Finite Group Databasebeta

← #46847969

Subgroups — #46847969

Order 34688.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576184C211yesmaximal, characteristic, complemented
2#31774541C211yesmaximal, characteristic, complemented
3#31774535C211yesmaximal, characteristic, complemented
4#31774278C211yesmaximal, characteristic, complemented
5#31774143C211yesmaximal, characteristic, complemented
6#31774140C211yesmaximal, characteristic, complemented
7#31774114C211yesmaximal, characteristic, complemented
8#31774110C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774056C211yesmaximal, characteristic, complemented
11#31774031C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13D8×Dic271C211yesmaximal, characteristic, complemented
14#31773973C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×D4×D271C2211yescharacteristic, complemented
18#22576184—24no—
19C2×#16864644C2211yescharacteristic, complemented
20C2×#16864643C2211yescharacteristic, complemented
21C2×#16864642C2211yescharacteristic, complemented
22C2×#16864639C2211yescharacteristic
23C2×#16864639C2211yescharacteristic
24C2×#16864637C2211yescharacteristic, complemented
25#22576149C2211yescharacteristic, complemented
26#22576147C2211yescharacteristic, complemented
27#22576143—21no—
28#22576141—21no—
29#22576139C2211yescharacteristic, complemented
30#22576136C2211yescharacteristic, complemented
31#22576134C2211yescharacteristic
32#22576132—21no—
33#22576132—21no—
34#22576131C2211yescharacteristic
35#22576129—21no—
36#22576128—21no—
37D4×Dic271C2211yescharacteristic, complemented
38D4×Dic271C2211yescharacteristic, complemented
39C4×C22⋊D271C2211yescharacteristic
40C2×C22⋊Dic271—21no—
41C2×(C2×D271)⋊C4—21no—
42#22576094C2211yescharacteristic, complemented
43Dic271⋊C8C2211yescharacteristic, complemented
44#22576085C2211yescharacteristic, complemented
45#22576084C2211yescharacteristic, complemented
46#22576082C2211yescharacteristic, complemented
47#22576081C2211yescharacteristic, complemented
48#22576081C2211yescharacteristic, complemented
49#22576081C2211yescharacteristic, complemented
50C22⋊(C271⋊C8)C2211yescharacteristic, complemented
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52C8×Dic271C2211yescharacteristic, complemented
53C2×#41×C271C2211yescharacteristic
54C22×D4×C271C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C271—24no—
56C2×SD16×C271C2211yescharacteristic, complemented
57C2×SD16×C271C2211yescharacteristic, complemented
58C2×D8×C271C2211yescharacteristic
59C2×D8×C271C2211yescharacteristic
60C2×M16×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864653—22no—
63#16864653—41no—
64C2×Dic542C2311yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271D411yescharacteristic
71D4×D271—22no—
72D4×D271—41no—
73C22×Dic271—21no—
74C22×Dic271—21no—
75C22×Dic271—21no—
76C2×C4×D271C2311yescharacteristic
77#16864644—41no—
78#16864643—22no—
79#16864642—22no—
80#16864639—22no—
81#16864639—41no—
82#16864637—22no—
83C2×C271⋊C8C2311yescharacteristic
84C2×C271⋊C8C2311yescharacteristic
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⋊Dic271D411yescharacteristic
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4D411yescharacteristic
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×C271—22no—
118D8×C271—22no—
119D8×C271D414yescomplemented
120M16×C271—22no—
121C2×C8×C271C2311yescharacteristic
122C2×C8×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—42no—
125C22×D271—42no—
126C22×D271—81no—
127C22×D271C2×D411yescharacteristic
