Finite Group Databasebeta

← #46839808

Subgroups — #46839808

Order 34688.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576138C211yesmaximal, characteristic, complemented
2C2×#22576137C211yesmaximal, characteristic, complemented
3C2×(C22⋊D271)⋊C4C211yesmaximal, characteristic
4C2×C4⋊C4×D271C211yesmaximal, characteristic, complemented
5#31773830C211yesmaximal, characteristic, complemented
6#31773827C211yesmaximal, characteristic, complemented
7#31773820C211yesmaximal, characteristic, complemented
8#31773801C211yesmaximal, characteristic, complemented
9#31773792C211yesmaximal, characteristic, complemented
10#31773773C211yesmaximal, characteristic
11#31773773C211yesmaximal, characteristic
12#31773756C211yesmaximal, characteristic, complemented
13#31773755C211yesmaximal, characteristic, complemented
14#31773754C211yesmaximal, characteristic, complemented
15#396×C271C211yesmaximal, characteristic, complemented
16C22×Dic542C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C23×Dic271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22#22576138—24no—
23#22576137—24no—
24(C22⋊D271)⋊C4—24no—
25C2×C4⋊Dic271C2211yescharacteristic
26C2×C4⋊Dic271C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×Dic271⋊C4C2211yescharacteristic
32C2×Dic271⋊C4C2211yescharacteristic
33C2×Dic271⋊C4C2211yescharacteristic
34C4⋊C4×D271—24no—
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×C4×Dic271C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C2×C4⋊C4×C271C2211yescharacteristic
49C2×C4⋊C4×C271C2211yescharacteristic
50C2×C4⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54#95×C271C2211yescharacteristic
55C23×D271C2311yescharacteristic
56C2×Dic542—22no—
57C2×Dic542—22no—
58C2×Dic542—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271Q814yescomplemented
62C22×Dic271—22no—
63C22×Dic271—22no—
64C22×Dic271—22no—
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C2×C4×D271—22no—
73C2×C4×D271—22no—
74C2×C4×D271—22no—
75C2×C4×D271—22no—
76C2×C4×D271—22no—
77C2×C4×D271—22no—
78C2×C4×D271—41no—
79C2×C4×D271—41no—
80C2×C4×D271D414yes—
81C4⋊Dic271—22no—
82C4⋊Dic271—22no—
83Dic271⋊C4—22no—
84Dic271⋊C4—22no—
85Dic271⋊C4—22no—
86Dic271⋊C4—22no—
87Dic271⋊C4—22no—
88Dic271⋊C4—41no—
89Dic271⋊C4D414yescomplemented
90C22⋊Dic271—22no—
91C22⋊Dic271—41no—
92(C2×D271)⋊C4—22no—
93(C2×D271)⋊C4—22no—
94(C2×D271)⋊C4—22no—
95C4×Dic271—22no—
96C24×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102C22×C4×C271C2311yescharacteristic
103C4⋊C4×C271—22no—
104C4⋊C4×C271—22no—
105C4⋊C4×C271—41no—
106C22⋊C4×C271—22no—
107C22⋊C4×C271—22no—
108C22⋊C4×C271—41no—
109C22×D271—21no—
110C22×D271—21no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271C2×D412yes—
114C22×D271C2×Q812yes—
115C22×D271#4112yes—
116Dic542—44no—
117C22⋊D271—28no—
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×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—44no—
135C2×Dic271C2×D412yes—
136C2×Dic271C2×D412yes—
137C2×Dic271C2×D412yes—
138C2×Dic271C2×Q812yes—
139C2×Dic271#4112yes—
