Finite Group Databasebeta

← #46846610

Subgroups — #46846610

Order 34688.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774507C211yesmaximal, characteristic, complemented
2#31774372C211yesmaximal, characteristic
3#31774239C211yesmaximal, characteristic, complemented
4#31774235C211yesmaximal, characteristic, complemented
5#31774199C211yesmaximal, characteristic, complemented
6#31774195C211yesmaximal, characteristic, complemented
7#31774100C211yesmaximal, characteristic, complemented
8#31774100C211yesmaximal, characteristic, complemented
9#31774048C211yesmaximal, characteristic
10#31773998C211yesmaximal, characteristic, complemented
11#31773994C211yesmaximal, characteristic, complemented
12#31773988C211yesmaximal, characteristic, complemented
13SD16×Dic271C211yesmaximal, characteristic, complemented
14#31773955C211yesmaximal, characteristic, complemented
15#450×C271C211yesmaximal, characteristic
16C2×#16864642C2211yescharacteristic
17C2×#16864642C2211yescharacteristic
18#22576158C2211yescharacteristic, complemented
19#22576153—21no—
20#22576153—21no—
21#22576151—21no—
22#22576151C2211yescharacteristic, complemented
23#22576150—21no—
24#22576139C2211yescharacteristic
25Q8×Dic271C2211yescharacteristic, complemented
26#22576125C2211yescharacteristic, complemented
27D4×Dic271—21no—
28D4×Dic271—21no—
29D4×Dic271C2211yescharacteristic
30D4×Dic271C2211yescharacteristic
31D4×Dic271C2211yescharacteristic
32D4×Dic271C2211yescharacteristic
33#22576119C2211yescharacteristic
34#22576119C2211yescharacteristic
35C2×C22⋊Dic271—21no—
36C2×C22⋊Dic271—21no—
37#22576094C2211yescharacteristic, complemented
38#22576093C2211yescharacteristic, complemented
39#22576092C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41Dic271⋊C8C2211yescharacteristic, complemented
42Dic271⋊C8C2211yescharacteristic, complemented
43#22576085C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576084C2211yescharacteristic, complemented
46#22576084C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C22×D4×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59C2×Dic542C2311yescharacteristic
60C22×Dic271—21no—
61C22×Dic271—21no—
62C22×Dic271—21no—
63C22×Dic271—21no—
64#16864642—22no—
65#16864642—22no—
66C2×C271⋊C8C2311yescharacteristic
67C2×C271⋊C8C2311yescharacteristic
68C4⋊Dic271—21no—
69C4⋊Dic271—21no—
70C4⋊Dic271D411yescharacteristic
71C4⋊Dic271D411yescharacteristic
72C4⋊Dic271C2311yescharacteristic
73C4⋊Dic271C2311yescharacteristic
74C4⋊Dic271C2311yescharacteristic
75Dic271⋊C4—21no—
76Dic271⋊C4—21no—
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79Dic271⋊C4—21no—
80C22⋊Dic271—21no—
81C22⋊Dic271—21no—
82C22⋊Dic271—21no—
83C22⋊Dic271—21no—
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271D411yescharacteristic
87C22⋊Dic271D411yescharacteristic
88C4×Dic271—21no—
89C4×Dic271—21no—
90C4×Dic271—21no—
91C4×Dic271C2311yescharacteristic
92C4×Dic271C2311yescharacteristic
93C24×C271—21no—
94C2×Q8×C271C2311yescharacteristic
95C2×D4×C271—22no—
96C2×D4×C271—22no—
97C2×D4×C271—21no—
98C2×D4×C271C2311yescharacteristic
99C2×D4×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101SD16×C271—22no—
102SD16×C271—22no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271—21no—
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108Dic542—21no—
109Dic542—21no—
110C2×Dic271—21no—
111C2×Dic271—21no—
112C2×Dic271—21no—
113C2×Dic271—21no—
114C2×Dic271—81no—
115C2×Dic271—81no—
116C2×Dic271—81no—
117C2×Dic271—81no—
118C2×Dic271C2×D411yescharacteristic
119C2×Dic271C2×D411yescharacteristic
120C2×Dic271C2×D411yescharacteristic
121C2×Dic271C2×D411yescharacteristic
122C271⋊C8—21no—
123C271⋊C8—21no—
124C23×C271—24no—
125C23×C271—21no—
126C23×C271—21no—
127C23×C271—81no—
128C23×C271—81no—
129C23×C271C2×D411yescharacteristic
130Q8×C271—21no—
131Q8×C271—21no—
132D4×C271—24no—
133D4×C271—21no—
134D4×C271—21no—
135D4×C271#4112yes—
136D4×C271#4112yes—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271C2×D411yescharacteristic
142C2×C4×C271C2411yescharacteristic
143C8×C271—21no—
144C8×C271—21no—
145C22×C271—21no—
146C22×C271—22no—
147C22×C271—22no—
148C22×C271—21no—
149C22×C271—21no—
150C22×C271—41no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271—84no—
155C22×C271C22×D411yescharacteristic
156C4×C271—21no—
157C4×C271—41no—
158C4×C271—41no—
159C4×C271C2×#4111yescharacteristic
160C4×C271C2×#4111yescharacteristic
161Dic271—21no—
162Dic271—21no—
163Dic271—21no—
164Dic271—21no—
165Dic271—41no—
166Dic271—41no—
167Dic271—41no—
168Dic271—41no—
169C2×C271—41no—
