Finite Group Databasebeta

← #46839350

Subgroups — #46839350

Order 34688.

652 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×#22576141C211yesmaximal, characteristic
2C2×#22576116C211yesmaximal, characteristic, complemented
3#31773792C211yesmaximal, characteristic, complemented
4#31773791C211yesmaximal, characteristic, complemented
5#31773785C211yesmaximal, characteristic, complemented
6#31773781C211yesmaximal, characteristic, complemented
7#31773778C211yesmaximal, characteristic, complemented
8#31773777C211yesmaximal, characteristic
9#31773770C211yesmaximal, characteristic, complemented
10#31773770C211yesmaximal, characteristic, complemented
11C4×Dic271⋊C4C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic271C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic271C211yesmaximal, characteristic, complemented
14C2×C271⋊#95C211yesmaximal, characteristic, complemented
15C2×C4×D4×C271C211yesmaximal, characteristic
16C23×Dic271C2211yescharacteristic
17C23×Dic271C2211yescharacteristic
18#22576141C418yescomplemented
19#22576116—24no—
20C2×Dic271⋊C4C2211yescharacteristic
21C2×Dic271⋊C4C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×C22⋊Dic271C2211yescharacteristic
27C2×C22⋊Dic271C2211yescharacteristic
28C2×C22⋊Dic271C2211yescharacteristic
29C2×C22⋊Dic271C2211yescharacteristic
30C2×C22⋊Dic271C2211yescharacteristic
31C2×C22⋊Dic271C2211yescharacteristic
32C2×C4×Dic271C2211yescharacteristic
33C2×C4×Dic271C2211yescharacteristic
34C2×C4×Dic271C2211yescharacteristic
35C2×C4×Dic271C2211yescharacteristic
36C271⋊#95C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C22×D4×C271C2211yescharacteristic
47C23×C4×C271C2211yescharacteristic
48C23×C4×C271C2211yescharacteristic
49C4×D4×C271—24no—
50C2×C4⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C42×C271C2211yescharacteristic
54C22×Dic271—22no—
55C22×Dic271—22no—
56C22×Dic271—22no—
57C22×Dic271—22no—
58C22×Dic271—41no—
59C22×Dic271C2×C414yescomplemented
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C22×Dic271C2311yescharacteristic
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68Dic271⋊C4—22no—
69Dic271⋊C4—22no—
70Dic271⋊C4—22no—
71Dic271⋊C4—41no—
72Dic271⋊C4C2×C414yescomplemented
73Dic271⋊C4C2×C414yescomplemented
74C22⋊Dic271—22no—
75C22⋊Dic271—22no—
76C22⋊Dic271—41no—
77C22⋊Dic271C2×C414yescomplemented
78C22⋊Dic271C2×C414yescomplemented
79C22⋊Dic271C2×C414yescomplemented
80C4×Dic271—22no—
81C4×Dic271—22no—
82C4×Dic271—22no—
83C4×Dic271—41no—
84C24×C271C2311yescharacteristic
85C24×C271C2311yescharacteristic
86C2×D4×C271—22no—
87C2×D4×C271—22no—
88C2×D4×C271C2×C414yes—
89C22×C4×C271—22no—
90C22×C4×C271—22no—
91C22×C4×C271—22no—
92C22×C4×C271—22no—
93C22×C4×C271—41no—
94C22×C4×C271—41no—
95C22×C4×C271C2311yescharacteristic
96C22×C4×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C4⋊C4×C271—22no—
101C22⋊C4×C271—22no—
102C22⋊C4×C271D414yes—
103C42×C271—22no—
104C2×Dic271—28no—
105C2×Dic271—21no—
106C2×Dic271—21no—
107C2×Dic271—21no—
108C2×Dic271—21no—
