Finite Group Databasebeta

← #46847995

Subgroups — #46847995

Order 34688.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576189C211yesmaximal, characteristic, complemented
2#31774554C211yesmaximal, characteristic, complemented
3#31774541C211yesmaximal, characteristic, complemented
4#31774280C211yesmaximal, characteristic, complemented
5#31774155C211yesmaximal, characteristic, complemented
6#31774147C211yesmaximal, characteristic, complemented
7#31774117C211yesmaximal, characteristic, complemented
8#31774110C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774048C211yesmaximal, characteristic, complemented
11#31774039C211yesmaximal, characteristic, complemented
12#31774034C211yesmaximal, characteristic, complemented
13SD16×Dic271C211yesmaximal, characteristic, complemented
14#31773973C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic, complemented
18#22576189—24no—
19C2×Dic1084C2211yescharacteristic, complemented
20C2×#16864644C2211yescharacteristic, complemented
21C2×#16864642C2211yescharacteristic
22C2×#16864642C2211yescharacteristic
23C2×#16864642C2211yescharacteristic, complemented
24C2×#16864637C2211yescharacteristic, complemented
25#22576147—21no—
26#22576147C2211yescharacteristic
27#22576147C2211yescharacteristic, complemented
28#22576141—21no—
29#22576139—21no—
30#22576139C2211yescharacteristic
31#22576139C2211yescharacteristic
32#22576132—21no—
33#22576131—21no—
34#22576131C2211yescharacteristic
35#22576131C2211yescharacteristic, complemented
36#22576129—21no—
37Q8×Dic271C2211yescharacteristic, complemented
38D4×Dic271C2211yescharacteristic
39C4×C22⋊D271—21no—
40C4×C22⋊D271C2211yescharacteristic
41C2×C22⋊Dic271—21no—
42#22576092C2211yescharacteristic, complemented
43Dic271⋊C8C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576085C2211yescharacteristic, complemented
46#22576084C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576081C2211yescharacteristic, complemented
50C22⋊(C271⋊C8)C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52C8×Dic271C2211yescharacteristic, complemented
53C2×#41×C271C2211yescharacteristic
54C22×D4×C271C2211yescharacteristic
55AGL(1,ℤ/8)×C271—24no—
56C2×SD16×C271C2211yescharacteristic
57C2×SD16×C271C2211yescharacteristic
58C2×D8×C271C2211yescharacteristic
59C2×D8×C271C2211yescharacteristic
60C2×M16×C271C2211yescharacteristic
61#16864653—22no—
62#16864653—41no—
63C2×Dic542—21no—
64C2×Dic542C2311yescharacteristic
65C2×Dic542C2311yescharacteristic
66Q8×D271—22no—
67Q8×D271—41no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271D411yescharacteristic
72C22×Dic271—21no—
73C22×Dic271—21no—
74C2×C4×D271—21no—
75C2×C4×D271C2311yescharacteristic
76Dic1084—22no—
77#16864644—22no—
78#16864642—22no—
79#16864642—41no—
80#16864642—41no—
81#16864637—22no—
82C2×C271⋊C8C2311yescharacteristic
83C2×C271⋊C8C2311yescharacteristic
84C4⋊Dic271—21no—
85C4⋊Dic271C2311yescharacteristic
86C4⋊Dic271C2311yescharacteristic
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4D411yescharacteristic
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271—21no—
95C22⋊Dic271—21no—
96C22⋊Dic271—21no—
97C22⋊Dic271—21no—
98C22⋊Dic271—21no—
99C22⋊Dic271D411yescharacteristic
100(C2×D271)⋊C4D411yescharacteristic
101C4×Dic271—21no—
102C4×Dic271C2311yescharacteristic
103C24×C271—21no—
104#41×C271—22no—
105#41×C271—41no—
106C2×Q8×C271C2311yescharacteristic
107C2×D4×C271—22no—
108C2×D4×C271—21no—
109C2×D4×C271—21no—
110C2×D4×C271—41no—
