Finite Group Databasebeta

← #46846542

Subgroups — #46846542

Order 34688.

406 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#31774371C211yesmaximal, characteristic, complemented
2#31774235C211yesmaximal, characteristic, complemented
3#31774218C211yesmaximal, characteristic, complemented
4#31774184C211yesmaximal, characteristic, complemented
5#31774174C211yesmaximal, characteristic, complemented
6#31774109C211yesmaximal, characteristic, complemented
7#31774109C211yesmaximal, characteristic, complemented
8#31774047C211yesmaximal, characteristic, complemented
9#31774030C211yesmaximal, characteristic
10#31774007C211yesmaximal, characteristic, complemented
11#31773995C211yesmaximal, characteristic, complemented
12C4×#16864641C211yesmaximal, characteristic, complemented
13C4×#16864639C211yesmaximal, characteristic, complemented
14#31773958C211yesmaximal, characteristic, complemented
15#542×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic
17C2×#16864641C2211yescharacteristic
18C2×#16864641C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21C4×D1084C2211yescharacteristic
22C4×D1084C2211yescharacteristic
23C4×D1084C2211yescharacteristic
24C4×D1084C2211yescharacteristic
25#22576122—21no—
26#22576122—21no—
27#22576119C2211yescharacteristic
28C22⋊C4×D271—21no—
29C22⋊C4×D271—21no—
30#22576093C2211yescharacteristic, complemented
31#22576093C2211yescharacteristic, complemented
32#22576091C2211yescharacteristic, complemented
33#22576091C2211yescharacteristic, complemented
34C4⋊(C271⋊C8)C2211yescharacteristic, complemented
35C4⋊(C271⋊C8)C2211yescharacteristic, complemented
36#22576084C2211yescharacteristic, complemented
37#22576084C2211yescharacteristic, complemented
38#22576081C2211yescharacteristic, complemented
39#22576081C2211yescharacteristic, complemented
40#22576080C2211yescharacteristic
41#22576080C2211yescharacteristic
42#22576080C2211yescharacteristic
43#22576080C2211yescharacteristic
44C22⋊(C271⋊C8)C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46#22576071C2211yescharacteristic, complemented
47C4×C271⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C271C2211yescharacteristic, complemented
49#126×C271—21no—
50#124×C271—21no—
51#124×C271C2211yescharacteristic, complemented
52#123×C271—21no—
53C22⋊Q8×C271—21no—
54C22⋊Q8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56C4×Q8×C271C2211yescharacteristic, complemented
57C4×D4×C271C2211yescharacteristic, complemented
58#117×C271C2211yescharacteristic
59C23×D271—21no—
60C2×D1084—22no—
61C2×D1084—22no—
62C2×D1084—21no—
63C2×D1084C2311yescharacteristic
64C2×D1084C2311yescharacteristic
65C2×C4×D271—21no—
66C2×C4×D271—21no—
67C2×C4×D271—21no—
68C2×C4×D271—21no—
69#16864641—22no—
70#16864641—22no—
71#16864639—22no—
72#16864639—22no—
73C2×C271⋊C8C2311yescharacteristic
74C2×C271⋊C8C2311yescharacteristic
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78C4⋊Dic271C2311yescharacteristic
79C22⋊Dic271—21no—
80(C2×D271)⋊C4—21no—
81(C2×D271)⋊C4—21no—
82(C2×D271)⋊C4—21no—
83(C2×D271)⋊C4—21no—
84C4×Dic271—21no—
85C2×Q8×C271—21no—
86C2×Q8×C271C2311yescharacteristic
87C2×D4×C271—21no—
88C2×D4×C271C2311yescharacteristic
89C22×C4×C271—21no—