128Dic542—21no—
129Dic542—21no—
130D1084—21no—
131D1084—21no—
132C22⋊D271—41no—
133C22⋊D271—42no—
134C22⋊D271—42no—
135C22⋊D271—81no—
136C22⋊D271—81no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—42no—
141C2×Dic271—81no—
142C2×Dic271—81no—
143C2×Dic271C2×D411yescharacteristic
144C2×Dic271C2×D411yescharacteristic
145C2×Dic271C2×D411yescharacteristic
146C4×D271—21no—
147C4×D271—21no—
148C271⋊C8—21no—
149C271⋊C8—21no—
150C23×C271—21no—
151C23×C271—21no—
152C23×C271—21no—
153C23×C271—42no—
154C23×C271—42no—
155C23×C271—81no—
156C23×C271C2×D411yescharacteristic
157Q8×C271—21no—
158Q8×C271—21no—
159D4×C271—21no—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271—21no—
163D4×C271—42no—
164D4×C271—42no—
165D4×C271C2×D412yes—
166D4×C271C2×D412yes—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—42no—
171C2×C4×C271C2×D411yescharacteristic
172C2×C4×C271C2411yescharacteristic
173C8×C271—21no—
174C8×C271C2×D412yes—
175C22×C271—21no—
176C22×C271—22no—
177C22×C271—22no—
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—84no—
188C22×C271C22×D411yescharacteristic
189C2×D271—21no—
190C2×D271—21no—
191C2×D271—21no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—84no—
196C4×C271—21no—
197C4×C271—41no—
198C4×C271C22×D411yescharacteristic
199C4×C271H(5,𝔽2)11yescharacteristic
200Dic271—21no—
201Dic271—21no—
202Dic271—21no—
203Dic271—41no—
204Dic271—41no—
205Dic271—41no—
206C2×C271—42no—
207C2×C271—41no—
208C2×C271—42no—
209C2×C271—81no—
210C2×C271D4211yescharacteristic
211C2×C271#57511yescharacteristic
212C2×C271#57611yescharacteristic
213D271—41no—
214D271—81no—
215C271#327311yescharacteristic, complemented, Sylow
216#3273—2711nomaximal, Sylow
217C2×AGL(1,ℤ/8)—2711no—
218C2×AGL(1,ℤ/8)D27111yescharacteristic, complemented
219#546—2711no—
220#546—2711no—
221#495—2711no—
222#468—2711no—
223#468—2711no—
224#463—2711no—
225#463—2711no—
226#452—2711no—
227#452—2711no—
228#447—2711no—
229#447—2711no—
230C4×D8—2711no—
231#436—2711no—
232C2×#41—2711no—
233C2×#41C2×D27111yescharacteristic
234C22×D4—2711no—
235C22×D4C2×D27111yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—5424no—
238C2×Q16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16C2×D27111yescharacteristic, complemented
242C2×SD16C2×D27111yescharacteristic, complemented
243C2×D8—2711no—
244C2×D8—2711no—
245C2×D8C2×D27111yescharacteristic
246C2×D8C2×D27111yescharacteristic
247C2×M16—2711no—
248C2×M16C2×D27111yescharacteristic, complemented
249#124—2711no—
250#124—2711no—
251#123—5421no—
252#123—5421no—
253C22⋊Q8—2711no—
254C22⋊Q8—2711no—
255#121—2711no—
256#121—2711no—
257#121—5421no—
258#121—5421no—
259#120—5421no—
260#120—5421no—
261C4×D4—2711no—
262C4×D4—2711no—
263C4×D4—2711no—
264C2×C22⋊C4—5421no—
265C2×C22⋊C4—5421no—
266#107—2711no—
267C4⋊C8—2711no—
268#103—2711no—
269#103—2711no—
270#102—2711no—
271#102—2711no—
272#102—2711no—
273#102—2711no—
274C22⋊C8—2711no—
275C22⋊C8—2711no—
276C4×C8—2711no—
277C24—21no—
278C24—5421no—
279#41—22no—
280#41—41no—
281#41—5422no—
282#41—10841no—
283C2×Q8—2711no—
284C2×Q8C22×D27111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—2711no—
290C2×D4—2711no—
291C2×D4—5421no—
292C2×D4—5422no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—10841no—
297C2×D4C22×D27111yescharacteristic
298C2×D4C22×D27111yescharacteristic
299C2×D4C22×D27111yescharacteristic
300C22×C4—21no—
301C22×C4—2711no—
302C22×C4—5421no—
303C22×C4—5421no—
304C22×C4—5421no—
305C22×C4C22×D27111yescharacteristic
306Q16—10841no—
307SD16—22no—
