89(C2×D271)⋊C4C2×C414yescomplemented
90C4×Dic271—22no—
91C4×Dic271—41no—
92C4×Dic271—41no—
93C4×Dic271C2×C414yescomplemented
94C22×C4×C271C2311yescharacteristic
95C22×C4×C271C2311yescharacteristic
96C22×C4×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C4⋊C4×C271—22no—
102C4⋊C4×C271C2×C414yescomplemented
103C42×C271—22no—
104C22×D271—21no—
105C22×D271—21no—
106C22×D271—21no—
107C22×D271—21no—
108C22×D271—21no—
109C22×D271—21no—
110C22×D271—21no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—21no—
114C22×D271C22×C412yes—
115C22×D271C22×C412yes—
116C22×D271#4112yes—
117C22×D271#4112yes—
118D1084—28no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271C22×C412yes—
133C2×Dic271C22×C412yes—
134C2×Dic271C2×D412yes—
135C2×Dic271C2×D412yes—
136C2×Dic271#4112yes—
137C4×D271—28no—
138C4×D271—28no—
139C4×D271—44no—
140C4×D271—44no—
141C23×C271C2411yescharacteristic
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271—21no—
154C2×C4×C271—21no—
155C2×C4×C271—21no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271C22×C412yes—
160C2×C4×C271C22×C412yes—
161C2×C4×C271C22×C412yes—
162C2×C4×C271C2×D412yes—
163C22×C271C23×C411yescharacteristic
164C22×C271C22×D411yescharacteristic
165C22×C271C2×#4111yescharacteristic
166C22×C271C2×#4111yescharacteristic
167C22×C271C2×#4111yescharacteristic
168C22×C271H(5,𝔽2)11yescharacteristic
169C22×C2712−1+411yescharacteristic
170C2×D271—22no—
171C2×D271—22no—
172C2×D271—22no—
173C2×D271—22no—
174C2×D271—22no—
175C2×D271—22no—
176C2×D271—22no—
177C2×D271—22no—
178C2×D271—22no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271#11714yescomplemented
184C4×C271—22no—
185C4×C271—22no—
186C4×C271—22no—
187C4×C271—22no—
188C4×C271—41no—
189C4×C271—41no—
190C4×C271—41no—
191Dic271—22no—
192Dic271—22no—
193Dic271—41no—
194Dic271—41no—
195Dic271—41no—
196Dic271C4×D414yes—
197C2×C271C2×#11711yescharacteristic
198C2×C271C2×C4×D411yescharacteristic
199C2×C271#52011yescharacteristic
200C2×C271#54611yescharacteristic
201C2×C271#54811yescharacteristic
202C2×C271#55211yescharacteristic
203C2×C271#55311yescharacteristic
204D271—24no—
205D271—42no—
206C271#235211yescharacteristic, complemented, Sylow
207#2352—2711nomaximal, Sylow
208C2×C4×D4—2711no—
209C2×#117—2711no—
210C22×C4⋊C4—2711no—
211#388—2711no—
212#388—2711no—
213#387—2711no—
214#387D27111yescharacteristic, complemented
215#386—2711no—
216#386—2711no—
217#382—2711no—
218#380—2711no—
219#380—2711no—
220C4×C4⋊C4—2711no—
221C4×C22⋊C4—2711no—
222C4×C22⋊C4—2711no—
223C22×D4—2711no—
224C23×C4—2711no—
225C23×C4—2711no—
226C23×C4—2711no—
227C23×C4—2711no—
228C4×D4—2718no—
229#117—5424no—
230C2×C4⋊C4—2711no—
231C2×C4⋊C4—2711no—
232C2×C4⋊C4—2711no—
233C2×C4⋊C4—2711no—
234C2×C4⋊C4—2711no—
235C2×C4⋊C4—5424no—
236C2×C4⋊C4C2×D27111yescharacteristic
237C2×C4⋊C4C2×D27111yescharacteristic
238C2×C22⋊C4—2711no—
239C2×C22⋊C4—2711no—
240C2×C22⋊C4—2711no—
241C2×C22⋊C4—2711no—
242C2×C22⋊C4—2711no—
243C2×C22⋊C4—2711no—
244C2×C22⋊C4—2711no—
245C2×C22⋊C4—2711no—
246C2×C42—2711no—
247C2×C42—2711no—
248C2×C42—2711no—
249C2×C42—2711no—
250C2×C42C2×D27111yescharacteristic
251#95—2711no—
252#95—2711no—
253#95—2711no—
254#95—2711no—
255#95—2711no—
256#95—2711no—
257#95C2×D27111yescharacteristic
258#95C2×D27111yescharacteristic
259#95C2×D27111yescharacteristic
260#95C2×D27111yescharacteristic
261C24—2711no—
262C24—2711no—
263C2×D4—2714no—
264C2×D4—5422no—
265C2×D4—5422no—
266C22×C4—2711no—
267C22×C4—2711no—
268C22×C4—2711no—
269C22×C4—2714no—
