Finite Group Databasebeta

← #46839996

Subgroups — #46839996

Order 34688.

1348 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D271C211yesmaximal, characteristic, complemented
2C2×#22576156C212yesmaximal, complemented
3C2×#22576133C211yesmaximal, characteristic, complemented
4C2×#22576130C212yesmaximal, complemented
5C2×#22576128C212yesmaximal, complemented
6#31773807C211yesmaximal, characteristic, complemented
7#31773806C211yesmaximal, characteristic, complemented
8#31773805C211yesmaximal, characteristic, complemented
9#31773797C212yesmaximal, complemented
10#31773744C211yesmaximal, characteristic, complemented
11#392×C271C211yesmaximal, characteristic, complemented
12C24×D271C2212yes—
13C22×D1084C2211yescharacteristic
14C22×D1084C2211yescharacteristic
15C22×D1084C2212yescomplemented
16C22×C22⋊D271C2211yescharacteristic
17C22×C22⋊D271C2212yes—
18C22×C22⋊D271C2211yescharacteristic, complemented
19C22×C22⋊D271C2212yescomplemented
20C22×C22⋊D271C2212yescomplemented
21C2×D4×D271—28no—
22C2×D4×D271—24no—
23C22×C4×D271C2211yescharacteristic
24#22576156—28no—
25#22576133—24no—
26#22576130—28no—
27#22576128—28no—
28C2×Dic271⋊C4C2211yescharacteristic, complemented
29C2×C22⋊Dic271C2212yescomplemented
30C2×(C2×D271)⋊C4C2212yes—
31C2×(C2×D271)⋊C4C2212yescomplemented
32C2×(C2×D271)⋊C4C2212yescomplemented
33C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
34C2×C4×Dic271C2212yescomplemented
35C271⋊#95C2211yescharacteristic, complemented
36C22×D4×C271C2211yescharacteristic
37C22×D4×C271C2212yescomplemented
38C2×C22⋊C4×C271C2212yescomplemented
39C2×C22⋊C4×C271C2211yescharacteristic, complemented
40#95×C271C2211yescharacteristic, complemented
41C23×D271—24no—
42C23×D271—24no—
43C23×D271—24no—
44C23×D271—24no—
45C23×D271—24no—
46C23×D271—42no—
47C23×D271—42no—
48C23×D271C2311yescharacteristic
49C23×D271C2312yes—
50C23×D271C2311yescharacteristic
51C2×D1084—24no—
52C2×D1084—24no—
53C2×D1084—22no—
54C2×D1084—24no—
55C2×D1084—22no—
56C2×D1084—24no—
57C2×D1084—42no—
58C2×C22⋊D271—24no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—24no—
61C2×C22⋊D271—24no—
62C2×C22⋊D271—24no—
63C2×C22⋊D271—22no—
64C2×C22⋊D271—24no—
65C2×C22⋊D271—24no—
66C2×C22⋊D271—24no—
67C2×C22⋊D271—24no—
68C2×C22⋊D271—42no—
69C2×C22⋊D271—42no—
70C2×C22⋊D271D418yescomplemented
71D4×D271—232no—
72C22×Dic271C2311yescharacteristic
73C22×Dic271C2312yes—
74C22×Dic271C2311yescharacteristic
75C2×C4×D271—24no—
76C2×C4×D271D414yescomplemented
77Dic271⋊C4—22no—
78C22⋊Dic271—24no—
79(C2×D271)⋊C4—24no—
80(C2×D271)⋊C4—22no—
81(C2×D271)⋊C4—24no—
82(C2×D271)⋊C4D418yescomplemented
83C4×Dic271—24no—
84C24×C271C2311yescharacteristic
85C24×C271C2312yes—
86C2×D4×C271—24no—
87C2×D4×C271—24no—
88C2×D4×C271—22no—
89C2×D4×C271—24no—
90C2×D4×C271—42no—
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C22×C4×C271C2312yes—
94C22⋊C4×C271—22no—
95C22⋊C4×C271—24no—
96C22×D271—22no—
97C22×D271—21no—
98C22×D271—21no—
99C22×D271—216no—
100C22×D271—22no—
101C22×D271—22no—
102C22×D271—22no—
103C22×D271—22no—
104C22×D271—22no—
105C22×D271—22no—
106C22×D271—21no—
107C22×D271—22no—
108C22×D271—22no—
109C22×D271—21no—
110C22×D271—21no—
111C22×D271—48no—
112C22×D271—48no—
113C22×D271—48no—
114C22×D271—48no—
115C22×D271—48no—
116C22×D271—48no—
117C22×D271C2×D414yes—
118C22×D271C2×D414yes—
119C22×D271C2×D412yes—
120D1084—28no—
121D1084—48no—
122D1084—44no—
123C22⋊D271—216no—
124C22⋊D271—48no—