140C4×D271—28no—
141C4×D271—44no—
142C4×D271—44no—
143C23×C271—21no—
144C23×C271—21no—
145C23×C271—21no—
146C23×C271—21no—
147C23×C271—21no—
148C23×C271—21no—
149C23×C271C2×Q812yes—
150C23×C271C2411yescharacteristic
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×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271C2×D412yes—
168C2×C4×C271C2×D412yes—
169C22×C271—22no—
170C22×C271—22no—
171C22×C271—22no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271C22×D411yescharacteristic
177C22×C271C22×D411yescharacteristic
178C22×C271C22×Q811yescharacteristic
179C22×C271C2×#4111yescharacteristic
180C22×C271C2×#4111yescharacteristic
181C22×C271H(5,𝔽2)11yescharacteristic
182C22×C2712−1+411yescharacteristic
183C2×D271—22no—
184C2×D271—22no—
185C2×D271—22no—
186C2×D271—22no—
187C2×D271—22no—
188C2×D271—22no—
189C2×D271C22⋊Q814yescomplemented
190C4×C271—22no—
191C4×C271—22no—
192C4×C271—41no—
193C4×C271—41no—
194C4×C271—41no—
195C4×C271—41no—
196Dic271—22no—
197Dic271—22no—
198Dic271—41no—
199Dic271—41no—
200Dic271—41no—
201Dic271—41no—
202Dic271#12114yes—
203C2×C271—42no—
204C2×C271C2×#12111yescharacteristic
205C2×C271C2×C22⋊Q811yescharacteristic
206C2×C271#53711yescharacteristic
207C2×C271#54611yescharacteristic
208C2×C271D4×Q811yescharacteristic
209C2×C271#55211yescharacteristic
210C2×C271#55411yescharacteristic
211D271—24no—
212C271#241711yescharacteristic, complemented, Sylow
213#2417—2711nomaximal, Sylow
214C2×C22⋊Q8—2711no—
215C2×C22⋊Q8—2711no—
216C2×C4×D4—2711no—
217C22×C4⋊C4—2711no—
218#398—2711no—
219#398—2711no—
220#396—2711no—
221#396D27111yescharacteristic, complemented
222#391—2711no—
223#389—2711no—
224#386—2711no—
225#386—2711no—
226#380—2711no—
227#380—2711no—
228#380—2711no—
229C22×Q8—2711no—
230C22×D4—2711no—
231C23×C4—2711no—
232C23×C4—2711no—
233C23×C4—2711no—
234C23×C4—2711no—
235C22⋊Q8—5424no—
236C22⋊Q8—5424no—
237C4×D4—5424no—
238C2×C4⋊C4—2711no—
239C2×C4⋊C4—2711no—
240C2×C4⋊C4—2711no—
241C2×C4⋊C4—2711no—
242C2×C4⋊C4—2711no—
243C2×C4⋊C4—2711no—
244C2×C4⋊C4—2711no—
245C2×C4⋊C4—2711no—
246C2×C4⋊C4—2711no—
247C2×C4⋊C4—5424no—
248C2×C4⋊C4C2×D27111yescharacteristic
249C2×C4⋊C4C2×D27111yescharacteristic
250C2×C4⋊C4C2×D27111yescharacteristic
251C2×C22⋊C4—2711no—
252C2×C22⋊C4—2711no—
253C2×C22⋊C4—2711no—
254C2×C22⋊C4—2711no—
255C2×C22⋊C4—2711no—
256C2×C22⋊C4C2×D27111yescharacteristic
257C2×C22⋊C4C2×D27111yescharacteristic
258C2×C22⋊C4C2×D27111yescharacteristic
259C2×C42—2711no—
260#95—2711no—
261#95—2711no—
262#95—2711no—
263#95—2711no—
264#95—2711no—
265#95—2711no—
266#95—2711no—
267#95C2×D27111yescharacteristic
268C24—2711no—
269C24C22×D27111yescharacteristic
270C2×Q8—5422no—
271C2×Q8—5422no—
272C2×Q8—5422no—
273C2×D4—2714no—
274C2×D4—5422no—
275C2×D4—5422no—
276C22×C4—2711no—
277C22×C4—2711no—
278C22×C4—2711no—
279C22×C4—2714no—
280C22×C4—2711no—
281C22×C4—2711no—
282C22×C4—2711no—
283C22×C4—2711no—
284C22×C4—5422no—
285C22×C4—5422no—
286C22×C4—5422no—
287C22×C4—5422no—
288C22×C4—5422no—
289C22×C4—5422no—
290C22×C4—5422no—