109C2×Dic271—21no—
110C2×Dic271—21no—
111C2×Dic271—21no—
112C2×Dic271—21no—
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—44no—
125C2×Dic271C22×C412yes—
126C2×Dic271C22×C412yes—
127C2×Dic271C22×C412yes—
128C2×Dic271C22×C412yes—
129C2×Dic271#4112yes—
130C23×C271—21no—
131C23×C271—21no—
132C23×C271—21no—
133C23×C271—21no—
134C23×C271—21no—
135C23×C271—21no—
136C23×C271—21no—
137C23×C271—21no—
138C23×C271—21no—
139C23×C271—21no—
140C23×C271C22×C412yes—
141C23×C271C22×C412yes—
142C23×C271C2×D412yes—
143C23×C271#4112yes—
144C23×C271C2411yescharacteristic
145D4×C271—28no—
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—44no—
158C2×C4×C271—44no—
159C2×C4×C271C22×C412yes—
160C2×C4×C271C2×D412yes—
161C2×C4×C271C2×D412yes—
162C2×C4×C271#4112yes—
163C22×C271—22no—
164C22×C271—22no—
165C22×C271—22no—
166C22×C271—22no—
167C22×C271—22no—
168C22×C271—22no—
169C22×C271—22no—
170C22×C271—22no—
171C22×C271—22no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271C4×D414yes—
177C22×C271C23×C411yescharacteristic
178C22×C271C22×D411yescharacteristic
179C22×C271C2×#4111yescharacteristic
180C22×C271C2×#4111yescharacteristic
181C22×C271C2×#4111yescharacteristic
182C22×C271H(5,𝔽2)11yescharacteristic
183C22×C2712−1+411yescharacteristic
184C4×C271—22no—
185C4×C271—22no—
186C4×C271—22no—
187C4×C271—22no—
188C4×C271—41no—
189Dic271—22no—
190Dic271—22no—
191Dic271—22no—
192Dic271—22no—
193Dic271—22no—
194Dic271—41no—
195Dic271—41no—
196Dic271—41no—
197C2×C271—24no—
198C2×C271—42no—
199C2×C271C2×C4×D411yescharacteristic
200C2×C271C4×#4111yescharacteristic
201C2×C271#52011yescharacteristic
202C2×C271#54611yescharacteristic
203C2×C271#54711yescharacteristic
204C2×C271#55111yescharacteristic
205C2×C271#55211yescharacteristic
206C271#235911yescharacteristic, complemented, Sylow
207#2359—2711nomaximal, Sylow
208C2×#123—2711no—
209C2×C4×D4D27111yescharacteristic
210C2×#117—2711no—
211#389—2711no—
212#388—2711no—
213#388—2711no—
214#387—2711no—
215#387—2711no—
216#386—2711no—
217#385—2711no—
218#385—2711no—
219C4×C4⋊C4—2711no—
220C4×C22⋊C4—2711no—
221C4×C22⋊C4—2711no—
222C2×#95—2711no—
223C22×D4C2×D27111yescharacteristic
224C23×C4—2711no—
225C23×C4—2711no—
226C23×C4C2×D27111yescharacteristic
227C23×C4C2×D27111yescharacteristic
228#123—2718no—
229C4×D4—24no—
230#117—5424no—
231C2×C4⋊C4—2711no—
232C2×C4⋊C4—2711no—
233C2×C4⋊C4—2711no—
234C2×C4⋊C4—2711no—
235C2×C4⋊C4—2711no—
236C2×C4⋊C4—2711no—
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⋊C4C2×D27111yescharacteristic
245C2×C22⋊C4C2×D27111yescharacteristic
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#95—2711no—
258#95—2711no—
259#95—2711no—
260#95—2711no—
261C24C22×D27111yescharacteristic
262C24C22×D27111yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4C4×D27114yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—2714no—
273C22×C4—2711no—
274C22×C4—2711no—
275C22×C4—2711no—
276C22×C4—2711no—
277C22×C4—2711no—
278C22×C4—2711no—
279C22×C4—2711no—
280C22×C4—2711no—
281C22×C4—5422no—
