Finite Group Databasebeta

← #46846639

Subgroups — #46846639

Order 34688.

384 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#31774519C211yesmaximal, characteristic, complemented
2#31774379C211yesmaximal, characteristic, complemented
3#31774226C211yesmaximal, characteristic, complemented
4#31774217C211yesmaximal, characteristic, complemented
5#31774186C211yesmaximal, characteristic, complemented
6#31774180C211yesmaximal, characteristic, complemented
7#31774121C211yesmaximal, characteristic, complemented
8#31774115C211yesmaximal, characteristic, complemented
9#31774058C211yesmaximal, characteristic, complemented
10#31774045C211yesmaximal, characteristic
11#31774010C211yesmaximal, characteristic, complemented
12C4×#16864641C211yesmaximal, characteristic, complemented
13C4×#16864641C211yesmaximal, characteristic, complemented
14#31773959C211yesmaximal, characteristic, complemented
15#441×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864641C2211yescharacteristic
18C2×#16864641C2211yescharacteristic
19C2×D2168C2211yescharacteristic
20#22576159C2211yescharacteristic
21#22576154—21no—
22#22576149C2211yescharacteristic
23#22576148—21no—
24#22576145—21no—
25#22576138—21no—
26#22576136C2211yescharacteristic
27#22576135C2211yescharacteristic
28Q8×Dic271C2211yescharacteristic
29C4×D1084—21no—
30C4×D1084C2211yescharacteristic
31C4×D1084C2211yescharacteristic
32#22576122—21no—
33#22576122C2211yescharacteristic
34#22576117—21no—
35#22576117C2211yescharacteristic
36C4⋊C4×D271—21no—
37C4⋊C4×D271C2211yescharacteristic
38#22576092C2211yescharacteristic
39#22576091C2211yescharacteristic
40#22576091C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C4×C271⋊C8C2211yescharacteristic
52C2×Q16×C271C2211yescharacteristic
53C4×Q8×C271C2211yescharacteristic
54C4×Q8×C271C2211yescharacteristic
55#106×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—22no—
61C2×Dic542—21no—
62C2×D1084—21no—
63C2×D1084—21no—
64C2×D1084C2311yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×C4×D271—21no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271C2311yescharacteristic
72#16864641—22no—
73#16864641—22no—
74D2168—22no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C4⋊Dic271—21no—
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84Dic271⋊C4D411yescharacteristic
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4D411yescharacteristic
93(C2×D271)⋊C4D411yescharacteristic
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96C2×Q8×C271C2311yescharacteristic
97C2×Q8×C271C2311yescharacteristic
98Q16×C271—22no—
99C2×C8×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C4⋊C4×C271—21no—
102C4⋊C4×C271—21no—
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C42×C271—21no—
106C42×C271—21no—
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109C22×D271—21no—
110C22×D271C2×D411yescharacteristic
111Dic542—81no—
112D1084—24no—
113D1084—21no—
114D1084—21no—
115D1084—21no—
116D1084—81no—
117D1084#4112yes—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271C2×D411yescharacteristic
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C4×D271—24no—
124C4×D271—21no—
125C4×D271—21no—
126C4×D271—81no—
127C4×D271—81no—
128C4×D271—81no—
129C4×D271—81no—
130C271⋊C8—21no—
131C271⋊C8—21no—
132Q8×C271—21no—
133Q8×C271—21no—
134Q8×C271—21no—
135Q8×C271#4112yes—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2×D411yescharacteristic
142C2×C4×C271C2411yescharacteristic
143C8×C271—21no—
144C8×C271—21no—
145C22×C271C22×D411yescharacteristic
146C2×D271—21no—
147C2×D271—22no—
148C2×D271—21no—
149C2×D271—21no—
150C2×D271—41no—
151C2×D271—41no—
152C2×D271—41no—
153C2×D271—41no—
154C2×D271—41no—
155C4×C271—22no—
156C4×C271—21no—
157C4×C271—21no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271—41no—
161C4×C271C2×#4111yescharacteristic
162C4×C271C2×#4111yescharacteristic
163Dic271—21no—
164Dic271—21no—
165Dic271—21no—
166Dic271—41no—
167Dic271—41no—
168C2×C271#52511yescharacteristic
169C2×C271#57711yescharacteristic