90C22×C4×C271C2311yescharacteristic
91C4⋊C4×C271—21no—
92C4⋊C4×C271—21no—
93C4⋊C4×C271—21no—
94C4⋊C4×C271D411yescharacteristic
95C4⋊C4×C271D411yescharacteristic
96C4⋊C4×C271C2311yescharacteristic
97C4⋊C4×C271C2311yescharacteristic
98C22⋊C4×C271—21no—
99C22⋊C4×C271—21no—
100C22⋊C4×C271—21no—
101C22⋊C4×C271—21no—
102C22⋊C4×C271—21no—
103C22⋊C4×C271D411yescharacteristic
104C22⋊C4×C271D411yescharacteristic
105C42×C271—21no—
106C42×C271C2311yescharacteristic
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109C22×D271—21no—
110C22×D271—24no—
111C22×D271—81no—
112C22×D271—81no—
113D1084—24no—
114D1084—21no—
115D1084—21no—
116D1084#4112yes—
117D1084#4112yes—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C4×D271—81no—
121C4×D271—81no—
122C4×D271—81no—
123C4×D271—81no—
124C271⋊C8—21no—
125C271⋊C8—21no—
126C271⋊C8—21no—
127C271⋊C8—21no—
128C23×C271—21no—
129C23×C271C2×D411yescharacteristic
130Q8×C271—21no—
131Q8×C271—21no—
132Q8×C271—81no—
133D4×C271—21no—
134D4×C271—21no—
135D4×C271—81no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—81no—
142C2×C4×C271C2×D411yescharacteristic
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C22×C271—21no—
149C22×C271—21no—
150C22×C271—21no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271C22×D411yescharacteristic
155C2×D271—22no—
156C2×D271—22no—
157C2×D271—41no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—84no—
162C4×C271—21no—
163C4×C271—21no—
164C4×C271—21no—
165C4×C271—21no—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271—41no—
170C4×C271C2×#4111yescharacteristic
171C4×C271C2×#4111yescharacteristic
172Dic271—41no—
173Dic271—41no—
174C2×C271—41no—
175C2×C271—81no—
176C2×C271#52511yescharacteristic
177C2×C271#57611yescharacteristic
178C2×C271#57711yescharacteristic
179D271—42no—
180D271—42no—
181C271#311811yescharacteristic, complemented, Sylow
182#3118—2711nomaximal, Sylow
183#542D27111yescharacteristic, complemented
184#518—2711no—
185#485—2711no—
186#482—2711no—
187#476—2711no—
188#474—2711no—
189#463—2711no—
190#463—2711no—
191#450—2711no—
192#447—2711no—
193#442—2711no—
194#440—2711no—
195C4×SD16—2711no—
196C4×D8—2711no—
197#433—2711no—
198C22×D4—2711no—
199C2×SD16—2711no—
200C2×SD16—2711no—
201C2×D8—2711no—
202C2×D8—2711no—
203C4⋊Q8C2×D27111yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D27111yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D27111yescharacteristic
210#121C2×D27111yescharacteristic
211C4×Q8C2×D27111yescharacteristic, complemented
212C4×D4—2711no—
213C4×D4—2711no—
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4—5421no—
217C4×D4—5421no—
218C4×D4C2×D27111yescharacteristic, complemented
219#117—2711no—
220#117C2×D27111yescharacteristic
221C2×C22⋊C4—5421no—
222C2×C22⋊C4—5421no—
223#107—2711no—
224#107—2711no—
225#106—2711no—
226#106—2711no—
227C4⋊C8—2711no—
228C4⋊C8—2711no—
229#103—2711no—
230#103—2711no—
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102—2711no—
236#102—2711no—
237C22⋊C8—2711no—
238C22⋊C8—2711no—
239#97—2711no—
240C4×C8—2711no—
241C24—5421no—
242C2×Q8—21no—
243C2×Q8C22×D27111yescharacteristic
244C2×D4—21no—
245C2×D4—2711no—
246C2×D4—2711no—
247C2×D4—5422no—
248C2×D4—5422no—
249C2×D4—5421no—