282C22×C4—5422no—
283C22×C4—5422no—
284C22×C4—5422no—
285C22×C4—10841no—
286C22×C4C22×D27111yescharacteristic
287C22×C4C22×D27111yescharacteristic
288C22×C4C22×D27111yescharacteristic
289C22×C4C22×D27111yescharacteristic
290C22×C4C22×D27111yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—2714no—
293C4⋊C4—2714no—
294C4⋊C4—5422no—
295C4⋊C4—5422no—
296C4⋊C4—5422no—
297C4⋊C4—10841no—
298C22⋊C4—22no—
299C22⋊C4—2714no—
300C22⋊C4—2714no—
301C22⋊C4—2714no—
302C22⋊C4—5422no—
303C22⋊C4—5422no—
304C22⋊C4—10841no—
305C22⋊C4C22⋊D27114yes—
306C42—22no—
307C42—5422no—
308C42—5422no—
309C42—5422no—
310C42—10841no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C4×D27112yes—
322C23C2×C4×D27112yes—
323C23C2×C22⋊D27112yes—
324C23#1686465412yes—
325C23C23×D27111yescharacteristic
326D4—28no—
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—44no—
339C2×C4—44no—
340C2×C4—2712no—
341C2×C4—2712no—
342C2×C4—2712no—
343C2×C4—2712no—
344C2×C4—2712no—
345C2×C4—5421no—
346C2×C4—5421no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—5428no—
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—5421no—
365C2×C4—10844no—
366C2×C4C2×C4×D27112yes—
367C2×C4C2×C22⋊D27112yes—
368C2×C4C2×C22⋊D27112yes—
369C2×C4#1686465312yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22C4×C22⋊D27114yes—
384C22C22×C4×D27111yescharacteristic
385C22C22×C22⋊D27111yescharacteristic
386C22#41×D27111yescharacteristic
387C22C2×#1686465311yescharacteristic
388C22C2×#1686465411yescharacteristic
389C22#2257620511yescharacteristic
390C22#2257620711yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—5422no—
397C4—5422no—
398C4—5422no—
399C4—5422no—
400C4—5422no—
401C4—10841no—
402C4—10841no—
403C4—10841no—
404C2—24no—
405C2—42no—
406C2C2×C4×C22⋊D27111yescharacteristic
407C2C4×#1686465311yescharacteristic
408C2#3177437611yescharacteristic
409C2#3177454111yescharacteristic
410C2#3177454611yescharacteristic
411C2#3177457611yescharacteristic
412C2#3177458711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
FittingC2×C4×D4×C271C2
socle, cosocleC23×C271C24
prime radicalC22×D4×C271C22
hypercenterC2×C4×D4D271
hypocenterC271#2359
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#22576116C2semidirect
3. #31773792C2semidirect
4. #31773791C2semidirect
5. #31773785C2semidirect
6. #31773781C2semidirect
7. #31773778C2semidirect
9. #31773770C2semidirect
10. #31773770C2semidirect
11. C4×Dic271⋊C4C2semidirect
12. C22⋊C4×Dic271C2semidirect
13. C4×C22⋊Dic271C2semidirect
14. C2×C271⋊#95C2semidirect
18. #22576141C4semidirect
59. C22×Dic271C2×C4semidirect
72. Dic271⋊C4C2×C4semidirect
73. Dic271⋊C4C2×C4semidirect
77. C22⋊Dic271C2×C4semidirect
78. C22⋊Dic271C2×C4semidirect
79. C22⋊Dic271C2×C4semidirect
206. C271#2359semidirect

Maximal subgroups

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

subgroupindex
C2×#225761412
C2×#225761162
#317737922
#317737912
#317737852
#317737812
#317737782
#317737772
#317737702
C4×Dic271⋊C42
C22⋊C4×Dic2712
C4×C22⋊Dic2712
C2×C271⋊#952
C2×C4×D4×C2712
#2359271