170C2×C271#57711yescharacteristic
171D271—41no—
172D271—42no—
173D271—81no—
174C271#312511yescharacteristic, complemented, Sylow
175#3125—2711nomaximal, Sylow
176#542—2711no—
177#520—2711no—
178#483—2711no—
179#481—2711no—
180#476—2711no—
181#475—2711no—
182#464—2711no—
183#463—2711no—
184#452—2711no—
185#449—2711no—
186#442—2711no—
187#441D27111yescharacteristic, complemented
188C4×SD16—2711no—
189C4×SD16—2711no—
190#433—2711no—
191C2×#41—2711no—
192C2×Q16C2×D27111yescharacteristic
193C2×SD16—2711no—
194C2×SD16—2711no—
195C2×D8—2711no—
196C4⋊Q8—2711no—
197#126—5421no—
198#124—2711no—
199#124—5421no—
200#123—5421no—
201C22⋊Q8—2711no—
202C22⋊Q8—5421no—
203#121—2711no—
204C4×Q8—2711no—
205C4×Q8C2×D27111yescharacteristic
206C4×Q8C2×D27111yescharacteristic
207C4×D4—2711no—
208C4×D4—2711no—
209C4×D4—2711no—
210C4×D4—5421no—
211C4×D4—5421no—
212#117—2711no—
213#117—5421no—
214C2×C4⋊C4—2711no—
215C2×C4⋊C4—5421no—
216#106—2711no—
217#106—2711no—
218#106—2711no—
219#106C2×D27111yescharacteristic
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⋊C8—2711no—
232#97C2×D27111yescharacteristic
233C4×C8—2711no—
234#41—5422no—
235#41—5422no—
236C2×Q8—5421no—
237C2×Q8C22×D27111yescharacteristic
238C2×Q8C22×D27111yescharacteristic
239C2×D4—2711no—
240C2×D4—2711no—
241C2×D4—5421no—
242C2×D4—5421no—
243C22×C4—2711no—
244C22×C4—5421no—
245C22×C4—5421no—
246C22×C4—5421no—
247C22×C4—5421no—
248C22×C4—5421no—
249Q16—22no—
250SD16—5422no—
251SD16—5422no—
252D8—5422no—
253C2×C8—2711no—
254C2×C8—2711no—
255C2×C8C22×D27111yescharacteristic
256C2×C8C22×D27111yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—2711no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—2711no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4—5421no—
267C4⋊C4C22×D27111yescharacteristic
268C4⋊C4C22×D27111yescharacteristic
269C22⋊C4—2711no—
270C22⋊C4—2711no—
271C22⋊C4—5421no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C22⋊C4—5421no—
277C22⋊C4—5421no—
278C42—21no—
279C42—21no—
280C42—2711no—
281C42—5421no—
282C42C22×D27111yescharacteristic
283C23—2711no—
284C23—5421no—
285C23—5421no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21681no—
290Q8#1686465412yes—
291D4—2712no—
292D4—5424no—
293D4—5421no—
294D4—5421no—
295D4—5421no—
296D4—21681no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—2711no—
302C2×C4—2711no—
303C2×C4—2711no—
304C2×C4—5424no—
305C2×C4—5421no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—21681no—
310C2×C4—21681no—
311C2×C4—21681no—
312C2×C4—21681no—
313C2×C4D4×D27111yescharacteristic
314C2×C4D4×D27111yescharacteristic
315C2×C4C23×D27111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5421no—
319C8—5421no—
320C22—5422no—
321C22—5421no—
322C22—5421no—
323C22—5421no—
324C22—10841no—
325C22—10841no—
326C22—10841no—
327C22—10841no—
328C22—10841no—
329C22C2×D4×D27111yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—5421no—
337C4—5421no—
338C4—5421no—
339C4—10841no—
340C4—10841no—
341C4#41×D27111yescharacteristic
342C4C2×#1686465411yescharacteristic
343C2—10841no—
344C2—10842no—
345C2—21681no—
346C2#3177440611yescharacteristic
347C2#3177470611yescharacteristic
348C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#441×C271C2
socleC22×C271C22×D4
prime radical#31774045C2
hypercenter#441D271
hypocenterC271#3125
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #31774519C2semidirect
2. #31774379C2semidirect
3. #31774226C2semidirect
4. #31774217C2semidirect
5. #31774186C2semidirect
6. #31774180C2semidirect
7. #31774121C2semidirect
8. #31774115C2semidirect
9. #31774058C2semidirect
11. #31774010C2semidirect
12. C4×#16864641C2semidirect
13. C4×#16864641C2semidirect
14. #31773959C2semidirect
15. #441×C271C2semidirect
174. C271#3125semidirect
187. #441D271semidirect

Maximal subgroups

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

subgroupindex
#317745192
#317743792
#317742262
#317742172
#317741862
#317741802
#317741212
#317741152
#317740582
#317740452
#317740102
C4×#168646412
#317739592
#441×C2712
#3125271