Finite Group Databasebeta

← #46839176

Subgroups — #46839176

Order 34688.

540 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576149C211yesmaximal, characteristic
2#31773804C211yesmaximal, characteristic, complemented
3#31773789C211yesmaximal, characteristic, complemented
4#31773788C211yesmaximal, characteristic, complemented
5#31773787C211yesmaximal, characteristic, complemented
6#31773786C211yesmaximal, characteristic, complemented
7#31773775C211yesmaximal, characteristic
8#31773775C211yesmaximal, characteristic
9#31773771C211yesmaximal, characteristic, complemented
10#31773768C211yesmaximal, characteristic, complemented
11#31773764C211yesmaximal, characteristic, complemented
12#31773748C211yesmaximal, characteristic, complemented
13C4×(C2×D271)⋊C4C211yesmaximal, characteristic, complemented
14#31773744C211yesmaximal, characteristic, complemented
15#391×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C22×C4×D271C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19#22576149C418yescomplemented
20C2×C4⋊Dic271C2211yescharacteristic
21C2×Dic271⋊C4C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×(C2×D271)⋊C4C2211yescharacteristic
24C2×(C2×D271)⋊C4C2211yescharacteristic
25C2×(C2×D271)⋊C4C2211yescharacteristic
26C2×(C2×D271)⋊C4C2211yescharacteristic
27C2×(C2×D271)⋊C4C2211yescharacteristic
28C2×(C2×D271)⋊C4C2211yescharacteristic
29C2×(C2×D271)⋊C4C2211yescharacteristic
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×C4×Dic271C2211yescharacteristic
34C2×C4×Dic271C2211yescharacteristic
35C2×C4×Dic271C2211yescharacteristic
36C2×C4×Dic271C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C22×Q8×C271C2211yescharacteristic
46C2×C4⋊C4×C271C2211yescharacteristic
47C2×C42×C271C2211yescharacteristic
48#95×C271C2211yescharacteristic
49#95×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52C23×D271C2311yescharacteristic
53C23×D271C2311yescharacteristic
54C2×D1084—41no—
55C2×D1084—41no—
56C2×D1084C2×C414yes—
57C22×Dic271C2311yescharacteristic
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C2×C4×D271—22no—
64C2×C4×D271—22no—
65C2×C4×D271—41no—
66C2×C4×D271—41no—
67C2×C4×D271—41no—
68C2×C4×D271—41no—
69C4⋊Dic271—41no—
70Dic271⋊C4—22no—
71Dic271⋊C4—22no—
72(C2×D271)⋊C4—22no—
73(C2×D271)⋊C4—22no—
74(C2×D271)⋊C4—41no—
75(C2×D271)⋊C4—41no—
76(C2×D271)⋊C4—41no—
77(C2×D271)⋊C4—41no—
78(C2×D271)⋊C4C2×C414yescomplemented
79(C2×D271)⋊C4C2×C414yescomplemented
80(C2×D271)⋊C4C2×C414yescomplemented
81(C2×D271)⋊C4C2×C414yescomplemented
82C4×Dic271—41no—
83C4×Dic271—41no—
84C4×Dic271—41no—
85C4×Dic271C2×C414yescomplemented
86C2×Q8×C271—41no—
87C2×Q8×C271—41no—
88C2×Q8×C271C2×C414yescomplemented
89C22×C4×C271C2311yescharacteristic
90C22×C4×C271C2311yescharacteristic
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C22×C4×C271C2311yescharacteristic
94C22×C4×C271C2311yescharacteristic
95C22×C4×C271C2311yescharacteristic
96C4⋊C4×C271—41no—
97C42×C271—41no—
98C22×D271—21no—
99C22×D271—21no—
100C22×D271—21no—
101C22×D271—21no—
102C22×D271—21no—
103C22×D271—21no—
104C22×D271—21no—
105C22×D271—21no—
106C22×D271—21no—
107C22×D271—21no—
108C22×D271—21no—
