Finite Group Databasebeta

← #46846343

Subgroups — #46846343

Order 34688.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774477C211yesmaximal, characteristic
2#31774378C211yesmaximal, characteristic, complemented
3#31774236C211yesmaximal, characteristic, complemented
4#31774235C211yesmaximal, characteristic, complemented
5#31774231C211yesmaximal, characteristic, complemented
6#31774224C211yesmaximal, characteristic, complemented
7#31774222C211yesmaximal, characteristic, complemented
8#31774212C211yesmaximal, characteristic, complemented
9#31774210C211yesmaximal, characteristic, complemented
10#31773938C211yesmaximal, characteristic, complemented
11#107×D271C211yesmaximal, characteristic, complemented
12#31773897C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14#31773862C211yesmaximal, characteristic, complemented
15#484×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654C2211yescharacteristic
18C2×#16864637C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576138—21no—
21#22576136C2211yescharacteristic
22#22576135—21no—
23#22576135C2211yescharacteristic
24#22576134C2211yescharacteristic
25#22576133—21no—
26#22576133—21no—
27#22576125—21no—
28#22576122—21no—
29C4×C22⋊D271—21no—
30C4×C22⋊D271—21no—
31#22576119C2211yescharacteristic
32#22576118C2211yescharacteristic
33#22576118C2211yescharacteristic
34C4⋊C4×D271C2211yescharacteristic
35C4⋊C4×D271C2211yescharacteristic
36C4⋊C4×D271C2211yescharacteristic
37#22576094C2211yescharacteristic
38#22576093C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576093C2211yescharacteristic
41#22576092C2211yescharacteristic
42#22576091C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576080C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52C22⋊Q8×C271C2211yescharacteristic
53C2×C4⋊C4×C271C2211yescharacteristic
54#107×C271C2211yescharacteristic
55#107×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—22no—
61#16864654—22no—
62#16864654—22no—
63C2×Dic542D411yescharacteristic
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084D411yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271D411yescharacteristic
70C2×C22⋊D271D411yescharacteristic
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271C2311yescharacteristic
76C2×C4×D271C2311yescharacteristic
77#16864637—22no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C8×D271—22no—
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88C22⋊Dic271—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94C4×Dic271—21no—
95C4×Dic271—21no—
96C4×Dic271—21no—
97C2×Q8×C271C2311yescharacteristic
98C22×C4×C271—21no—
99C22×C4×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C2×C8×C271C2311yescharacteristic
102C4⋊C4×C271—21no—
103C4⋊C4×C271—21no—
104C4⋊C4×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C4⋊C4×C271C2311yescharacteristic
107C22⋊C4×C271—21no—
108C22×D271—21no—
109C22×D271C2×D411yescharacteristic
110C22×D271C2×D411yescharacteristic
111Dic542—22no—
112D1084—22no—
113D1084—21no—
114D1084—21no—
115D1084—42no—
116D1084—81no—
117C22⋊D271—22no—
118C22⋊D271—22no—
119C22⋊D271—81no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C4×D271—21no—
126C4×D271—21no—
127C4×D271—42no—
128C4×D271—81no—
129C4×D271—81no—
130C4×D271—81no—
131C4×D271#4112yes—
132C4×D271#4112yes—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271C2×D411yescharacteristic
136Q8×C271—21no—
137Q8×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—81no—
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2411yescharacteristic
147C8×C271—21no—
148C8×C271—21no—
149C22×C271—21no—
150C22×C271—21no—
151C22×C271—21no—
152C22×C271C22×D411yescharacteristic
153C2×D271—21no—
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—21no—
157C2×D271—21no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271#12312yes—
162C4×C271—21no—
163C4×C271—41no—
164C4×C271—41no—
165C4×C271—41no—
166C4×C271—41no—
167C4×C271C2×#4111yescharacteristic
168C4×C271C2×#4111yescharacteristic
169Dic271—21no—
170Dic271—41no—
171Dic271—41no—
172Dic271—41no—
173Dic271#12312yes—
174C2×C271—41no—
175C2×C271C2×#12311yescharacteristic
176C2×C271#57611yescharacteristic
177C2×C271#57811yescharacteristic