111C2×D4×C271C2311yescharacteristic
112C2×D4×C271C2311yescharacteristic
113C2×D4×C271C2311yescharacteristic
114C22×C4×C271—21no—
115C22×C4×C271C2311yescharacteristic
116SD16×C271—22no—
117SD16×C271D414yescomplemented
118D8×C271—22no—
119D8×C271—22no—
120M16×C271—22no—
121C2×C8×C271C2311yescharacteristic
122C2×C8×C271C2311yescharacteristic
123C22×D271C2×D411yescharacteristic
124Dic542—21no—
125Dic542—21no—
126Dic542—21no—
127Dic542—21no—
128Dic542—42no—
129C22⋊D271—41no—
130C22⋊D271—42no—
131C22⋊D271—42no—
132C22⋊D271—81no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—42no—
138C2×Dic271—81no—
139C2×Dic271C2×D411yescharacteristic
140C2×Dic271C2×D411yescharacteristic
141C2×Dic271C2×D411yescharacteristic
142C4×D271—21no—
143C4×D271—21no—
144C4×D271—42no—
145C271⋊C8—21no—
146C271⋊C8—21no—
147C23×C271—21no—
148C23×C271—21no—
149C23×C271—21no—
150C23×C271—42no—
151C23×C271—42no—
152C23×C271—81no—
153C23×C271C2×D411yescharacteristic
154Q8×C271—21no—
155Q8×C271C2×D412yes—
156D4×C271—21no—
157D4×C271—21no—
158D4×C271—21no—
159D4×C271—21no—
160D4×C271—21no—
161D4×C271—42no—
162D4×C271—42no—
163D4×C271C2×D412yes—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—42no—
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2411yescharacteristic
170C8×C271—21no—
171C8×C271C2×D412yes—
172C22×C271—21no—
173C22×C271—22no—
174C22×C271—21no—
175C22×C271—21no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271—84no—
185C22×C271C22×D411yescharacteristic
186C2×D271—21no—
187C2×D271—21no—
188C2×D271—21no—
189C4×C271—21no—
190C4×C271—22no—
191C4×C271C22×D411yescharacteristic
192C4×C271H(5,𝔽2)11yescharacteristic
193Dic271—21no—
194Dic271—21no—
195Dic271—21no—
196Dic271—41no—
197Dic271—41no—
198Dic271—41no—
199Dic271—41no—
200C2×C271—41no—
201C2×C271—42no—
202C2×C271—81no—
203C2×C271—81no—
204C2×C271D4211yescharacteristic
205C2×C271#57511yescharacteristic
206C2×C271#57711yescharacteristic
207D271—41no—
208C271#327511yescharacteristic, complemented, Sylow
209#3275—2711nomaximal, Sylow
210C2×#137—2711no—
211C2×AGL(1,ℤ/8)D27111yescharacteristic, complemented
212#548—2711no—
213#546—2711no—
214#495—2711no—
215#470—2711no—
216#469—2711no—
217#464—2711no—
218#463—2711no—
219#452—2711no—
220#450—2711no—
221#449—2711no—
222#448—2711no—
223C4×SD16—2711no—
224#436—2711no—
225C2×#41—2711no—
226C2×#41C2×D27111yescharacteristic
227C22×Q8—2711no—
228C22×D4C2×D27111yescharacteristic
229#137—5424no—
230AGL(1,ℤ/8)—24no—
231C2×Q16—2711no—
232C2×Q16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—2711no—
236C2×SD16C2×D27111yescharacteristic
237C2×SD16C2×D27111yescharacteristic
238C2×D8C2×D27111yescharacteristic
239C2×D8C2×D27111yescharacteristic
240C2×M16—2711no—
241C2×M16C2×D27111yescharacteristic
242#124—2711no—
243#124—2711no—
244#124—5421no—
245#123—5421no—
246C22⋊Q8—2711no—
247C22⋊Q8—2711no—
248C22⋊Q8—5421no—
249#121—2711no—
250#121—2711no—
251#121—5421no—
252#121—5421no—
253#120—5421no—
254C4×Q8—2711no—
255C4×D4—2711no—
256C4×D4—2711no—
257C4×D4—5421no—
258C2×C22⋊C4—5421no—
259#106—2711no—
260C4⋊C8—2711no—
261#103—2711no—
262#103—2711no—
263#103—2711no—
264#102—2711no—
265#102—2711no—
266#102—2711no—
267C22⋊C8—2711no—
268C22⋊C8—2711no—
269C4×C8—2711no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—5422no—
274#41—10841no—
275C2×Q8—2711no—
276C2×Q8—2711no—
277C2×Q8—5421no—
278C2×Q8—5422no—
279C2×Q8—10841no—
280C2×Q8C22×D27111yescharacteristic
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—2711no—
