Finite Group Databasebeta

← #46840063

Subgroups — #46840063

Order 34688.

608 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
1C2×#22576158C211yesmaximal, characteristic, complemented
2C2×#22576137C211yesmaximal, characteristic
3C22×Dic271⋊C4C211yesmaximal, characteristic
4#31773828C211yesmaximal, characteristic, complemented
5#31773827C211yesmaximal, characteristic, complemented
6#31773817C211yesmaximal, characteristic, complemented
7#31773816C211yesmaximal, characteristic, complemented
8#31773792C211yesmaximal, characteristic, complemented
9#31773792C211yesmaximal, characteristic, complemented
10#31773770C211yesmaximal, characteristic
11#31773769C211yesmaximal, characteristic
12#31773766C211yesmaximal, characteristic, complemented
13#31773755C211yesmaximal, characteristic
14C22⋊C4×Dic271C211yesmaximal, characteristic
15C2×C22⋊Q8×C271C211yesmaximal, characteristic
16C22×Dic542C2211yescharacteristic
17C23×Dic271C2211yescharacteristic
18C23×Dic271C2211yescharacteristic
19#22576158—24no—
20#22576137—24no—
21C2×C4⋊Dic271C2211yescharacteristic
22C2×Dic271⋊C4—24no—
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×Dic271⋊C4C2211yescharacteristic
32C2×Dic271⋊C4C2211yescharacteristic
33C2×Dic271⋊C4C2211yescharacteristic
34C2×Dic271⋊C4C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×C4×Dic271C2211yescharacteristic
38C2×C4×Dic271C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C22×Q8×C271C2211yescharacteristic
48C23×C4×C271C2211yescharacteristic
49C22⋊Q8×C271—24no—
50C2×C4⋊C4×C271C2211yescharacteristic
51C2×C4⋊C4×C271C2211yescharacteristic
52C2×C4⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic
55C2×Dic542—22no—
56C2×Dic542—22no—
57C2×Dic542—41no—
58C22×Dic271—22no—
59C22×Dic271—22no—
60C22×Dic271—22no—
61C22×Dic271—22no—
62C22×Dic271—41no—
63C22×Dic271Q814yescomplemented
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C4⋊Dic271—22no—
73Dic271⋊C4—22no—
74Dic271⋊C4—22no—
75Dic271⋊C4—22no—
76Dic271⋊C4—22no—
77Dic271⋊C4—22no—
78Dic271⋊C4—22no—
79Dic271⋊C4—22no—
80Dic271⋊C4—22no—
81Dic271⋊C4—22no—
82Dic271⋊C4—41no—
83Dic271⋊C4—41no—
84Dic271⋊C4D414yes—
85C22⋊Dic271—22no—
86C22⋊Dic271Q814yescomplemented
87C4×Dic271—22no—
88C4×Dic271—22no—
89C24×C271C2311yescharacteristic
90C2×Q8×C271—22no—
91C2×Q8×C271—22no—
92C2×Q8×C271—41no—
93C22×C4×C271—22no—
94C22×C4×C271—22no—
95C22×C4×C271—41no—
96C22×C4×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102C4⋊C4×C271—22no—
103C4⋊C4×C271—22no—
104C4⋊C4×C271—22no—
105C22⋊C4×C271—22no—
106C22⋊C4×C271D414yes—
107Dic542—44no—
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—28no—
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—44no—
127C2×Dic271C2×D412yes—
128C2×Dic271C2×D412yes—
129C2×Dic271C2×Q812yes—
130C2×Dic271C2×Q812yes—
131C2×Dic271C2×Q812yes—
132C2×Dic271C2×Q812yes—
133C2×Dic271#4112yes—
134C23×C271—21no—
135C23×C271—21no—
136C23×C271—21no—
137C23×C271—21no—
138C23×C271C2×D412yes—
139C23×C271C2×Q812yes—
140C23×C271C2×Q812yes—
141C23×C271C2411yescharacteristic
142Q8×C271—44no—
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—44no—
159C2×C4×C271C2×D412yes—
160C2×C4×C271C2×D412yes—
161C2×C4×C271C2×D412yes—
162C22×C271—22no—
163C22×C271—22no—
164C22×C271—22no—
165C22×C271—22no—
166C22×C271—22no—
167C22×C271—22no—
168C22×C271C4⋊Q814yescomplemented
169C22×C271C22×D411yescharacteristic
170C22×C271C22×D411yescharacteristic
171C22×C271C22×Q811yescharacteristic
172C22×C271C22×Q811yescharacteristic
173C22×C271C2×#4111yescharacteristic
174C22×C271H(5,𝔽2)11yescharacteristic
175C22×C2712−1+411yescharacteristic
176C4×C271—22no—
177C4×C271—22no—
178C4×C271—22no—
179C4×C271—41no—
180C4×C271—41no—
181C4×C271—41no—
182Dic271—22no—
183Dic271—22no—
184Dic271—22no—
185Dic271—22no—
186Dic271—41no—
187Dic271—41no—
188Dic271—41no—
189Dic271C22⋊Q814yescomplemented
190C2×C271—24no—
191C2×C271C2×C22⋊Q811yescharacteristic
192C2×C271C2×C4⋊Q811yescharacteristic
193C2×C271#54411yescharacteristic
194C2×C271D4211yescharacteristic
195C2×C271#54711yescharacteristic
196C2×C271D4×Q811yescharacteristic
197C2×C271#55411yescharacteristic
198C271#244011yescharacteristic, complemented, Sylow
199#2440—2711nomaximal, Sylow
200C2×C4⋊Q8—2711no—
201C2×C22⋊Q8—2711no—
202C2×C22⋊Q8D27111yescharacteristic