291C22×C4—5422no—
292C22×C4—5422no—
293C22×C4—10841no—
294C22×C4—10841no—
295C22×C4C22×D27111yescharacteristic
296C22×C4C22×D27111yescharacteristic
297C22×C4C22×D27111yescharacteristic
298C22×C4C22×D27111yescharacteristic
299C22×C4C22×D27111yescharacteristic
300C22×C4C22×D27111yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—2714no—
305C4⋊C4—5422no—
306C4⋊C4—5422no—
307C4⋊C4—5422no—
308C4⋊C4—5422no—
309C4⋊C4—5422no—
310C4⋊C4—5422no—
311C4⋊C4—5422no—
312C4⋊C4—10841no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—5422no—
317C22⋊C4—5422no—
318C22⋊C4—5422no—
319C22⋊C4—5422no—
320C22⋊C4—10841no—
321C42—5422no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—2712no—
329C23—2712no—
330C23—2712no—
331C23—5421no—
332C23—5421no—
333C23—5421no—
334C23—5421no—
335C23Q8×D27112yes—
336C23C23×D27111yescharacteristic
337Q8—10844no—
338D4—5428no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—2712no—
356C2×C4—2712no—
357C2×C4—2712no—
358C2×C4—2712no—
359C2×C4—2712no—
360C2×C4—5428no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—5421no—
364C2×C4—5421no—
365C2×C4—5421no—
366C2×C4—5421no—
367C2×C4—5421no—
368C2×C4—5421no—
369C2×C4—5421no—
370C2×C4—5421no—
371C2×C4—5421no—
372C2×C4—5421no—
373C2×C4—5421no—
374C2×C4—5421no—
375C2×C4—5421no—
376C2×C4—5421no—
377C2×C4—10844no—
378C2×C4—10844no—
379C2×C4—10844no—
380C2×C4D4×D27112yes—
381C2×C4D4×D27112yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—2714no—
390C22—5422no—
391C22—5422no—
392C22—5422no—
393C22—5422no—
394C22—5422no—
395C22—5422no—
396C22C2×D4×D27111yescharacteristic
397C22C2×D4×D27111yescharacteristic
398C22C2×Q8×D27111yescharacteristic
399C22#41×D27111yescharacteristic
400C22#41×D27111yescharacteristic
401C22#2257620511yescharacteristic
402C22#2257620711yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—2714no—
410C4—5422no—
411C4—5422no—
412C4—10841no—
413C4—10841no—
414C4—10841no—
415C4—10841no—
416C2—42no—
417C2—5424no—
418C2#121×D27111yescharacteristic
419C2C22⋊Q8×D27111yescharacteristic
420C2#3177447611yescharacteristic
421C2#3177453611yescharacteristic
422C2#3177456011yescharacteristic
423C2#3177458311yescharacteristic
424C2#3177460111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#396×C271C2
prime radicalC22×C22⋊D271C22
hypercenter#396D271
hypocenterC271#2417
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#22576138C2semidirect
2. C2×#22576137C2semidirect
4. C2×C4⋊C4×D271C2semidirect
5. #31773830C2semidirect
6. #31773827C2semidirect
7. #31773820C2semidirect
8. #31773801C2semidirect
9. #31773792C2semidirect
12. #31773756C2semidirect
13. #31773755C2semidirect
14. #31773754C2semidirect
15. #396×C271C2semidirect
61. C2×C22⋊D271Q8semidirect
89. Dic271⋊C4D4semidirect
189. C2×D271C22⋊Q8semidirect
212. C271#2417semidirect
221. #396D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761382
C2×#225761372
C2×(C22⋊D271)⋊C42
C2×C4⋊C4×D2712
#317738302
#317738272
#317738202
#317738012
#317737922
#317737732
#317737562
#317737552
#317737542
#396×C2712
#2417271