125C22⋊D271—48no—
126C22⋊D271—44no—
127C22⋊D271—44no—
128C2×Dic271—22no—
129C2×Dic271—22no—
130C2×Dic271—22no—
131C2×Dic271—21no—
132C2×Dic271C2×D414yes—
133C2×Dic271C2×D412yes—
134C2×Dic271C2×D414yescomplemented
135C4×D271—28no—
136C23×C271—22no—
137C23×C271—22no—
138C23×C271—22no—
139C23×C271—22no—
140C23×C271—22no—
141C23×C271—22no—
142C23×C271—21no—
143C23×C271—22no—
144C23×C271—22no—
145C23×C271—21no—
146C23×C271—21no—
147C23×C271C2×D414yes—
148C23×C271C2411yescharacteristic
149D4×C271—28no—
150D4×C271—48no—
151C2×C4×C271—22no—
152C2×C4×C271—22no—
153C2×C4×C271—22no—
154C2×C4×C271—22no—
155C2×C4×C271—21no—
156C2×C4×C271C2×D412yes—
157C2×C4×C271C2×D414yescomplemented
158C22×C271—24no—
159C22×C271—24no—
160C22×C271—24no—
161C22×C271—42no—
162C22×C271—42no—
163C22×C271—42no—
164C22×C271—42no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271—41no—
168C22×C271—42no—
169C22×C271—42no—
170C22×C271C22×D411yescharacteristic
171C22×C271C22×D412yes—
172C22×C271C22×D412yes—
173C22×C271H(5,𝔽2)11yescharacteristic
174C22×C271H(5,𝔽2)11yescharacteristic
175C2×D271—22no—
176C2×D271—22no—
177C2×D271—24no—
178C2×D271—24no—
179C2×D271—24no—
180C2×D271—24no—
181C2×D271—24no—
182C2×D271—432no—
183C2×D271—42no—
184C2×D271—42no—
185C2×D271—42no—
186C2×D271—42no—
187C2×D271—41no—
188C2×D271—41no—
189C2×D271—41no—
190C2×D271—42no—
191C2×D271—42no—
192C2×D271#12714yescomplemented
193C4×C271—22no—
194C4×C271—24no—
195C4×C271—41no—
196Dic271—24no—
197Dic271—41no—
198Dic271#12014yescomplemented
199C2×C271—44no—
200C2×C271—81no—
201C2×C271C2×#12011yescharacteristic
202C2×C271C2×#12711yescharacteristic
203C2×C271#53511yescharacteristic
204C2×C271D4212yes—
205C2×C271D4212yes—
206D271—24no—
207D271—44no—
208D271—81no—
209C271#243311yescharacteristic, complemented, Sylow
210#2433—2711nomaximal, Sylow
211C23×D4—2711no—
212C2×#127—2712no—
213C2×#121—2711no—
214C2×#120—2712no—
215C2×#120—2712no—
216#392—2711no—
217#392—2711no—
218#392—2711no—
219#392D27111yescharacteristic, complemented
220#390—2712no—
221C4×C22⋊C4—2711no—
222C25—2712no—
223C22×D4—2712no—
224C22×D4—2711no—
225C22×D4—2711no—
226C22×D4—2712no—
227C22×D4—2712no—
228C22×D4—2711no—
229C22×D4—2711no—
230C22×D4—2712no—
231C22×D4—5424no—
232C22×D4—5428no—
233C22×D4C2×D27111yescharacteristic
234C22×D4C2×D27112yescomplemented
235C23×C4—2711no—
236#127—5428no—
237#121—5424no—
238#120—5428no—
239#120—5428no—
240C2×C4⋊C4—2711no—
241C2×C22⋊C4—2712no—
242C2×C22⋊C4—2712no—
243C2×C22⋊C4—2712no—
244C2×C22⋊C4—2712no—
245C2×C22⋊C4—2711no—
246C2×C22⋊C4C2×D27112yescomplemented
247C2×C22⋊C4C2×D27111yescharacteristic, complemented
248C2×C42—2712no—
249#95—2711no—
250#95C2×D27111yescharacteristic, complemented
251C24—2711no—
252C24—2712no—
253C24—2711no—
254C24—5424no—
255C24—5424no—
256C24—5424no—
257C24—5424no—
258C24—5424no—
259C24—10842no—
260C24—10842no—
261C24C22×D27111yescharacteristic
262C24C22×D27112yes—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—42no—
268C2×D4—2718no—
269C2×D4—5424no—
270C2×D4—5424no—
271C2×D4—5422no—
272C2×D4—5424no—
273C2×D4—5424no—
274C2×D4—5424no—
275C2×D4—5424no—
276C2×D4—5424no—
277C2×D4—5422no—
278C2×D4—5424no—
279C2×D4—54232no—
280C2×D4—5424no—
281C2×D4—5422no—
282C2×D4—5424no—
283C2×D4—5424no—
284C2×D4—5422no—
285C2×D4—5424no—
286C2×D4—10842no—
287C2×D4—10842no—
288C2×D4—10842no—
289C22×C4—2711no—
290C22×C4—2712no—
291C22×C4—2714no—
292C22×C4—2711no—
293C22×C4—5424no—