286C2×D4—5421no—
287C2×D4—5421no—
288C2×D4—5421no—
289C2×D4C22×D27111yescharacteristic
290C2×D4C22×D27111yescharacteristic
291C2×D4C22×D27111yescharacteristic
292C22×C4—21no—
293C22×C4—2711no—
294C22×C4—5421no—
295C22×C4—5421no—
296C22×C4—5421no—
297C22×C4C22×D27111yescharacteristic
298Q16—5422no—
299Q16—5422no—
300SD16—22no—
301SD16—5422no—
302SD16—10841no—
303SD16—10841no—
304SD16C22⋊D27114yescomplemented
305D8—22no—
306D8—22no—
307M16—22no—
308M16—5422no—
309C2×C8—2711no—
310C2×C8—2711no—
311C2×C8C22×D27111yescharacteristic
312C2×C8C22×D27111yescharacteristic
313C4⋊C4—2711no—
314C4⋊C4—2711no—
315C4⋊C4—2711no—
316C4⋊C4—5421no—
317C4⋊C4—5421no—
318C4⋊C4—5421no—
319C22⋊C4—2711no—
320C22⋊C4—2711no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4—5421no—
325C22⋊C4—5421no—
326C22⋊C4—5421no—
327C22⋊C4—5421no—
328C22⋊C4—5421no—
329C22⋊C4—5421no—
330C42—2711no—
331C42—5421no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—2711no—
339C23D4×D27111yescharacteristic
340Q8—21no—
341Q8—5421no—
342Q8—5421no—
343Q8—5421no—
344Q8—5421no—
345Q8—10842no—
346Q8C2×C22⋊D27112yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—42no—
354D4—10841no—
355D4—10842no—
356D4—10842no—
357D4—21681no—
358D4C2×C22⋊D27112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—2711no—
364C2×C4—2711no—
365C2×C4—2711no—
366C2×C4—5421no—
367C2×C4—5421no—
368C2×C4—5421no—
369C2×C4—5421no—
370C2×C4—5421no—
371C2×C4—5421no—
372C2×C4—10842no—
373C2×C4—10842no—
374C2×C4—21681no—
375C2×C4D4×D27111yescharacteristic
376C2×C4C23×D27111yescharacteristic
377C8—21no—
378C8—5421no—
379C8—5421no—
380C8C2×C22⋊D27112yes—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—5421no—
395C22—5421no—
396C22—5421no—
397C22C2×D4×D27111yescharacteristic
398C4—21no—
399C4—22no—
400C4—5421no—
401C4—5421no—
402C4—5421no—
403C4—10841no—
404C4—10841no—
405C4—10841no—
406C4—10841no—
407C4C22×C22⋊D27111yescharacteristic
408C4#2257620511yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—10841no—
414C2D4×C22⋊D27111yescharacteristic
415C2#3177469311yescharacteristic
416C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×AGL(1,ℤ/8)×C271C2
socleC22×C271C22×D4
hypercenterC2×AGL(1,ℤ/8)D271
hypocenterC271#3275
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22576189C2semidirect
2. #31774554C2semidirect
3. #31774541C2semidirect
4. #31774280C2semidirect
5. #31774155C2semidirect
6. #31774147C2semidirect
7. #31774117C2semidirect
8. #31774110C2semidirect
9. #31774059C2semidirect
10. #31774048C2semidirect
11. #31774039C2semidirect
12. #31774034C2semidirect
13. SD16×Dic271C2semidirect
14. #31773973C2semidirect
15. C2×AGL(1,ℤ/8)×C271C2semidirect
17. C2×Q8×D271C22semidirect
19. C2×Dic1084C22semidirect
20. C2×#16864644C22semidirect
23. C2×#16864642C22semidirect
24. C2×#16864637C22semidirect
27. #22576147C22semidirect
35. #22576131C22semidirect
37. Q8×Dic271C22semidirect
42. #22576092C22semidirect
43. Dic271⋊C8C22semidirect
44. #22576085C22semidirect
45. #22576085C22semidirect
46. #22576084C22semidirect
49. #22576081C22semidirect
51. (C2×D271)⋊C8C22semidirect
52. C8×Dic271C22semidirect
117. SD16×C271D4semidirect
208. C271#3275semidirect
211. C2×AGL(1,ℤ/8)D271semidirect
304. SD16C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761892
#317745542
#317745412
#317742802
#317741552
#317741472
#317741172
#317741102
#317740592
#317740482
#317740392
#317740342
SD16×Dic2712
#317739732
C2×AGL(1,ℤ/8)×C2712
#3275271