Finite Group Databasebeta

← #46848184

Subgroups — #46848184

Order 34688.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774602C211yesmaximal, characteristic, complemented
2D4×C22⋊D271C211yesmaximal, characteristic
3#31774242C211yesmaximal, characteristic, complemented
4#31774215C211yesmaximal, characteristic, complemented
5#31774214C211yesmaximal, characteristic, complemented
6#31774185C211yesmaximal, characteristic, complemented
7#31774126C211yesmaximal, characteristic, complemented
8#31774125C211yesmaximal, characteristic, complemented
9#31774096C211yesmaximal, characteristic, complemented
10#31774049C211yesmaximal, characteristic, complemented
11#31774048C211yesmaximal, characteristic, complemented
12#31773988C211yesmaximal, characteristic, complemented
13C8×C22⋊D271C211yesmaximal, characteristic, complemented
14#102×D271C211yesmaximal, characteristic, complemented
15C2×#102×C271C211yesmaximal, characteristic, complemented
16C22×C22⋊D271—21no—
17C2×D4×D271C2211yescharacteristic
18C2×#16864643C2211yescharacteristic, complemented
19C2×#16864642C2211yescharacteristic
20C2×#16864642C2211yescharacteristic, complemented
21C2×C8×D271C2211yescharacteristic, complemented
22#22576159C2211yescharacteristic, complemented
23#22576156C2211yescharacteristic
24#22576150—21no—
25#22576140C2211yescharacteristic, complemented
26#22576139—21no—
27#22576139C2211yescharacteristic, complemented
28#22576138—21no—
29#22576134C2211yescharacteristic
30#22576132—21no—
31#22576131C2211yescharacteristic
32#22576129—21no—
33#22576128—21no—
34#22576125C2211yescharacteristic, complemented
35D4×Dic271C2211yescharacteristic
36C4×C22⋊D271—21no—
37C4×C22⋊D271C2211yescharacteristic
38C4⋊C4×D271C2211yescharacteristic, complemented
39#22576092C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41#22576091C2211yescharacteristic, complemented
42Dic271⋊C8C2211yescharacteristic, complemented
43#22576085C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic, complemented
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic, complemented
47#22576080C2211yescharacteristic, complemented
48#22576080C2211yescharacteristic, complemented
49C22⋊(C271⋊C8)C2211yescharacteristic, complemented
50(C2×D271)⋊C8C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C22×D4×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic, complemented
54C2×C4⋊C4×C271C2211yescharacteristic, complemented
55#102×C271C2211yescharacteristic
56#102×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic, complemented
58#102×C271C2211yescharacteristic, complemented
59C23×D271—21no—
60C2×Dic542—21no—
61C2×Dic542C2311yescharacteristic
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—44no—
69C2×C22⋊D271D411yescharacteristic
70D4×D271—22no—
71D4×D271—41no—
72C22×Dic271—21no—
73C2×C4×D271—21no—
74C2×C4×D271C2311yescharacteristic
75#16864643—41no—
76#16864642—22no—
77#16864642—41no—
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C8×D271—41no—
81C4⋊Dic271—21no—
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4D411yescharacteristic
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271D411yescharacteristic
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4D411yescharacteristic
96C4×Dic271—21no—
97C4×Dic271C2311yescharacteristic
98C24×C271—21no—
99C2×D4×C271—22no—
100C2×D4×C271—21no—
101C2×D4×C271—41no—
102C2×D4×C271C2311yescharacteristic
103C2×D4×C271C2311yescharacteristic
104C22×C4×C271—21no—
105C22×C4×C271C2311yescharacteristic
106C2×C8×C271—41no—
107C2×C8×C271C2311yescharacteristic
108C2×C8×C271C2311yescharacteristic