170C2×C271—42no—
171C2×C271—42no—
172C2×C271#52511yescharacteristic
173C2×C271#57711yescharacteristic
174C2×C271#57711yescharacteristic
175C271#312311yescharacteristic, complemented, Sylow
176#3123—2711nomaximal, Sylow
177#541—2711no—
178#518—2711no—
179#485—2711no—
180#485—2711no—
181#478—2711no—
182#478—2711no—
183#461—2711no—
184#461—2711no—
185#450—2711no—
186#450D27111yescharacteristic
187#440—2711no—
188#440—2711no—
189C4×SD16—2711no—
190C4×SD16—2711no—
191#433—2711no—
192C22×D4C2×D27111yescharacteristic
193C2×SD16—2711no—
194C2×SD16—2711no—
195C2×SD16C2×D27111yescharacteristic
196C2×SD16C2×D27111yescharacteristic
197C4⋊Q8—2711no—
198#126—5421no—
199#126—5421no—
200#125—2711no—
201#125—5421no—
202#125—5421no—
203C22⋊Q8—2711no—
204C22⋊Q8C2×D27111yescharacteristic
205C4×Q8—2711no—
206C4×Q8—2711no—
207C4×D4—2711no—
208C4×D4—2711no—
209C4×D4—2711no—
210C4×D4—2711no—
211C4×D4—5421no—
212C4×D4—5421no—
213#117—2711no—
214#117—2711no—
215C2×C22⋊C4—5421no—
216C2×C22⋊C4—5421no—
217#107—2711no—
218#107—2711no—
219#106—2711no—
220#106—2711no—
221C4⋊C8—2711no—
222C4⋊C8—2711no—
223#103—2711no—
224#103—2711no—
225#103—2711no—
226#103—2711no—
227#102—2711no—
228#102—2711no—
229#102C2×D27111yescharacteristic
230#102C2×D27111yescharacteristic
231C22⋊C8—2711no—
232C22⋊C8C2×D27111yescharacteristic
233#97—2711no—
234C4×C8—2711no—
235C24—21no—
236C2×Q8—2711no—
237C2×Q8C22×D27111yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D27111yescharacteristic
242C2×D4C22×D27111yescharacteristic
243C22×C4—5421no—
244C22×C4—5421no—
245C22×C4—5421no—
246C22×C4—5421no—
247C22×C4C22×D27111yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—5422no—
251SD16—5422no—
252C2×C8—2711no—
253C2×C8—2711no—
254C2×C8C22×D27111yescharacteristic
255C2×C8C22×D27111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—2711no—
258C4⋊C4—2711no—
259C4⋊C4—2711no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—5421no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4—5421no—
267C4⋊C4—5421no—
268C4⋊C4—5421no—
269C4⋊C4C22×D27111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—2711no—
272C22⋊C4—2711no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C22⋊C4—5421no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C42—2711no—
280C42—2711no—
281C42—5421no—
282C42—5421no—
283C42—5421no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D27111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—5421no—
293Q8—5421no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1686465312yes—
298D4#1686465312yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—2711no—
304C2×C4—2711no—
305C2×C4—2711no—
306C2×C4—2711no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—5421no—
310C2×C4—5421no—
311C2×C4—21681no—
312C2×C4—21681no—
313C2×C4—21681no—
314C2×C4—21681no—
315C2×C4D4×D27111yescharacteristic
316C2×C4C23×D27111yescharacteristic
317C8—21no—
318C8—21no—
319C8—5421no—
320C8—5421no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D27111yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—5421no—
336C4—5421no—
337C4—5421no—
338C4—5421no—
339C4—10841no—
340C4—10841no—
341C4—10841no—
342C4—10841no—
343C4C2×#1686465311yescharacteristic
344C4C2×#1686465311yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#3177440311yescharacteristic
349C2#3177470311yescharacteristic
350C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#450×C271C2
socleC22×C271C22×D4
prime radicalC22×D4×C271C22
hypercenter#450D271
hypocenterC271#3123
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #31774507C2semidirect
3. #31774239C2semidirect
4. #31774235C2semidirect
5. #31774199C2semidirect
6. #31774195C2semidirect
7. #31774100C2semidirect
8. #31774100C2semidirect
10. #31773998C2semidirect
11. #31773994C2semidirect
12. #31773988C2semidirect
13. SD16×Dic271C2semidirect
14. #31773955C2semidirect
18. #22576158C22semidirect
22. #22576151C22semidirect
25. Q8×Dic271C22semidirect
26. #22576125C22semidirect
37. #22576094C22semidirect
38. #22576093C22semidirect
39. #22576092C22semidirect
40. #22576091C22semidirect
41. Dic271⋊C8C22semidirect
42. Dic271⋊C8C22semidirect
43. #22576085C22semidirect
44. #22576085C22semidirect
45. #22576084C22semidirect
46. #22576084C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
175. C271#3123semidirect

Maximal subgroups

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

subgroupindex
#317745072
#317743722
#317742392
#317742352
#317741992
#317741952
#317741002
#317740482
#317739982
#317739942
#317739882
SD16×Dic2712
#317739552
#450×C2712
#3123271