Finite Group Databasebeta

← #46846096

Subgroups — #46846096

Order 34688.

446 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774693C211yesmaximal, characteristic
2C2×#22576189C211yesmaximal, characteristic, complemented
3#137×D271C211yesmaximal, characteristic, complemented
4#31774662C211yesmaximal, characteristic, complemented
5#31774173C211yesmaximal, characteristic, complemented
6#31774172C211yesmaximal, characteristic, complemented
7#31774171C211yesmaximal, characteristic, complemented
8#31774170C211yesmaximal, characteristic, complemented
9#31774159C211yesmaximal, characteristic, complemented
10#31774158C211yesmaximal, characteristic, complemented
11#31774157C211yesmaximal, characteristic, complemented
12#31773944C211yesmaximal, characteristic, complemented
13#31773878C211yesmaximal, characteristic, complemented
14#101×D271C211yesmaximal, characteristic, complemented
15#471×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17#41×D271C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19#22576193C2211yescharacteristic
20#22576193C2211yescharacteristic
21#22576192—21no—
22#22576191C2211yescharacteristic
23#22576190C2211yescharacteristic
24#22576190C2211yescharacteristic
25#22576189—21no—
26#22576189—21no—
27#22576189—21no—
28#22576189—21no—
29#22576189—21no—
30#22576189—21no—
31#22576186C2211yescharacteristic
32#22576184—21no—
33#22576181—21no—
34#22576179—21no—
35C2×Dic1084C2211yescharacteristic, complemented
36C2×#16864644C2211yescharacteristic, complemented
37Q16×D271—21no—
38C2×#16864642C2211yescharacteristic, complemented
39SD16×D271—21no—
40#22576168—21no—
41#22576168C2211yescharacteristic
42#22576167C2211yescharacteristic
43#22576166—21no—
44#22576166C2211yescharacteristic
45C2×#16864637—21no—
46C2×#16864637C2211yescharacteristic
47M16×D271—21no—
48M16×D271C2211yescharacteristic
49M16×D271C2211yescharacteristic
50M16×D271C2211yescharacteristic
51#22576096C2211yescharacteristic, complemented
52#22576095C2211yescharacteristic, complemented
53#22576095C2211yescharacteristic, complemented
54#22576079C2211yescharacteristic, complemented
55#22576078C2211yescharacteristic, complemented
56#22576078C2211yescharacteristic, complemented
57#22576078C2211yescharacteristic, complemented
58#22576078C2211yescharacteristic, complemented
59#22576077C2211yescharacteristic, complemented
60#137×C271C2211yescharacteristic
61AGL(1,ℤ/8)×C271C2211yescharacteristic
62C2×SD16×C271C2211yescharacteristic, complemented
63#131×C271C2211yescharacteristic
64#108×C271C2211yescharacteristic, complemented
65#101×C271C2211yescharacteristic, complemented
66#100×C271C2211yescharacteristic, complemented
67#16864655D411yescharacteristic
68#16864654—21no—
69#16864654C2311yescharacteristic
70#16864653—21no—
71#16864653—21no—
72#16864653—21no—
73#16864653—21no—
74#16864653—21no—
75#16864653D411yescharacteristic
76C2×Dic542—21no—
77C2×Dic542C2311yescharacteristic
78C2×Dic542C2311yescharacteristic
79Q8×D271—21no—
80Q8×D271—21no—
81Q8×D271—21no—
82Q8×D271—21no—
83Q8×D271D411yescharacteristic
84C2×C22⋊D271—21no—
85D4×D271—21no—
86D4×D271D411yescharacteristic
87C22×Dic271—21no—
88C2×C4×D271—21no—
89C2×C4×D271—21no—
90C2×C4×D271C2311yescharacteristic
91Dic1084—21no—
92Dic1084—21no—
93Dic1084—21no—
