Finite Group Databasebeta

← #46846649

Subgroups — #46846649

Order 34688.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774500C211yesmaximal, characteristic
2#31774375C211yesmaximal, characteristic, complemented
3#31774222C211yesmaximal, characteristic, complemented
4#31774218C211yesmaximal, characteristic, complemented
5#31774206C211yesmaximal, characteristic, complemented
6#31774202C211yesmaximal, characteristic, complemented
7#31774095C211yesmaximal, characteristic, complemented
8#31774095C211yesmaximal, characteristic, complemented
9#31774036C211yesmaximal, characteristic, complemented
10#31774000C211yesmaximal, characteristic, complemented
11#31773998C211yesmaximal, characteristic, complemented
12SD16×Dic271C211yesmaximal, characteristic, complemented
13#31773982C211yesmaximal, characteristic, complemented
14#31773955C211yesmaximal, characteristic, complemented
15#448×C271C211yesmaximal, characteristic, complemented
16C2×#16864641C2211yescharacteristic
17C2×#16864641C2211yescharacteristic
18#22576156C2211yescharacteristic
19#22576151C2211yescharacteristic
20#22576144—21no—
21#22576142—21no—
22#22576134C2211yescharacteristic
23#22576132—21no—
24#22576131—21no—
25Q8×Dic271—21no—
26Q8×Dic271—21no—
27Q8×Dic271C2211yescharacteristic
28Q8×Dic271C2211yescharacteristic
29Q8×Dic271C2211yescharacteristic
30Q8×Dic271C2211yescharacteristic
31D4×Dic271C2211yescharacteristic
32#22576122C2211yescharacteristic
33#22576119—21no—
34#22576119—21no—
35#22576119C2211yescharacteristic
36#22576119C2211yescharacteristic
37#22576094C2211yescharacteristic
38#22576093C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576091C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50#22576070C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C22×Q8×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59C2×D1084C2311yescharacteristic
60C2×C22⋊D271—21no—
61C2×C22⋊D271—21no—
62C2×C22⋊D271—21no—
63C22×Dic271—21no—
64C2×C4×D271—21no—
65#16864641—22no—
66#16864641—22no—
67C2×C271⋊C8C2311yescharacteristic
68C2×C271⋊C8C2311yescharacteristic
69C4⋊Dic271—21no—
70C4⋊Dic271—21no—
71C4⋊Dic271—21no—
72C4⋊Dic271—21no—
73C4⋊Dic271—21no—
74C4⋊Dic271D411yescharacteristic
75C4⋊Dic271D411yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79Dic271⋊C4—21no—
80C22⋊Dic271—21no—
81C22⋊Dic271—21no—
82C22⋊Dic271D411yescharacteristic
83C22⋊Dic271D411yescharacteristic
84(C2×D271)⋊C4—21no—
85(C2×D271)⋊C4—21no—
86C4×Dic271—21no—
87C4×Dic271—21no—
88C4×Dic271—21no—
89C4×Dic271—21no—
90C4×Dic271—21no—
91C4×Dic271C2311yescharacteristic
92C4×Dic271C2311yescharacteristic
93C2×Q8×C271—22no—
94C2×Q8×C271—22no—
95C2×Q8×C271—21no—
96C2×Q8×C271C2311yescharacteristic
97C2×Q8×C271C2311yescharacteristic
98C2×D4×C271—21no—
99C2×D4×C271C2311yescharacteristic
100C22×C4×C271—21no—
101C22×C4×C271C2311yescharacteristic
102SD16×C271—22no—
103SD16×C271—22no—
104C2×C8×C271C2311yescharacteristic
105C2×C8×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108C22×D271—21no—
109D1084—21no—
110D1084—21no—
111C22⋊D271—81no—
112C22⋊D271—81no—
113C22⋊D271—81no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—81no—
118C2×Dic271C2×D411yescharacteristic
119C2×Dic271C2×D411yescharacteristic
120C2×Dic271C2×D411yescharacteristic
121C2×Dic271C2×D411yescharacteristic
122C4×D271—81no—
123C271⋊C8—21no—
124C271⋊C8—21no—
125C23×C271—21no—
126C23×C271C2×D411yescharacteristic
127Q8×C271—24no—
128Q8×C271—21no—
129Q8×C271—21no—
130Q8×C271#4112yes—
131Q8×C271#4112yes—
132D4×C271—21no—
133D4×C271—21no—
134D4×C271—81no—
135C2×C4×C271—24no—
136C2×C4×C271—21no—
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—21no—
147C22×C271—21no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271—41no—
151C22×C271C22×D411yescharacteristic
152C2×D271—41no—
153C2×D271—41no—
154C2×D271—41no—
155C4×C271—22no—
156C4×C271—22no—
157C4×C271—21no—
158C4×C271—41no—
159C4×C271C2×#4111yescharacteristic
160C4×C271C2×#4111yescharacteristic
161Dic271—21no—
162Dic271—21no—
163Dic271—21no—
164Dic271—21no—
165Dic271—41no—
166Dic271—41no—
167Dic271—41no—