109C22×D271—21no—
110C22×D271C22×C412yes—
111C22×D271C22×C412yes—
112D1084—44no—
113C2×Dic271—21no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
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×Dic271C22×C412yes—
129C2×Dic271C22×C412yes—
130C2×Dic271#4112yes—
131C4×D271—44no—
132C4×D271—44no—
133C23×C271C2411yescharacteristic
134Q8×C271—44no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
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×C271C22×C412yes—
153C2×C4×C271C22×C412yes—
154C2×C4×C271C22×C412yes—
155C2×C4×C271#4112yes—
156C22×C271C23×C411yescharacteristic
157C22×C271C2×#4111yescharacteristic
158C22×C271C2×#4111yescharacteristic
159C22×C271H(5,𝔽2)11yescharacteristic
160C22×C271H(5,𝔽2)11yescharacteristic
161C22×C271H(5,𝔽2)11yescharacteristic
162C22×C2712−1+411yescharacteristic
163C2×D271—22no—
164C2×D271—22no—
165C2×D271—22no—
166C2×D271—22no—
167C2×D271—22no—
168C2×D271—22no—
169C2×D271—41no—
170C2×D271—41no—
171C2×D271—41no—
172C2×D271—41no—
173C2×D271—41no—
174C2×D271—41no—
175C2×D271—41no—
176C2×D271—41no—
177C4×C271—22no—
178C4×C271—22no—
179C4×C271—22no—
180C4×C271—22no—
181C4×C271—41no—
182C4×C271—41no—
183C4×C271—41no—
184Dic271—22no—
185Dic271—22no—
186Dic271—22no—
187Dic271—41no—
188Dic271—41no—
189Dic271—41no—
190C2×C271C4×#4111yescharacteristic
191C2×C271#51811yescharacteristic
192C2×C271#52011yescharacteristic
193C2×C271#53811yescharacteristic
194C2×C271#53811yescharacteristic
195C2×C271#54211yescharacteristic
196C2×C271#54211yescharacteristic
197D271—42no—
198D271—42no—
199C271#233711yescharacteristic, complemented, Sylow
200#2337—2711nomaximal, Sylow
201C2×#124—2711no—
202#391—2711no—
203#391D27111yescharacteristic, complemented
204#388—2711no—
205#388—2711no—
206#388—2711no—
207#388—2711no—
208#386—2711no—
209#386—2711no—
210#385—2711no—
211#385—2711no—
212#383—2711no—
213C4×C4⋊C4—2711no—
214C4×C22⋊C4—2711no—
215C4×C22⋊C4—2711no—
216C22×Q8C2×D27111yescharacteristic
217C22×D4—2711no—
218C23×C4—2711no—
219C23×C4—2711no—
220#124—2718no—
221C2×C4⋊C4—2711no—
222C2×C4⋊C4—2711no—
223C2×C4⋊C4—2711no—
224C2×C4⋊C4C2×D27111yescharacteristic
225C2×C22⋊C4—2711no—
226C2×C22⋊C4—2711no—
227C2×C22⋊C4—2711no—
228C2×C22⋊C4—2711no—
229C2×C22⋊C4—2711no—
230C2×C22⋊C4—2711no—
231C2×C22⋊C4—2711no—
232C2×C22⋊C4—2711no—
233C2×C22⋊C4—2711no—
234C2×C22⋊C4—2711no—
235C2×C42—2711no—
236C2×C42—2711no—
237C2×C42—2711no—
238C2×C42—2711no—
239C2×C42C2×D27111yescharacteristic
240#95—2711no—
241#95—2711no—
242#95—2711no—
243#95—2711no—
244#95—2711no—
245#95—2711no—
246#95—2711no—
247#95—2711no—
248#95C2×D27111yescharacteristic
249#95C2×D27111yescharacteristic
250#95C2×D27111yescharacteristic
251#95C2×D27111yescharacteristic
252C24—2711no—
253C24—2711no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8C4×D27114yescomplemented
257C2×D4—2714no—
258C2×D4—10841no—
259C2×D4—10841no—
260C22×C4—2711no—
261C22×C4—2711no—
262C22×C4—2711no—
263C22×C4—2711no—
264C22×C4—2711no—
265C22×C4—2711no—
266C22×C4—5422no—
267C22×C4—5422no—
268C22×C4—10841no—
269C22×C4—10841no—
270C22×C4—10841no—
271C22×C4—10841no—
272C22×C4C22×D27111yescharacteristic
