Finite Group Databasebeta

← #46839158

Subgroups — #46839158

Order 34688.

532 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31773800C211yesmaximal, characteristic
2#31773796C211yesmaximal, characteristic, complemented
3#31773790C211yesmaximal, characteristic, complemented
4#31773790C211yesmaximal, characteristic, complemented
5#31773790C211yesmaximal, characteristic, complemented
6#31773790C211yesmaximal, characteristic, complemented
7#31773783C211yesmaximal, characteristic, complemented
8#31773783C211yesmaximal, characteristic, complemented
9#31773777C211yesmaximal, characteristic
10#31773772C211yesmaximal, characteristic, complemented
11#31773757C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic271C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic271C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic271C211yesmaximal, characteristic, complemented
15C2×#123×C271C211yesmaximal, characteristic
16C23×Dic271C2211yescharacteristic
17C2×C4⋊Dic271C2211yescharacteristic
18C2×C4⋊Dic271C2211yescharacteristic
19C2×C4⋊Dic271C2211yescharacteristic
20C2×C4⋊Dic271C2211yescharacteristic
21C2×C4⋊Dic271C2211yescharacteristic
22C2×C22⋊Dic271C2211yescharacteristic
23C2×C22⋊Dic271C2211yescharacteristic
24C2×C22⋊Dic271C2211yescharacteristic
25C2×C22⋊Dic271C2211yescharacteristic
26C2×C22⋊Dic271C2211yescharacteristic
27C2×C22⋊Dic271C2211yescharacteristic
28C2×C22⋊Dic271C2211yescharacteristic
29C2×C4×Dic271C2211yescharacteristic
30C2×C4×Dic271C2211yescharacteristic
31C2×C4×Dic271C2211yescharacteristic
32C2×C4×Dic271C2211yescharacteristic
33C2×C4×Dic271C2211yescharacteristic
34C271⋊#95C2211yescharacteristic
35C271⋊#95C2211yescharacteristic
36C271⋊#95C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C22×D4×C271C2211yescharacteristic
45C23×C4×C271C2211yescharacteristic
46#123×C271C418yescomplemented
47C2×C4⋊C4×C271C2211yescharacteristic
48C2×C4⋊C4×C271C2211yescharacteristic
49C2×C22⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C22×Dic271—22no—
53C22×Dic271—41no—
54C22×Dic271—41no—
55C22×Dic271C2311yescharacteristic
56C22×Dic271C2311yescharacteristic
57C22×Dic271C2311yescharacteristic
58C22×Dic271C2311yescharacteristic
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C4⋊Dic271—22no—
64C4⋊Dic271—41no—
65C4⋊Dic271—41no—
66C4⋊Dic271—41no—
67C4⋊Dic271—41no—
68C22⋊Dic271—22no—
69C22⋊Dic271—22no—
70C22⋊Dic271—22no—
71C22⋊Dic271—41no—
72C22⋊Dic271—41no—
73C22⋊Dic271—41no—
74C22⋊Dic271—41no—
75C4×Dic271—22no—
76C4×Dic271—41no—
77C4×Dic271—41no—
78C4×Dic271—41no—
79C4×Dic271—41no—
80C24×C271C2311yescharacteristic
81C24×C271C2311yescharacteristic
82C2×D4×C271—41no—
83C2×D4×C271—41no—
84C2×D4×C271C2×C414yes—
85C22×C4×C271—41no—
86C22×C4×C271—41no—
87C22×C4×C271C2×C414yescomplemented
88C22×C4×C271C2311yescharacteristic
89C22×C4×C271C2311yescharacteristic
90C22×C4×C271C2311yescharacteristic
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C4⋊C4×C271C2×C414yescomplemented
94C4⋊C4×C271C2×C414yescomplemented
95C22⋊C4×C271C2×C414yes—
96C22⋊C4×C271C2×C414yes—
97C22⋊C4×C271C2×C414yescomplemented
98C2×Dic271—21no—