250C2×D4C22×D27111yescharacteristic
251C22×C4—21no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4—5421no—
255C22×C4—5421no—
256C22×C4C22×D27111yescharacteristic
257SD16—5422no—
258SD16—5422no—
259D8—5422no—
260D8—5422no—
261C2×C8—2711no—
262C2×C8—2711no—
263C2×C8—2711no—
264C2×C8—2711no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—2711no—
269C4⋊C4—2711no—
270C4⋊C4C22⋊D27111yescharacteristic
271C4⋊C4C22⋊D27111yescharacteristic
272C4⋊C4C22×D27111yescharacteristic
273C4⋊C4C22×D27111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4C22⋊D27111yescharacteristic
285C22⋊C4C22⋊D27111yescharacteristic
286C42—21no—
287C42—5421no—
288C42C22×D27111yescharacteristic
289C42C22×D27111yescharacteristic
290C23—21no—
291C23—5424no—
292C23—5421no—
293C23—5421no—
294C23—21681no—
295C23—21681no—
296C23C2×C22⋊D27111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—2712no—
304D4—2712no—
305D4—5424no—
306D4—5421no—
307D4—5421no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—21681no—
317C2×C4—21681no—
318C2×C4—21681no—
319C2×C4—21681no—
320C2×C4C2×C22⋊D27111yescharacteristic
321C2×C4C2×C22⋊D27111yescharacteristic
322C2×C4C2×C22⋊D27111yescharacteristic
323C2×C4C2×C22⋊D27111yescharacteristic
324C2×C4C2×C22⋊D27111yescharacteristic
325C2×C4C23×D27111yescharacteristic
326C8—5421no—
327C8—5421no—
328C8—5421no—
329C8—5421no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—5422no—
337C22—5422no—
338C22—10841no—
339C22—10841no—
340C22—10841no—
341C22—10841no—
342C22—21684no—
343C22C22×C22⋊D27111yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—10841no—
353C4—10841no—
354C4#41×D27111yescharacteristic
355C4#41×D27111yescharacteristic
356C2—41no—
357C2—81no—
358C2—10842no—
359C2—10842no—
360C2#3177440411yescharacteristic
361C2#3177470011yescharacteristic
362C2#3177470511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#542×C271C2
socleC22×C271C22×D4
prime radical#31774030C2
hypercenter#542D271
hypocenterC271#3118
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #31774371C2semidirect
2. #31774235C2semidirect
3. #31774218C2semidirect
4. #31774184C2semidirect
5. #31774174C2semidirect
6. #31774109C2semidirect
7. #31774109C2semidirect
8. #31774047C2semidirect
10. #31774007C2semidirect
11. #31773995C2semidirect
12. C4×#16864641C2semidirect
13. C4×#16864639C2semidirect
14. #31773958C2semidirect
15. #542×C271C2semidirect
30. #22576093C22semidirect
31. #22576093C22semidirect
32. #22576091C22semidirect
33. #22576091C22semidirect
34. C4⋊(C271⋊C8)C22semidirect
35. C4⋊(C271⋊C8)C22semidirect
36. #22576084C22semidirect
37. #22576084C22semidirect
38. #22576081C22semidirect
39. #22576081C22semidirect
46. #22576071C22semidirect
47. C4×C271⋊C8C22semidirect
48. C4⋊Q8×C271C22semidirect
51. #124×C271C22semidirect
56. C4×Q8×C271C22semidirect
57. C4×D4×C271C22semidirect
181. C271#3118semidirect
183. #542D271semidirect
203. C4⋊Q8C2×D271semidirect
206. #124C2×D271semidirect
211. C4×Q8C2×D271semidirect
218. C4×D4C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317743712
#317742352
#317742182
#317741842
#317741742
#317741092
#317740472
#317740302
#317740072
#317739952
C4×#168646412
C4×#168646392
#317739582
#542×C2712
#3118271