Finite Group Databasebeta

← #46839925

Subgroups — #46839925

Order 34688.

632 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576146C211yesmaximal, characteristic, complemented
2C2×#22576138C211yesmaximal, characteristic, complemented
3C2×#22576133C211yesmaximal, characteristic
4C2×#22576125C211yesmaximal, characteristic, complemented
5#31773821C211yesmaximal, characteristic, complemented
6#31773814C211yesmaximal, characteristic, complemented
7#31773812C211yesmaximal, characteristic, complemented
8#31773798C211yesmaximal, characteristic, complemented
9#31773797C211yesmaximal, characteristic, complemented
10#31773797C211yesmaximal, characteristic, complemented
11#31773792C211yesmaximal, characteristic, complemented
12#31773787C211yesmaximal, characteristic, complemented
13#31773785C211yesmaximal, characteristic, complemented
14#31773744C211yesmaximal, characteristic, complemented
15#396×C271C211yesmaximal, characteristic, complemented
16C22×Dic542C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21#22576146—24no—
22#22576138—24no—
23#22576133—24no—
24#22576125—24no—
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×C22⋊Dic271C2211yescharacteristic
31C2×C22⋊Dic271C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C2×C4×Dic271C2211yescharacteristic
42C2×C4×Dic271C2211yescharacteristic
43C2×C4×Dic271C2211yescharacteristic
44C2×C4×Dic271C2211yescharacteristic
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
56C23×D271C2311yescharacteristic
57C2×Dic542—22no—
58C2×Dic542—22no—
59C2×Dic542D414yescomplemented
60C2×D1084—22no—
61C2×D1084—22no—
62C2×D1084—41no—
63C2×C22⋊D271—22no—
64C2×C22⋊D271—22no—
65C2×C22⋊D271—22no—
66C2×C22⋊D271—22no—
67C2×C22⋊D271—22no—
68C2×C22⋊D271—41no—
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C22×Dic271C2311yescharacteristic
73C22×Dic271C2311yescharacteristic
74C22×Dic271C2311yescharacteristic
75C2×C4×D271—22no—
76C2×C4×D271—22no—
77C2×C4×D271—22no—
78Dic271⋊C4—22no—
79Dic271⋊C4—22no—
80Dic271⋊C4—22no—
81Dic271⋊C4—41no—
82Dic271⋊C4D414yes—
83C22⋊Dic271—22no—
84C22⋊Dic271—22no—
85(C2×D271)⋊C4—22no—
86(C2×D271)⋊C4—22no—
87(C2×D271)⋊C4—22no—
88(C2×D271)⋊C4—22no—
89(C2×D271)⋊C4—22no—
90(C2×D271)⋊C4—22no—
91(C2×D271)⋊C4—22no—
92(C2×D271)⋊C4—22no—
93(C2×D271)⋊C4—41no—
94C4×Dic271—22no—
95C4×Dic271—22no—
96C4×Dic271—22no—
97C4×Dic271—22no—
98C24×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101C22×C4×C271C2311yescharacteristic
102C22×C4×C271C2311yescharacteristic
103C22×C4×C271C2311yescharacteristic
104C22×C4×C271C2311yescharacteristic
105C4⋊C4×C271—22no—
106C4⋊C4×C271—22no—
107C4⋊C4×C271—41no—
108C22⋊C4×C271—22no—
109C22⋊C4×C271—22no—
110C22⋊C4×C271—41no—
111C22×D271—21no—
112C22×D271—21no—
113C22×D271—21no—
114C22×D271—21no—
115C22×D271—21no—
116C22×D271—21no—
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—21no—
122C22×D271—21no—
123C22×D271—21no—
124C22×D271#4112yes—
125Dic542—28no—
126D1084—44no—
127C22⋊D271—44no—
128C22⋊D271—44no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271C2×D412yes—
142C2×Dic271C2×D412yes—
143C2×Dic271C2×D412yes—
144C2×Dic271C2×D412yes—