94Dic1084—21no—
95#16864644—21no—
96#16864644—21no—
97#16864644—21no—
98#16864644—21no—
99#16864644—21no—
100#16864643—21no—
101#16864643—21no—
102#16864642—21no—
103#16864642—21no—
104#16864642—21no—
105#16864642—21no—
106#16864642—21no—
107#16864641—21no—
108#16864639—21no—
109#16864638—21no—
110#16864638C2311yescharacteristic
111#16864638C2311yescharacteristic
112#16864638C2311yescharacteristic
113#16864637—21no—
114#16864637—21no—
115#16864637—21no—
116#16864637—21no—
117#16864637—21no—
118#16864637D411yescharacteristic
119#16864637D411yescharacteristic
120#16864637D411yescharacteristic
121#16864637D411yescharacteristic
122C2×C271⋊C8—21no—
123C2×C271⋊C8C2311yescharacteristic
124C8×D271—21no—
125C8×D271—21no—
126C8×D271—21no—
127C8×D271—21no—
128C8×D271—21no—
129#41×C271C2311yescharacteristic
130C2×Q8×C271C2311yescharacteristic
131C2×D4×C271C2311yescharacteristic
132Q16×C271—21no—
133SD16×C271—21no—
134SD16×C271—21no—
135SD16×C271—21no—
136SD16×C271—21no—
137D8×C271—21no—
138M16×C271—21no—
139M16×C271C2311yescharacteristic
140M16×C271C2311yescharacteristic
141M16×C271C2311yescharacteristic
142C2×C8×C271—21no—
143C2×C8×C271C2311yescharacteristic
144C22×D271—21no—
145C22×D271#4111yescharacteristic
146Dic542—21no—
147Dic542—21no—
148Dic542—21no—
149Dic542—21no—
150Dic542—21no—
151Dic542—41no—
152Dic542—41no—
153Dic542C2×D411yescharacteristic
154D1084—21no—
155D1084C2×D411yescharacteristic
156C22⋊D271—21no—
157C22⋊D271—21no—
158C22⋊D271—21no—
159C22⋊D271—41no—
160C22⋊D271—41no—
161C2×Dic271—21no—
162C2×Dic271—21no—
163C2×Dic271—21no—
164C2×Dic271—41no—
165C2×Dic271—41no—
166C2×Dic271#4111yescharacteristic
167C4×D271—21no—
168C4×D271—21no—
169C4×D271—21no—
170C4×D271—41no—
171C4×D271—41no—
172C4×D271C2×D411yescharacteristic
173C4×D271C2×D411yescharacteristic
174C4×D271C2×D411yescharacteristic
175C4×D271C2×D411yescharacteristic
176C271⋊C8—21no—
177C271⋊C8—21no—
178C271⋊C8—21no—
179C271⋊C8C2×D411yescharacteristic
180C271⋊C8C2×D411yescharacteristic
181C23×C271—21no—
182Q8×C271—21no—
183Q8×C271—21no—
184Q8×C271C2×D411yescharacteristic
185D4×C271—21no—
186D4×C271—21no—
187D4×C271—21no—
188D4×C271C2×D411yescharacteristic
189C2×C4×C271—21no—
190C2×C4×C271—21no—
191C2×C4×C271C2411yescharacteristic
192C8×C271—21no—
193C8×C271—21no—
194C8×C271—21no—
195C8×C271C2×D411yescharacteristic
196C8×C271C2×D411yescharacteristic
197C22×C271—21no—
198C22×C271—41no—
199C22×C271—81no—
200C22×C271C2×#4111yescharacteristic
201C2×D271—21no—
202C2×D271—41no—
203C2×D271—81no—
204C2×D271#12111yescharacteristic
205C2×D271#12111yescharacteristic
206C4×C271—21no—
207C4×C271—41no—
208C4×C271C22×D411yescharacteristic
209C4×C271C22×D411yescharacteristic
210Dic271—21no—
211Dic271—41no—
212Dic271—41no—
213Dic271#12111yescharacteristic
214Dic271#12111yescharacteristic
215C2×C271—21no—
216C2×C271—41no—
217C2×C271—81no—
218C2×C271C2×#12111yescharacteristic
219D271—21no—
220D271—21no—
221D271—41no—
222C271#306911yescharacteristic, complemented, Sylow
223#3069—2711nomaximal, Sylow
224#575—2711no—
225C2×#137—2711no—
226C2×#137—2711no—
227#568—2711no—
228#473—2711no—
229#473—2711no—
230#473—2711no—
231#473—2711no—
232#471—2711no—
233#471—2711no—
234#471—2711no—
235#471D27111yescharacteristic, complemented
236#430—2711no—