99C2×Dic271—21no—
100C2×Dic271—21no—
101C2×Dic271—21no—
102C2×Dic271—21no—
103C2×Dic271—21no—
104C2×Dic271—21no—
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—44no—
121C2×Dic271#4112yes—
122C2×Dic271#4112yes—
123C23×C271—21no—
124C23×C271—21no—
125C23×C271—21no—
126C23×C271—21no—
127C23×C271—21no—
128C23×C271—21no—
129C23×C271—21no—
130C23×C271—21no—
131C23×C271—21no—
132C23×C271—21no—
133C23×C271—21no—
134C23×C271—21no—
135C23×C271C22×C412yes—
136C23×C271C22×C412yes—
137C23×C271C2411yescharacteristic
138D4×C271—44no—
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—44no—
150C2×C4×C271C22×C412yes—
151C2×C4×C271C22×C412yes—
152C2×C4×C271C22×C412yes—
153C2×C4×C271C22×C412yes—
154C2×C4×C271C22×C412yes—
155C22×C271—22no—
156C22×C271—22no—
157C22×C271—22no—
158C22×C271—22no—
159C22×C271—22no—
160C22×C271—22no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271—41no—
164C22×C271—41no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271—41no—
168C22×C271—41no—
169C22×C271C23×C411yescharacteristic
170C22×C271C2×#4111yescharacteristic
171C22×C271C2×#4111yescharacteristic
172C22×C271H(5,𝔽2)11yescharacteristic
173C22×C271H(5,𝔽2)11yescharacteristic
174C22×C271H(5,𝔽2)11yescharacteristic
175C22×C2712−1+411yescharacteristic
176C4×C271—22no—
177C4×C271—22no—
178C4×C271—22no—
179C4×C271—22no—
180C4×C271—22no—
181Dic271—22no—
182Dic271—22no—
183Dic271—41no—
184Dic271—41no—
185Dic271—41no—
186Dic271—41no—
187Dic271—41no—
188Dic271—41no—
189C2×C271—42no—
190C2×C271—42no—
191C2×C271C4×#4111yescharacteristic
192C2×C271#51811yescharacteristic
193C2×C271#52011yescharacteristic
194C2×C271#53811yescharacteristic
195C2×C271#53911yescharacteristic
196C2×C271#54011yescharacteristic
197C2×C271#54211yescharacteristic
198C271#233511yescharacteristic, complemented, Sylow
199#2335—2711nomaximal, Sylow
200C2×#123D27111yescharacteristic
201#390—2711no—
202#389—2711no—
203#388—2711no—
204#388—2711no—
205#388—2711no—
206#388—2711no—
207#387—2711no—
208#387—2711no—
209#386—2711no—
210#385—2711no—
211#380—2711no—
212C4×C4⋊C4—2711no—
213C4×C22⋊C4—2711no—
214C4×C22⋊C4—2711no—
215C22×D4C2×D27111yescharacteristic
216C23×C4—2711no—
217C23×C4C2×D27111yescharacteristic
218#123Dic27118yescomplemented
219C2×C4⋊C4—2711no—
220C2×C4⋊C4—2711no—
221C2×C4⋊C4—2711no—
222C2×C4⋊C4—2711no—
223C2×C4⋊C4—2711no—
224C2×C4⋊C4C2×D27111yescharacteristic
225C2×C4⋊C4C2×D27111yescharacteristic
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⋊C4C2×D27111yescharacteristic
234C2×C22⋊C4C2×D27111yescharacteristic
235C2×C22⋊C4C2×D27111yescharacteristic
236C2×C42—2711no—
237C2×C42—2711no—
238C2×C42—2711no—
239C2×C42—2711no—
240C2×C42—2711no—
241#95—2711no—
242#95—2711no—
243#95—2711no—
244#95—2711no—
245#95—2711no—
246#95—2711no—
247#95—2711no—
248#95—2711no—
249#95—2711no—
250#95—2711no—
251C24C22×D27111yescharacteristic
252C24C22×D27111yescharacteristic
253C2×D4—41no—
254C2×D4—41no—
255C2×D4C2×Dic27114yes—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—2711no—
259C22×C4—2711no—
260C22×C4—2711no—
261C22×C4—2711no—
262C22×C4—2711no—
263C22×C4—2711no—
264C22×C4—2711no—