178D271—22no—
179D271—41no—
180D271—81no—
181C271#309811yescharacteristic, complemented, Sylow
182#3098—2711nomaximal, Sylow
183#537—2711no—
184#520—2711no—
185#485—2711no—
186#485—2711no—
187#484—2711no—
188#484D27111yescharacteristic, complemented
189#482—2711no—
190#482—2711no—
191#480—2711no—
192#480—2711no—
193#428—2711no—
194C2×#107—2711no—
195#417—2711no—
196#416—2711no—
197#408—2711no—
198C2×#41—2711no—
199C2×#41—2711no—
200C2×M16—2711no—
201C22×C8—2711no—
202C22⋊Q8—2711no—
203C22⋊Q8—5421no—
204C22⋊Q8C2×D27111yescharacteristic
205#121—2711no—
206#121—2711no—
207#121—5421no—
208#121—5421no—
209#121—5421no—
210C4×Q8—5421no—
211C4×D4—5421no—
212C4×D4—5421no—
213C4×D4—5421no—
214#117—2711no—
215#117—2711no—
216#117—2711no—
217C2×C4⋊C4—2711no—
218C2×C4⋊C4—2711no—
219C2×C4⋊C4—2711no—
220C2×C4⋊C4C2×D27111yescharacteristic
221#107—2711no—
222#107—2711no—
223#107—2711no—
224#107—2711no—
225#107C2×D27111yescharacteristic
226#107C2×D27111yescharacteristic
227#106—2711no—
228#106—2711no—
229#103—2711no—
230#103—2711no—
231#103C2×D27111yescharacteristic
232#103C2×D27111yescharacteristic
233#102—2711no—
234#102—2711no—
235#102—2711no—
236#102—2711no—
237C22⋊C8—2711no—
238C22⋊C8—2711no—
239C22⋊C8—2711no—
240C22⋊C8C2×D27111yescharacteristic
241#41—5422no—
242#41—5422no—
243#41—5422no—
244#41—5422no—
245C2×Q8—2711no—
246C2×Q8C22×D27111yescharacteristic
247C2×D4—2711no—
248C2×D4—2711no—
249C2×D4—2711no—
250C2×D4—2711no—
251C2×D4—5421no—
252C2×D4—5421no—
253C2×D4—5421no—
254C22×C4—21no—
255C22×C4—2711no—
256C22×C4—2711no—
257C22×C4—5421no—
258C22×C4—5421no—
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4C22×D27111yescharacteristic
262M16—5422no—
263C2×C8—2711no—
264C2×C8—2711no—
265C2×C8—5422no—
266C2×C8C22×D27111yescharacteristic
267C2×C8C22×D27111yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—2711no—
271C4⋊C4—2711no—
272C4⋊C4—2711no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4—5421no—
276C4⋊C4—5421no—
277C4⋊C4C22×D27111yescharacteristic
278C4⋊C4C22×D27111yescharacteristic
279C4⋊C4C22×D27111yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C22⋊C4—5421no—
287C42—5421no—
288C42—5421no—
289C42—5421no—
290C23—2711no—
291C23—2711no—
292C23—5421no—
293C23D4×D27111yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—5422no—
297D4—5422no—
298D4—5422no—
299D4—5422no—
300D4—5421no—
301D4—5421no—
302D4—10842no—
303D4—21681no—
304D4—21681no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—2711no—
313C2×C4—2712no—
314C2×C4—2712no—
315C2×C4—2711no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—10842no—
322C2×C4—21681no—
323C2×C4—21681no—
324C2×C4—21681no—
325C2×C4D4×D27111yescharacteristic
326C2×C4C23×D27111yescharacteristic
327C8—21no—
328C8—21no—
329C8—5421no—
330C8—5421no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—2712no—
335C22—5421no—
336C22—5421no—
337C22—5421no—
338C22—5421no—
339C22—5421no—
340C22—10841no—
341C22—10841no—
342C22—10841no—
343C22C2×D4×D27111yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—2712no—
350C4—5421no—
351C4—10841no—
352C4—10841no—
353C4—10841no—
354C4#41×D27111yescharacteristic
355C4#41×D27111yescharacteristic
356C2—41no—
357C2—5422no—
358C2—10841no—
359C2—21681no—
360C2#123×D27111yescharacteristic
361C2#3177470111yescharacteristic
362C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#484×C271C2
socleC22×C271C22×D4
prime radical#31774477C2
hypercenter#484D271
hypocenterC271#3098
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. #31774378C2semidirect
3. #31774236C2semidirect
4. #31774235C2semidirect
5. #31774231C2semidirect
6. #31774224C2semidirect
7. #31774222C2semidirect
8. #31774212C2semidirect
9. #31774210C2semidirect
10. #31773938C2semidirect
11. #107×D271C2semidirect
12. #31773897C2semidirect
13. #31773892C2semidirect
14. #31773862C2semidirect
15. #484×C271C2semidirect
181. C271#3098semidirect
188. #484D271semidirect

Maximal subgroups

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

subgroupindex
#317744772
#317743782
#317742362
#317742352
#317742312
#317742242
#317742222
#317742122
#317742102
#317739382
#107×D2712
#317738972
#317738922
#317738622
#484×C2712
#3098271