145C2×Dic271#4112yes—
146C2×Dic271#4112yes—
147C4×D271—44no—
148C23×C271—21no—
149C23×C271—21no—
150C23×C271—21no—
151C23×C271—21no—
152C23×C271—21no—
153C23×C271—21no—
154C23×C271—21no—
155C23×C271C2411yescharacteristic
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×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271C2×D412yes—
173C2×C4×C271C2×D412yes—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271C22×D411yescharacteristic
182C22×C271C22×D411yescharacteristic
183C22×C271C2×#4111yescharacteristic
184C22×C271C2×#4111yescharacteristic
185C22×C271C2×#4111yescharacteristic
186C22×C271H(5,𝔽2)11yescharacteristic
187C22×C2712−1+411yescharacteristic
188C2×D271—22no—
189C2×D271—22no—
190C2×D271—22no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—41no—
196C2×D271—41no—
197C2×D271—41no—
198C2×D271—41no—
199C2×D271—41no—
200C2×D271—41no—
201C2×D271—41no—
202C4×C271—22no—
203C4×C271—22no—
204C4×C271—41no—
205C4×C271—41no—
206C4×C271—41no—
207C4×C271—41no—
208Dic271—22no—
209Dic271—22no—
210Dic271—22no—
211Dic271—41no—
212Dic271—41no—
213Dic271#12114yes—
214C2×C271—81no—
215C2×C271C2×#12111yescharacteristic
216C2×C271#52911yescharacteristic
217C2×C271#54211yescharacteristic
218C2×C271#54611yescharacteristic
219C2×C271#54811yescharacteristic
220C2×C271#54811yescharacteristic
221C2×C271#55011yescharacteristic
222D271—42no—
223D271—81no—
224C271#242611yescharacteristic, complemented, Sylow
225#2426—2711nomaximal, Sylow
226C2×#124—2711no—
227C2×C22⋊Q8—2711no—
228C2×#121—2711no—
229C2×C4×Q8—2711no—
230#396—2711no—
231#396D27111yescharacteristic, complemented
232#394—2711no—
233#394—2711no—
234#390—2711no—
235#390—2711no—
236#390—2711no—
237#389—2711no—
238#388—2711no—
239#388—2711no—
240C4×C22⋊C4—2711no—
241C22×Q8—2711no—
242C22×D4—2711no—
243C22×D4—2711no—
244C22×D4—2711no—
245C23×C4—2711no—
246#124—5424no—
247C22⋊Q8—5424no—
248#121—5424no—
249C4×Q8—5424no—
250C2×C4⋊C4—2711no—
251C2×C4⋊C4—2711no—
252C2×C4⋊C4—2711no—
253C2×C4⋊C4—2711no—
254C2×C4⋊C4—2711no—
255C2×C4⋊C4C2×D27111yescharacteristic
256C2×C4⋊C4C2×D27111yescharacteristic
257C2×C4⋊C4C2×D27111yescharacteristic
258C2×C22⋊C4—2711no—
259C2×C22⋊C4—2711no—
260C2×C22⋊C4—2711no—
261C2×C22⋊C4—2711no—
262C2×C22⋊C4—2711no—
263C2×C22⋊C4—2711no—
264C2×C22⋊C4—2711no—
265C2×C22⋊C4—2711no—
266C2×C22⋊C4—2711no—
267C2×C22⋊C4—2711no—
268C2×C22⋊C4—2711no—
269C2×C22⋊C4C2×D27111yescharacteristic
270C2×C22⋊C4C2×D27111yescharacteristic
271C2×C22⋊C4C2×D27111yescharacteristic
272C2×C42—2711no—
273C2×C42—2711no—
274C2×C42—2711no—
275C2×C42—2711no—
276#95—2711no—
277#95—2711no—
278#95—2711no—
279#95C2×D27111yescharacteristic
280C24—2711no—
281C24—2711no—
282C24C22×D27111yescharacteristic
283C2×Q8—2714no—
284C2×Q8—5422no—
285C2×Q8—5422no—
286C2×D4—5422no—
287C2×D4—5422no—
288C2×D4—5422no—
289C2×D4—5422no—
290C2×D4—5422no—
291C2×D4—5422no—
292C2×D4—5422no—
293C2×D4—10841no—
294C2×D4—10841no—
295C22×C4—2711no—
296C22×C4—2711no—
297C22×C4—2711no—
298C22×C4—2711no—
299C22×C4—2711no—
300C22×C4—2711no—
301C22×C4—5422no—
302C22×C4—5422no—
303C22×C4—5422no—
304C22×C4C22×D27111yescharacteristic