109C4⋊C4×C271—21no—
110C4⋊C4×C271C2311yescharacteristic
111C4⋊C4×C271C2311yescharacteristic
112C22×D271—21no—
113C22×D271—42no—
114C22×D271—42no—
115C22×D271—81no—
116C22×D271C2×D411yescharacteristic
117Dic542—21no—
118Dic542—21no—
119Dic542—81no—
120D1084—21no—
121D1084—21no—
122C22⋊D271—42no—
123C22⋊D271—42no—
124C22⋊D271—42no—
125C22⋊D271—42no—
126C22⋊D271—81no—
127C22⋊D271SD1614yescomplemented
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—42no—
132C2×Dic271C2×D411yescharacteristic
133C2×Dic271C2×D411yescharacteristic
134C2×Dic271C2×D411yescharacteristic
135C4×D271—21no—
136C4×D271—21no—
137C4×D271—81no—
138C271⋊C8—21no—
139C271⋊C8—21no—
140C23×C271—21no—
141C23×C271—21no—
142C23×C271—42no—
143C23×C271—42no—
144C23×C271—81no—
145C23×C271C2×D411yescharacteristic
146D4×C271—21no—
147D4×C271—21no—
148D4×C271—21no—
149D4×C271—42no—
150D4×C271#4112yes—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271—21no—
154C2×C4×C271—21no—
155C2×C4×C271—81no—
156C2×C4×C271C2×D411yescharacteristic
157C2×C4×C271C2411yescharacteristic
158C8×C271—21no—
159C8×C271—21no—
160C22×C271—21no—
161C22×C271—21no—
162C22×C271—22no—
163C22×C271—41no—
164C22×C271—41no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271—41no—
168C22×C271—84no—
169C22×C271C2×SD1612yes—
170C22×C271C22×D411yescharacteristic
171C2×D271—21no—
172C2×D271—21no—
173C2×D271—41no—
174C2×D271—41no—
175C2×D271—41no—
176C2×D271—84no—
177C2×D271C2×SD1612yes—
178C4×C271—21no—
179C4×C271—41no—
180C4×C271—41no—
181C4×C271C2×#4111yescharacteristic
182C4×C271H(5,𝔽2)11yescharacteristic
183Dic271—21no—
184Dic271—21no—
185Dic271—41no—
186Dic271—41no—
187Dic271—41no—
188Dic271C2×SD1612yes—
189C2×C271—22no—
190C2×C271—42no—
191C2×C271—81no—
192C2×C271#54611yescharacteristic
193C2×C271C22×SD1611yescharacteristic
194C2×C271#57611yescharacteristic
195D271—22no—
196D271—81no—
197C271#329511yescharacteristic, complemented, Sylow
198#3295—2711nomaximal, Sylow
199#554—2711no—
200D42—2711no—
201#486—2711no—
202#481—2711no—
203#481—2711no—
204#476—2711no—
205#465—2711no—
206#465—2711no—
207#460—2711no—
208#450—2711no—
209#450—2711no—
210C4×SD16—2711no—
211C8×D4—2711no—
212C2×#102—2711no—
213C2×#102D27111yescharacteristic, complemented
214C22×D4—2711no—
215C22×D4—5421no—
216C22×D4C2×D27111yescharacteristic
217C2×SD16—2711no—
218C2×SD16—2711no—
219C2×SD16—2711no—
220C22×C8—2711no—
221C22×C8C2×D27111yescharacteristic, complemented
222C4⋊Q8—2711no—
223#127—2711no—
224#125—5421no—
225C22⋊Q8—2711no—
226C22⋊Q8—2711no—
227C22⋊Q8—5421no—
228C22⋊Q8—5421no—
229#121—2711no—
230#121—2711no—
231#121—5421no—
232#120—5421no—
233#120—5421no—
234C4×Q8—2711no—
235C4×D4—2711no—
236C4×D4—2711no—
237C4×D4—5421no—
238C2×C4⋊C4—2711no—
239C2×C4⋊C4C2×D27111yescharacteristic, complemented
240#106—2711no—
241#106—2711no—
242#106—2711no—
243C4⋊C8—2711no—
244#103—2711no—
245#102—2711no—
246#102—2711no—
247#102—2711no—
248#102—2711no—
249#102—2711no—
250#102C2×D27111yescharacteristic
251#102C2×D27111yescharacteristic
252#102C2×D27111yescharacteristic, complemented
253#102C2×D27111yescharacteristic, complemented
254C22⋊C8—2711no—
255C22⋊C8—2711no—
256C4×C8—2711no—
257C24—21no—
258C24—5421no—
259C2×Q8—2711no—
260C2×Q8—5421no—
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—2711no—
265C2×D4—2711no—
266C2×D4—5421no—
267C2×D4—5421no—
268C2×D4—5421no—
269C2×D4—5422no—
270C2×D4—5421no—
271C2×D4—5421no—
272C2×D4—10841no—
273C2×D4—10844no—
274C2×D4C22×D27111yescharacteristic
275C2×D4C22×D27111yescharacteristic
276C22×C4—21no—
277C22×C4—2711no—
278C22×C4—5421no—