308SD16—22no—
309SD16—5422no—
310SD16—5422no—
311D8—22no—
312D8—5422no—
313D8—10841no—
314D8C22⋊D27114yescomplemented
315M16—22no—
316M16—5422no—
317C2×C8—2711no—
318C2×C8—2711no—
319C2×C8C22×D27111yescharacteristic
320C2×C8C22×D27111yescharacteristic
321C4⋊C4—2711no—
322C4⋊C4—2711no—
323C4⋊C4—2711no—
324C4⋊C4—5421no—
325C4⋊C4—5421no—
326C22⋊C4—2711no—
327C22⋊C4—2711no—
328C22⋊C4—5421no—
329C22⋊C4—5421no—
330C22⋊C4—5421no—
331C22⋊C4—5421no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C42—2711no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—2711no—
346C23—5421no—
347C23—10842no—
348C23—10842no—
349C23—21681no—
350C23D4×D27111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—5421no—
354Q8—5421no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—5421no—
362D4—5421no—
363D4—10841no—
364D4—10842no—
365D4—10842no—
366D4—21681no—
367D4—21681no—
368D4C2×C22⋊D27112yes—
369D4C2×C22⋊D27112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—2711no—
375C2×C4—2711no—
376C2×C4—2711no—
377C2×C4—5421no—
378C2×C4—5421no—
379C2×C4—5421no—
380C2×C4—5421no—
381C2×C4—5421no—
382C2×C4—10842no—
383C2×C4—21681no—
384C2×C4—21681no—
385C2×C4D4×D27111yescharacteristic
386C2×C4C23×D27111yescharacteristic
387C8—21no—
388C8—5421no—
389C8—5421no—
390C8C2×C22⋊D27112yes—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—5421no—
405C22—5421no—
406C22—5421no—
407C22—10841no—
408C22—10841no—
409C22—10841no—
410C22—21684no—
411C22C2×D4×D27111yescharacteristic
412C4—21no—
413C4—41no—
414C4—5421no—
415C4—5421no—
416C4—5421no—
417C4—10841no—
418C4—10841no—
419C4—10841no—
420C4C22×C22⋊D27111yescharacteristic
421C4#2257620511yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—10841no—
427C2—21681no—
428C2D4×C22⋊D27111yescharacteristic
429C2#3177469311yescharacteristic
430C2#3177470111yescharacteristic
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#3273
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×#22576184C2semidirect
2. #31774541C2semidirect
3. #31774535C2semidirect
4. #31774278C2semidirect
5. #31774143C2semidirect
6. #31774140C2semidirect
7. #31774114C2semidirect
8. #31774110C2semidirect
9. #31774059C2semidirect
10. #31774056C2semidirect
11. #31774031C2semidirect
12. #31774029C2semidirect
13. D8×Dic271C2semidirect
14. #31773973C2semidirect
15. C2×AGL(1,ℤ/8)×C271C2semidirect
17. C2×D4×D271C22semidirect
19. C2×#16864644C22semidirect
20. C2×#16864643C22semidirect
21. C2×#16864642C22semidirect
24. C2×#16864637C22semidirect
25. #22576149C22semidirect
26. #22576147C22semidirect
29. #22576139C22semidirect
30. #22576136C22semidirect
37. D4×Dic271C22semidirect
38. D4×Dic271C22semidirect
42. #22576094C22semidirect
43. Dic271⋊C8C22semidirect
44. #22576085C22semidirect
45. #22576084C22semidirect
46. #22576082C22semidirect
47. #22576081C22semidirect
48. #22576081C22semidirect
49. #22576081C22semidirect
50. C22⋊(C271⋊C8)C22semidirect
51. (C2×D271)⋊C8C22semidirect
52. C8×Dic271C22semidirect
54. C22×D4×C271C22semidirect
56. C2×SD16×C271C22semidirect
57. C2×SD16×C271C22semidirect
60. C2×M16×C271C22semidirect
119. D8×C271D4semidirect
215. C271#3273semidirect
218. C2×AGL(1,ℤ/8)D271semidirect
235. C22×D4C2×D271semidirect
241. C2×SD16C2×D271semidirect
242. C2×SD16C2×D271semidirect
248. C2×M16C2×D271semidirect
314. D8C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
#317745412
#317745352
#317742782
#317741432
#317741402
#317741142
#317741102
#317740592
#317740562
#317740312
#317740292
D8×Dic2712
#317739732
C2×AGL(1,ℤ/8)×C2712
#3273271