305C22×C4C22×D27111yescharacteristic
306C22×C4C22×D27111yescharacteristic
307C22×C4C22×D27111yescharacteristic
308C22×C4C22×D27111yescharacteristic
309C22×C4C22×D27111yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—2714no—
314C4⋊C4—5422no—
315C4⋊C4—5422no—
316C4⋊C4—5422no—
317C4⋊C4—10841no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—5422no—
322C22⋊C4—5422no—
323C22⋊C4—5422no—
324C22⋊C4—5422no—
325C22⋊C4—5422no—
326C22⋊C4—5422no—
327C22⋊C4—5422no—
328C22⋊C4—5422no—
329C22⋊C4—5422no—
330C22⋊C4—5422no—
331C22⋊C4—10841no—
332C42—5422no—
333C42—5422no—
334C42—5422no—
335C42—5422no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—2712no—
344C23—5421no—
345C23—5421no—
346C23—5421no—
347C23—5421no—
348C23—5421no—
349C23—5421no—
350C23—5421no—
351C23—5421no—
352C23—5421no—
353C23—5421no—
354C23—5421no—
355C23—5421no—
356C23—5421no—
357C23C23×D27111yescharacteristic
358Q8—5428no—
359D4—10844no—
360D4—10844no—
361D4—10844no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—2712no—
379C2×C4—2712no—
380C2×C4—2712no—
381C2×C4—2712no—
382C2×C4—2712no—
383C2×C4—2712no—
384C2×C4—5421no—
385C2×C4—5421no—
386C2×C4—5421no—
387C2×C4—5421no—
388C2×C4—5421no—
389C2×C4—5421no—
390C2×C4—5421no—
391C2×C4—5421no—
392C2×C4—5421no—
393C2×C4—5421no—
394C2×C4—5421no—
395C2×C4—5421no—
396C2×C4—10844no—
397C2×C4D4×D27112yes—
398C2×C4D4×D27112yes—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—5422no—
407C22—5422no—
408C22—5422no—
409C22—10841no—
410C22—10841no—
411C22—10841no—
412C22—10841no—
413C22—10841no—
414C22—10841no—
415C22—10841no—
416C22—10841no—
417C22—10841no—
418C22—10841no—
419C22—10841no—
420C22C2×D4×D27111yescharacteristic
421C22C2×D4×D27111yescharacteristic
422C22#41×D27111yescharacteristic
423C22#41×D27111yescharacteristic
424C22#41×D27111yescharacteristic
425C22#2257620511yescharacteristic
426C22#2257620811yescharacteristic
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—2714no—
434C4—5422no—
435C4—5422no—
436C4—5422no—
437C4—10841no—
438C4—10841no—
439C2—81no—
440C2—10842no—
441C2—21681no—
442C2#121×D27111yescharacteristic
443C2#3177443011yescharacteristic
444C2#3177451711yescharacteristic
445C2#3177453611yescharacteristic
446C2#3177455711yescharacteristic
447C2#3177455711yescharacteristic
448C2#3177456511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#396×C271C2
prime radicalC2×#22576133C2
hypercenter#396D271
hypocenterC271#2426
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#22576146C2semidirect
2. C2×#22576138C2semidirect
4. C2×#22576125C2semidirect
5. #31773821C2semidirect
6. #31773814C2semidirect
7. #31773812C2semidirect
8. #31773798C2semidirect
9. #31773797C2semidirect
10. #31773797C2semidirect
11. #31773792C2semidirect
12. #31773787C2semidirect
13. #31773785C2semidirect
14. #31773744C2semidirect
15. #396×C271C2semidirect
59. C2×Dic542D4semidirect
224. C271#2426semidirect
231. #396D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761462
C2×#225761382
C2×#225761332
C2×#225761252
#317738212
#317738142
#317738122
#317737982
#317737972
#317737922
#317737872
#317737852
#317737442
#396×C2712
#2426271