203C22×C4⋊C4—2711no—
204#398—2711no—
205#398—2711no—
206#395—2711no—
207#395—2711no—
208#389—2711no—
209#389—2711no—
210#385—2711no—
211#385—2711no—
212#384—2711no—
213#380—2711no—
214C4×C22⋊C4—2711no—
215C22×Q8—2711no—
216C22×Q8C2×D27111yescharacteristic
217C23×C4—2711no—
218C23×C4—2711no—
219C23×C4C2×D27111yescharacteristic
220C4⋊Q8—5424no—
221C22⋊Q8—24no—
222C22⋊Q8—5424no—
223C2×C4⋊C4—2711no—
224C2×C4⋊C4—2711no—
225C2×C4⋊C4—2711no—
226C2×C4⋊C4—2711no—
227C2×C4⋊C4—2711no—
228C2×C4⋊C4—2711no—
229C2×C4⋊C4—2711no—
230C2×C4⋊C4—2711no—
231C2×C4⋊C4—2711no—
232C2×C4⋊C4—2711no—
233C2×C4⋊C4—2711no—
234C2×C4⋊C4—2711no—
235C2×C4⋊C4—2711no—
236C2×C4⋊C4—5424no—
237C2×C4⋊C4C2×D27111yescharacteristic
238C2×C4⋊C4C2×D27111yescharacteristic
239C2×C4⋊C4C2×D27111yescharacteristic
240C2×C22⋊C4—2711no—
241C2×C22⋊C4—2711no—
242C2×C22⋊C4C2×D27111yescharacteristic
243C2×C22⋊C4C2×D27111yescharacteristic
244C2×C42—2711no—
245C2×C42—2711no—
246#95—2711no—
247#95—2711no—
248#95—2711no—
249#95—2711no—
250#95—2711no—
251#95—2711no—
252#95—2711no—
253#95—2711no—
254C24C22×D27111yescharacteristic
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—5422no—
259C2×Q8—5422no—
260C2×Q8—10841no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—2711no—
265C22×C4—2714no—
266C22×C4—2711no—
267C22×C4—2711no—
268C22×C4—2711no—
269C22×C4—2711no—
270C22×C4—2711no—
271C22×C4—2711no—
272C22×C4—2711no—
273C22×C4—5422no—
274C22×C4—5422no—
275C22×C4—5422no—
276C22×C4—5422no—
277C22×C4—10841no—
278C22×C4C22×D27111yescharacteristic
279C22×C4C22×D27111yescharacteristic
280C22×C4C22×D27111yescharacteristic
281C22×C4C22×D27111yescharacteristic
282C22×C4C22×D27111yescharacteristic
283C22×C4C22×D27111yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—2714no—
288C4⋊C4—5422no—
289C4⋊C4—5422no—
290C4⋊C4—5422no—
291C4⋊C4—5422no—
292C4⋊C4—5422no—
293C4⋊C4—5422no—
294C4⋊C4—5422no—
295C4⋊C4—5422no—
296C4⋊C4—5422no—
297C4⋊C4—5422no—
298C4⋊C4—10841no—
299C4⋊C4—10841no—
300C22⋊C4—22no—
301C22⋊C4—2714no—
302C22⋊C4—5422no—
303C22⋊C4C22⋊D27114yes—
304C42—5422no—
305C42—5422no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23C2×C22⋊D27112yes—
311C23Q8×D27112yes—
312C23Q8×D27112yes—
313C23C23×D27111yescharacteristic
314Q8—44no—
315Q8—10844no—
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—44no—
332C2×C4—2712no—
333C2×C4—2712no—
334C2×C4—2712no—
335C2×C4—2712no—
336C2×C4—2712no—
337C2×C4—2712no—
338C2×C4—2712no—
339C2×C4—5428no—
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—5421no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—5421no—
356C2×C4—5421no—
357C2×C4—10844no—
358C2×C4D4×D27112yes—
359C2×C4C2×C22⋊D27112yes—
360C2×C4C2×C22⋊D27112yes—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22#2257615814yescomplemented
368C22C2×D4×D27111yescharacteristic
369C22C22×C22⋊D27111yescharacteristic
370C22C2×Q8×D27111yescharacteristic
371C22C2×Q8×D27111yescharacteristic
372C22#41×D27111yescharacteristic
373C22#2257620511yescharacteristic
374C22#2257620711yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—2714no—
382C4—5422no—
383C4—5422no—
384C4—5422no—
385C4—5422no—
386C4—10841no—
387C4—10841no—
388C4—10841no—
389C2—24no—
390C2C22⋊Q8×D27111yescharacteristic
391C2C2×#2257615811yescharacteristic
392C2#3177452611yescharacteristic
393C2D4×C22⋊D27111yescharacteristic
394C2#3177454611yescharacteristic
395C2#3177456011yescharacteristic
396C2#3177460111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×C22⋊Q8×C271C2
prime radicalC24×C271C23
hypercenterC2×C22⋊Q8D271
hypocenterC271#2440
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22576158C2semidirect
4. #31773828C2semidirect
5. #31773827C2semidirect
6. #31773817C2semidirect
7. #31773816C2semidirect
8. #31773792C2semidirect
9. #31773792C2semidirect
12. #31773766C2semidirect
63. C22×Dic271Q8semidirect
86. C22⋊Dic271Q8semidirect
168. C22×C271C4⋊Q8semidirect
189. Dic271C22⋊Q8semidirect
198. C271#2440semidirect
367. C22#22576158semidirect

Maximal subgroups

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

subgroupindex
C2×#225761582
C2×#225761372
C22×Dic271⋊C42
#317738282
#317738272
#317738172
#317738162
#317737922
#317737702
#317737692
#317737662
#317737552
C22⋊C4×Dic2712
C2×C22⋊Q8×C2712
#2440271