273C22×C4C22×D27111yescharacteristic
274C22×C4C22×D27111yescharacteristic
275C22×C4C22×D27111yescharacteristic
276C22×C4C22×D27111yescharacteristic
277C22×C4C22×D27111yescharacteristic
278C22×C4C22×D27111yescharacteristic
279C4⋊C4—41no—
280C4⋊C4—5422no—
281C4⋊C4—5422no—
282C4⋊C4—10841no—
283C22⋊C4—2714no—
284C22⋊C4—2714no—
285C22⋊C4—2714no—
286C22⋊C4—2714no—
287C22⋊C4—5422no—
288C22⋊C4—5422no—
289C22⋊C4—10841no—
290C22⋊C4—10841no—
291C22⋊C4—10841no—
292C22⋊C4—10841no—
293C42—41no—
294C42—2714no—
295C42—10841no—
296C42—10841no—
297C42—10841no—
298C23—2712no—
299C23—2712no—
300C23—5421no—
301C23—5421no—
302C23—5421no—
303C23—5421no—
304C23—5421no—
305C23—5421no—
306C23—5421no—
307C23—5421no—
308C23—5421no—
309C23—5421no—
310C23—5421no—
311C23—5421no—
312C23C23×D27111yescharacteristic
313Q8—44no—
314D4—10844no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—2712no—
333C2×C4—2712no—
334C2×C4—2712no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—5421no—
338C2×C4—5421no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—5421no—
344C2×C4—5421no—
345C2×C4—5421no—
346C2×C4—5421no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—10844no—
351C2×C4—10844no—
352C2×C4C2×C4×D27112yes—
353C2×C4C2×C4×D27112yes—
354C2×C4C2×C4×D27112yes—
355C2×C4#1686465512yes—
356C22—5422no—
357C22—5422no—
358C22—5422no—
359C22—5422no—
360C22—5422no—
361C22—5422no—
362C22—10841no—
363C22—10841no—
364C22—10841no—
365C22—10841no—
366C22—10841no—
367C22—10841no—
368C22—10841no—
369C22—10841no—
370C22C22×C4×D27111yescharacteristic
371C22#41×D27111yescharacteristic
372C22C2×#1686465511yescharacteristic
373C22#2257620511yescharacteristic
374C22#2257620511yescharacteristic
375C22#2257620611yescharacteristic
376C22#2257620811yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—5422no—
385C4—5422no—
386C4—5422no—
387C4—10841no—
388C4—10841no—
389C4—10841no—
390C2—10842no—
391C2—10842no—
392C2C4×#1686465511yescharacteristic
393C2#3177437011yescharacteristic
394C2#3177437911yescharacteristic
395C2#3177448111yescharacteristic
396C2#3177448711yescharacteristic
397C2#3177451711yescharacteristic
398C2#3177452111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#391×C271C2
socle, cosocleC23×C271C24
prime radicalC22×D1084C22
hypercenter#391D271
hypocenterC271#2337
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #31773804C2semidirect
3. #31773789C2semidirect
4. #31773788C2semidirect
5. #31773787C2semidirect
6. #31773786C2semidirect
9. #31773771C2semidirect
10. #31773768C2semidirect
11. #31773764C2semidirect
12. #31773748C2semidirect
13. C4×(C2×D271)⋊C4C2semidirect
14. #31773744C2semidirect
15. #391×C271C2semidirect
19. #22576149C4semidirect
78. (C2×D271)⋊C4C2×C4semidirect
79. (C2×D271)⋊C4C2×C4semidirect
80. (C2×D271)⋊C4C2×C4semidirect
81. (C2×D271)⋊C4C2×C4semidirect
85. C4×Dic271C2×C4semidirect
88. C2×Q8×C271C2×C4semidirect
199. C271#2337semidirect
203. #391D271semidirect
256. C2×Q8C4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761492
#317738042
#317737892
#317737882
#317737872
#317737862
#317737752
#317737712
#317737682
#317737642
#317737482
C4×(C2×D271)⋊C42
#317737442
#391×C2712
#2337271