270C22×C4—2714no—
271C22×C4—2711no—
272C22×C4—2711no—
273C22×C4—2711no—
274C22×C4—5422no—
275C22×C4—5422no—
276C22×C4—5422no—
277C22×C4—5422no—
278C22×C4—5422no—
279C22×C4—5422no—
280C22×C4—5422no—
281C22×C4—5422no—
282C22×C4—10841no—
283C22×C4—10841no—
284C22×C4C22×D27111yescharacteristic
285C22×C4C22×D27111yescharacteristic
286C22×C4C22×D27111yescharacteristic
287C22×C4C22×D27111yescharacteristic
288C22×C4C22×D27111yescharacteristic
289C22×C4C22×D27111yescharacteristic
290C22×C4C22×D27111yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—2714no—
293C4⋊C4—5422no—
294C4⋊C4—5422no—
295C4⋊C4—5422no—
296C4⋊C4—5422no—
297C4⋊C4C4×D27114yescomplemented
298C22⋊C4—2714no—
299C22⋊C4—2714no—
300C22⋊C4—5422no—
301C22⋊C4—5422no—
302C22⋊C4—5422no—
303C22⋊C4—5422no—
304C22⋊C4—10841no—
305C22⋊C4—10841no—
306C42—22no—
307C42—2714no—
308C42—5422no—
309C42—10841no—
310C42—10841no—
311C23—2712no—
312C23—2712no—
313C23—2712no—
314C23—2712no—
315C23—5421no—
316C23—5421no—
317C23—5421no—
318C23—5421no—
319C23—5421no—
320C23—5421no—
321C23—5421no—
322C23—5421no—
323C23—5421no—
324C23—5421no—
325C23C23×D27111yescharacteristic
326D4—5428no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2712no—
345C2×C4—2712no—
346C2×C4—2712no—
347C2×C4—2712no—
348C2×C4—2712no—
349C2×C4—5428no—
350C2×C4—5428no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—5421no—
356C2×C4—5421no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—5421no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—5421no—
364C2×C4—10844no—
365C2×C4—10844no—
366C2×C4C2×C4×D27112yes—
367C2×C4C2×C4×D27112yes—
368C2×C4C2×C4×D27112yes—
369C2×C4D4×D27112yes—
370C22—2714no—
371C22—5422no—
372C22—5422no—
373C22—5422no—
374C22—5422no—
375C22—5422no—
376C22—5422no—
377C22—5422no—
378C22—5422no—
379C22—5422no—
380C22—10841no—
381C22—10841no—
382C22—10841no—
383C22—10841no—
384C22C22×C4×D27111yescharacteristic
385C22C2×D4×D27111yescharacteristic
386C22#41×D27111yescharacteristic
387C22#41×D27111yescharacteristic
388C22#41×D27111yescharacteristic
389C22#2257620611yescharacteristic
390C22#2257620811yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—2714no—
399C4—5422no—
400C4—5422no—
401C4—10841no—
402C4—10841no—
403C4—10841no—
404C2—5424no—
405C2—10842no—
406C2#117×D27111yescharacteristic
407C2C4×D4×D27111yescharacteristic
408C2#3177437911yescharacteristic
409C2#3177453911yescharacteristic
410C2#3177455711yescharacteristic
411C2#3177457911yescharacteristic
412C2#3177459811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#387×C271C2
socle, cosocleC23×C271C24
prime radicalC22×D1084C22
hypercenter#387D271
hypocenterC271#2352
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×#22576117C2semidirect
3. C2×C4⋊C4×D271C2semidirect
4. #31773789C2semidirect
5. #31773787C2semidirect
6. #31773783C2semidirect
9. #31773761C2semidirect
10. #31773756C2semidirect
11. #31773756C2semidirect
12. #31773748C2semidirect
13. C4×(C2×D271)⋊C4C2semidirect
14. #31773744C2semidirect
15. #387×C271C2semidirect
21. #22576122C4semidirect
76. C2×C4×D271C2×C4semidirect
81. Dic271⋊C4D4semidirect
89. (C2×D271)⋊C4C2×C4semidirect
93. C4×Dic271C2×C4semidirect
102. C4⋊C4×C271C2×C4semidirect
183. C2×D271#117semidirect
206. C271#2352semidirect
214. #387D271semidirect
297. C4⋊C4C4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761222
C2×#225761172
C2×C4⋊C4×D2712
#317737892
#317737872
#317737832
#317737752
#317737612
#317737562
#317737482
C4×(C2×D271)⋊C42
#317737442
#387×C2712
#2352271