294C22×C4C22×D27111yescharacteristic
295C22×C4C22×D27111yescharacteristic
296C22×C4C22×D27112yes—
297C4⋊C4—5422no—
298C22⋊C4—22no—
299C22⋊C4—24no—
300C22⋊C4—2718no—
301C22⋊C4—5424no—
302C22⋊C4—5424no—
303C22⋊C4—5422no—
304C22⋊C4—5424no—
305C42—5424no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—21no—
316C23—21no—
317C23—2714no—
318C23—2712no—
319C23—2714no—
320C23—54216no—
321C23—5422no—
322C23—5421no—
323C23—5421no—
324C23—5422no—
325C23—5422no—
326C23—5422no—
327C23—5422no—
328C23—5422no—
329C23—5422no—
330C23—5421no—
331C23—5422no—
332C23—5422no—
333C23—5421no—
334C23—5421no—
335C23—10848no—
336C23—10848no—
337C23—10848no—
338C23—10848no—
339C23—10848no—
340C23—10848no—
341C23D4×D27114yes—
342C23C23×D27111yescharacteristic
343D4—28no—
344D4—48no—
345D4—5428no—
346D4—54216no—
347D4—10848no—
348D4—10844no—
349D4—10844no—
350D4—10848no—
351D4—10848no—
352D4—10844no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—2714no—
359C2×C4—2712no—
360C2×C4—2714no—
361C2×C4—5428no—
362C2×C4—5422no—
363C2×C4—5422no—
364C2×C4—5422no—
365C2×C4—5421no—
366C2×C4D4×D27112yes—
367C2×C4D4×D27114yescomplemented
368C22—24no—
369C22—24no—
370C22—24no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—2714no—
381C22—5424no—
382C22—5424no—
383C22—5422no—
384C22—5422no—
385C22—5424no—
386C22—5424no—
387C22—5424no—
388C22—108432no—
389C22—10842no—
390C22—10842no—
391C22—10842no—
392C22—10842no—
393C22—10842no—
394C22—10842no—
395C22—10841no—
396C22—10841no—
397C22—10841no—
398C22C2×D4×D27112yes—
399C22C2×D4×D27112yes—
400C22C2×D4×D27111yescharacteristic
401C22#2257620511yescharacteristic
402C22#2257620611yescharacteristic
403C4—24no—
404C4—22no—
405C4—41no—
406C4—2714no—
407C4—5424no—
408C4—10841no—
409C2—44no—
410C2—81no—
411C2—5424no—
412C2—10844no—
413C2—21681no—
414C2#120×D27111yescharacteristic
415C2#127×D27111yescharacteristic
416C2#3177446311yescharacteristic
417C2#3177452912yes—
418C2#3177453112yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#392×C271C2
hypercenter#392D271
hypocenterC271#2433
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D271C2semidirect
2. C2×#22576156C2semidirect
3. C2×#22576133C2semidirect
4. C2×#22576130C2semidirect
5. C2×#22576128C2semidirect
6. #31773807C2semidirect
7. #31773806C2semidirect
8. #31773805C2semidirect
9. #31773797C2semidirect
10. #31773744C2semidirect
11. #392×C271C2semidirect
15. C22×D1084C22semidirect
18. C22×C22⋊D271C22semidirect
19. C22×C22⋊D271C22semidirect
20. C22×C22⋊D271C22semidirect
28. C2×Dic271⋊C4C22semidirect
29. C2×C22⋊Dic271C22semidirect
31. C2×(C2×D271)⋊C4C22semidirect
32. C2×(C2×D271)⋊C4C22semidirect
33. C2×(C2×D271)⋊C4C22semidirect
34. C2×C4×Dic271C22semidirect
35. C271⋊#95C22semidirect
37. C22×D4×C271C22semidirect
38. C2×C22⋊C4×C271C22semidirect
39. C2×C22⋊C4×C271C22semidirect
40. #95×C271C22semidirect
70. C2×C22⋊D271D4semidirect
76. C2×C4×D271D4semidirect
82. (C2×D271)⋊C4D4semidirect
134. C2×Dic271C2×D4semidirect
157. C2×C4×C271C2×D4semidirect
192. C2×D271#127semidirect
198. Dic271#120semidirect
209. C271#2433semidirect
219. #392D271semidirect
234. C22×D4C2×D271semidirect
246. C2×C22⋊C4C2×D271semidirect
247. C2×C22⋊C4C2×D271semidirect
250. #95C2×D271semidirect
367. C2×C4D4×D271semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2712
C2×#225761562
C2×#225761332
C2×#225761302
C2×#225761282
#317738072
#317738062
#317738052
#317737972
#317737442
#392×C2712
#2433271