265C22×C4—2711no—
266C22×C4—5422no—
267C22×C4—10841no—
268C22×C4—10841no—
269C22×C4C2×Dic27114yescomplemented
270C22×C4C22×D27111yescharacteristic
271C22×C4C22×D27111yescharacteristic
272C22×C4C22×D27111yescharacteristic
273C22×C4C22×D27111yescharacteristic
274C22×C4C22×D27111yescharacteristic
275C4⋊C4—5422no—
276C4⋊C4—10841no—
277C4⋊C4—10841no—
278C4⋊C4—10841no—
279C4⋊C4—10841no—
280C4⋊C4C2×Dic27114yescomplemented
281C4⋊C4C2×Dic27114yescomplemented
282C22⋊C4—5422no—
283C22⋊C4—5422no—
284C22⋊C4—5422no—
285C22⋊C4—10841no—
286C22⋊C4—10841no—
287C22⋊C4—10841no—
288C22⋊C4—10841no—
289C22⋊C4C2×Dic27114yes—
290C22⋊C4C2×Dic27114yes—
291C22⋊C4C2×Dic27114yescomplemented
292C42—5422no—
293C42—10841no—
294C42—10841no—
295C42—10841no—
296C42—10841no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C22×Dic27112yes—
310C23C22×Dic27112yes—
311C23C23×D27111yescharacteristic
312D4—44no—
313C2×C4—21no—
314C2×C4—21no—
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—44no—
324C2×C4—2712no—
325C2×C4—2712no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—5421no—
333C2×C4—5421no—
334C2×C4—5421no—
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—10844no—
349C2×C4C22×Dic27112yes—
350C2×C4C22×Dic27112yes—
351C2×C4C22×Dic27112yes—
352C2×C4C22×Dic27112yes—
353C2×C4C22×Dic27112yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C23×Dic27111yescharacteristic
369C22#41×D27111yescharacteristic
370C22#41×D27111yescharacteristic
371C22#2257620511yescharacteristic
372C22#2257620511yescharacteristic
373C22#2257620611yescharacteristic
374C22#2257620711yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—5422no—
381C4—5422no—
382C4—10841no—
383C4—10841no—
384C4—10841no—
385C4—10841no—
386C4—10841no—
387C4—10841no—
388C2—42no—
389C2—42no—
390C2#41×Dic27111yescharacteristic
391C2#3177437211yescharacteristic
392C2#3177438011yescharacteristic
393C2#3177448311yescharacteristic
394C2#3177449011yescharacteristic
395C2#3177449911yescharacteristic
396C2#3177451511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
FittingC2×#123×C271C2
socle, cosocleC23×C271C24
prime radicalC22×D4×C271C22
hypercenterC2×#123D271
hypocenterC271#2335
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #31773796C2semidirect
3. #31773790C2semidirect
4. #31773790C2semidirect
5. #31773790C2semidirect
6. #31773790C2semidirect
7. #31773783C2semidirect
8. #31773783C2semidirect
10. #31773772C2semidirect
11. #31773757C2semidirect
12. C4⋊C4×Dic271C2semidirect
13. C22⋊C4×Dic271C2semidirect
14. C4×C22⋊Dic271C2semidirect
46. #123×C271C4semidirect
87. C22×C4×C271C2×C4semidirect
93. C4⋊C4×C271C2×C4semidirect
94. C4⋊C4×C271C2×C4semidirect
97. C22⋊C4×C271C2×C4semidirect
198. C271#2335semidirect
218. #123Dic271semidirect
269. C22×C4C2×Dic271semidirect
280. C4⋊C4C2×Dic271semidirect
281. C4⋊C4C2×Dic271semidirect
291. C22⋊C4C2×Dic271semidirect

Maximal subgroups

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

subgroupindex
#317738002
#317737962
#317737902
#317737832
#317737772
#317737722
#317737572
C4⋊C4×Dic2712
C22⋊C4×Dic2712
C4×C22⋊Dic2712
C2×#123×C2712
#2335271