168C2×C271—41no—
169C2×C271—81no—
170C2×C271#52511yescharacteristic
171C2×C271#57711yescharacteristic
172C2×C271#57711yescharacteristic
173D271—81no—
174C271#312611yescharacteristic, complemented, Sylow
175#3126—2711nomaximal, Sylow
176#540—2711no—
177#519—2711no—
178#482—2711no—
179#482—2711no—
180#479—2711no—
181#479—2711no—
182#460—2711no—
183#460—2711no—
184#448—2711no—
185#448D27111yescharacteristic, complemented
186#440—2711no—
187#440—2711no—
188C4×SD16—2711no—
189C4×SD16—2711no—
190#433—2711no—
191C22×Q8C2×D27111yescharacteristic
192C2×SD16—2711no—
193C2×SD16—2711no—
194C2×SD16C2×D27111yescharacteristic
195C2×SD16C2×D27111yescharacteristic
196#127—2711no—
197#125—2711no—
198#123—5421no—
199#123—5421no—
200#121—2711no—
201#121—5421no—
202#121—5421no—
203#121C2×D27111yescharacteristic
204C4×Q8—2711no—
205C4×Q8—2711no—
206C4×Q8—2711no—
207C4×Q8—2711no—
208C4×Q8—5421no—
209C4×Q8—5421no—
210C4×D4—2711no—
211C4×D4—2711no—
212#117—2711no—
213#117—2711no—
214#117—5421no—
215#117—5421no—
216#107—2711no—
217#107—2711no—
218#106—2711no—
219#106—2711no—
220C4⋊C8—2711no—
221C4⋊C8—2711no—
222#103—2711no—
223#103—2711no—
224#103C2×D27111yescharacteristic
225#103C2×D27111yescharacteristic
226#102—2711no—
227#102—2711no—
228#102—2711no—
229#102—2711no—
230C22⋊C8—2711no—
231C22⋊C8C2×D27111yescharacteristic
232#97—2711no—
233C4×C8—2711no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D27111yescharacteristic
238C2×Q8C22×D27111yescharacteristic
239C2×D4—21no—
240C2×D4—2711no—
241C2×D4—5421no—
242C2×D4—5421no—
243C2×D4—5421no—
244C2×D4C22×D27111yescharacteristic
245C22×C4—21no—
246C22×C4—5421no—
247C22×C4—5421no—
248C22×C4C22×D27111yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—5422no—
252SD16—5422no—
253C2×C8—2711no—
254C2×C8—2711no—
255C2×C8C22×D27111yescharacteristic
256C2×C8C22×D27111yescharacteristic
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⋊C4C22×D27111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—2711no—
271C22⋊C4—2711no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C42—2711no—
277C42—2711no—
278C42—5421no—
279C42—5421no—
280C42—5421no—
281C42—5421no—
282C42—5421no—
283C23—21no—
284C23—5421no—
285C23D4×D27111yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1686465312yes—
290Q8#1686465312yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—5421no—
295D4—5421no—
296D4—21681no—
297D4—21681no—
298D4—21681no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—2711no—
306C2×C4—2711no—
307C2×C4—2711no—
308C2×C4—2711no—
309C2×C4—5421no—
310C2×C4—5421no—
311C2×C4—5421no—
312C2×C4—21681no—
313C2×C4—21681no—
314C2×C4D4×D27111yescharacteristic
315C2×C4C23×D27111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5421no—
319C8—5421no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—10841no—
327C22—10841no—
328C22—10841no—
329C22C2×D4×D27111yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—5421no—
335C4—5421no—
336C4—5421no—
337C4—5421no—
338C4—10841no—
339C4—10841no—
340C4—10841no—
341C4C2×#1686465311yescharacteristic
342C4C2×#1686465311yescharacteristic
343C2—41no—
344C2—81no—
345C2—21681no—
346C2#3177440311yescharacteristic
347C2#3177470311yescharacteristic
348C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#448×C271C2
socleC22×C271C22×D4
prime radical#31774500C2
hypercenter#448D271
hypocenterC271#3126
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #31774375C2semidirect
3. #31774222C2semidirect
4. #31774218C2semidirect
5. #31774206C2semidirect
6. #31774202C2semidirect
7. #31774095C2semidirect
8. #31774095C2semidirect
9. #31774036C2semidirect
10. #31774000C2semidirect
11. #31773998C2semidirect
12. SD16×Dic271C2semidirect
13. #31773982C2semidirect
14. #31773955C2semidirect
15. #448×C271C2semidirect
174. C271#3126semidirect
185. #448D271semidirect

Maximal subgroups

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

subgroupindex
#317745002
#317743752
#317742222
#317742182
#317742062
#317742022
#317740952
#317740362
#317740002
#317739982
SD16×Dic2712
#317739822
#317739552
#448×C2712
#3126271