279C22×C4—5421no—
280C22×C4C22×D27111yescharacteristic
281SD16—5422no—
282SD16—10841no—
283SD16—10841no—
284C2×C8—41no—
285C2×C8—2711no—
286C2×C8—2711no—
287C2×C8—10841no—
288C2×C8C22×D27111yescharacteristic
289C2×C8C22×D27111yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—2711no—
292C4⋊C4—2711no—
293C4⋊C4—2711no—
294C4⋊C4—5421no—
295C4⋊C4—5421no—
296C4⋊C4—5421no—
297C4⋊C4—5421no—
298C4⋊C4—5421no—
299C4⋊C4C22×D27111yescharacteristic
300C4⋊C4C22×D27111yescharacteristic
301C22⋊C4—2711no—
302C22⋊C4—2711no—
303C22⋊C4—5421no—
304C22⋊C4—5421no—
305C22⋊C4—5421no—
306C22⋊C4—5421no—
307C22⋊C4—5421no—
308C42—2711no—
309C42—5421no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—2711no—
316C23—5421no—
317C23—10842no—
318C23—10842no—
319C23—21681no—
320C23D4×D27111yescharacteristic
321Q8—5421no—
322Q8—5421no—
323Q8—21681no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—2714no—
329D4—5421no—
330D4—5421no—
331D4—10842no—
332D4—10842no—
333D4—10842no—
334D4—10842no—
335D4—21681no—
336D4#1686465412yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—81no—
342C2×C4—2711no—
343C2×C4—2711no—
344C2×C4—2711no—
345C2×C4—5421no—
346C2×C4—5421no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—10842no—
351C2×C4—21681no—
352C2×C4D4×D27111yescharacteristic
353C2×C4C23×D27111yescharacteristic
354C8—21no—
355C8—21no—
356C8—5421no—
357C8—5421no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—2712no—
368C22—5421no—
369C22—5421no—
370C22—10841no—
371C22—10841no—
372C22—10841no—
373C22—21684no—
374C22SD16×D27112yes—
375C22C2×D4×D27111yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—2712no—
380C4—5421no—
381C4—5421no—
382C4—10841no—
383C4—10841no—
384C4—10841no—
385C4C2×#1686465411yescharacteristic
386C4#2257620511yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—5422no—
391C2—21681no—
392C2#3177453711yescharacteristic
393C2C2×SD16×D27111yescharacteristic
394C2#3177469811yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#102×C271C2
socleC22×C271C22×D4
prime radicalD4×C22⋊D271C2
hypercenterC2×#102D271
hypocenterC271#3295
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #31774602C2semidirect
3. #31774242C2semidirect
4. #31774215C2semidirect
5. #31774214C2semidirect
6. #31774185C2semidirect
7. #31774126C2semidirect
8. #31774125C2semidirect
9. #31774096C2semidirect
10. #31774049C2semidirect
11. #31774048C2semidirect
12. #31773988C2semidirect
13. C8×C22⋊D271C2semidirect
14. #102×D271C2semidirect
15. C2×#102×C271C2semidirect
18. C2×#16864643C22semidirect
20. C2×#16864642C22semidirect
21. C2×C8×D271C22semidirect
22. #22576159C22semidirect
25. #22576140C22semidirect
27. #22576139C22semidirect
34. #22576125C22semidirect
38. C4⋊C4×D271C22semidirect
39. #22576092C22semidirect
40. #22576091C22semidirect
41. #22576091C22semidirect
42. Dic271⋊C8C22semidirect
43. #22576085C22semidirect
44. #22576082C22semidirect
46. #22576081C22semidirect
47. #22576080C22semidirect
48. #22576080C22semidirect
49. C22⋊(C271⋊C8)C22semidirect
50. (C2×D271)⋊C8C22semidirect
51. C8×Dic271C22semidirect
53. C22×C8×C271C22semidirect
54. C2×C4⋊C4×C271C22semidirect
57. #102×C271C22semidirect
58. #102×C271C22semidirect
127. C22⋊D271SD16semidirect
197. C271#3295semidirect
213. C2×#102D271semidirect
221. C22×C8C2×D271semidirect
239. C2×C4⋊C4C2×D271semidirect
252. #102C2×D271semidirect
253. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746022
D4×C22⋊D2712
#317742422
#317742152
#317742142
#317741852
#317741262
#317741252
#317740962
#317740492
#317740482
#317739882
C8×C22⋊D2712
#102×D2712
C2×#102×C2712
#3295271