237#413—2711no—
238C2×#101—2711no—
239C2×#41—2711no—
240C2×#41—2711no—
241C22×Q8—2711no—
242#137—2711no—
243#137—2711no—
244#137—2711no—
245#137—2711no—
246#137—2711no—
247#137—5421no—
248#137—5421no—
249#137—5421no—
250#137—5421no—
251#137—5421no—
252#137—5421no—
253#137—5421no—
254#137C2×D27111yescharacteristic
255AGL(1,ℤ/8)—2711no—
256AGL(1,ℤ/8)—5421no—
257AGL(1,ℤ/8)C2×D27111yescharacteristic
258#135—5421no—
259#135—5421no—
260C2×Q16—2711no—
261C2×Q16—2711no—
262C2×Q16—5421no—
263C2×SD16—2711no—
264C2×SD16—5421no—
265C2×SD16C2×D27111yescharacteristic, complemented
266#131—2711no—
267#131—2711no—
268#131—2711no—
269#131—5421no—
270#131—5421no—
271#131C2×D27111yescharacteristic
272C2×M16—2711no—
273C2×M16—2711no—
274C2×M16—2711no—
275C2×M16—2711no—
276C2×M16—5421no—
277C2×M16—5421no—
278#108—2711no—
279#108—2711no—
280#108—2711no—
281#108C2×D27111yescharacteristic, complemented
282#101—2711no—
283#101—2711no—
284#101—2711no—
285#101—2711no—
286#101—2711no—
287#101C2×D27111yescharacteristic, complemented
288#100—2711no—
289#100C2×D27111yescharacteristic, complemented
290#41—2711no—
291#41—2711no—
292#41—2711no—
293#41—5421no—
294#41—5421no—
295#41—5421no—
296#41—5421no—
297#41—5421no—
298#41—5421no—
299#41C22×D27111yescharacteristic
300C2×Q8—2711no—
301C2×Q8—2711no—
302C2×Q8—2711no—
303C2×Q8—5421no—
304C2×Q8—5421no—
305C2×Q8—5421no—
306C2×Q8—5421no—
307C2×Q8—5421no—
308C2×Q8C22×D27111yescharacteristic
309C2×D4—2711no—
310C2×D4—5421no—
311C2×D4—5421no—
312C2×D4C22×D27111yescharacteristic
313C22×C4—2711no—
314C22×C4—5421no—
315C22×C4—5421no—
316C22×C4—5421no—
317Q16—21no—
318Q16—5421no—
319Q16—5421no—
320Q16—5421no—
321Q16—5421no—
322Q16—5421no—
323Q16—5421no—
324Q16—5421no—
325Q16—5421no—
326Q16—5421no—
327SD16—21no—
328SD16—21no—
329SD16—21no—
330SD16—21no—
331SD16—5421no—
332SD16—5421no—
333SD16—5421no—
334SD16—5421no—
335SD16—5421no—
336SD16—5421no—
337SD16—5421no—
338SD16—5421no—
339D8—21no—
340D8—5421no—
341M16—21no—
342M16—2711no—
343M16—2711no—
344M16—2711no—
345M16—2711no—
346M16—2711no—
347M16—2711no—
348M16—2711no—
349M16—5421no—
350M16—5421no—
351M16—5421no—
352M16—5421no—
353M16—5421no—
354M16—5421no—
355M16C22×D27111yescharacteristic
356M16C22×D27111yescharacteristic
357M16C22×D27111yescharacteristic
358C2×C8—21no—
359C2×C8—2711no—
360C2×C8—5421no—
361C2×C8—5421no—
362C2×C8—5421no—
363C2×C8—5421no—
364C2×C8—5421no—
365C2×C8—5421no—
366C2×C8C22×D27111yescharacteristic
367C23—21no—
368C23—2711no—
369C23—5421no—
370Q8—21no—
371Q8—21no—
372Q8—2711no—
373Q8—5421no—
374Q8—5421no—
375Q8—5421no—
376Q8—5421no—
377Q8—5421no—
378Q8—10841no—
379Q8—10841no—
380Q8D4×D27111yescharacteristic
381D4—21no—
382D4—21no—
383D4—21no—
384D4—2711no—
385D4—5421no—
386D4—5421no—
387D4—5421no—
388D4—5421no—
389D4—10841no—
390D4—10841no—
391D4D4×D27111yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—2711no—
395C2×C4—2711no—
396C2×C4—2711no—
397C2×C4—2711no—
398C2×C4—2711no—
399C2×C4—5421no—
400C2×C4—5421no—
401C2×C4—5421no—
402C2×C4—5421no—
403C2×C4—5421no—
404C2×C4—5421no—
405C2×C4—10841no—
406C2×C4—10841no—
407C2×C4—10841no—
408C2×C4—10841no—
409C2×C4C23×D27111yescharacteristic
410C8—21no—
411C8—21no—
412C8—21no—
413C8—2711no—
414C8—2711no—
415C8—5421no—
416C8—5421no—
417C8—5421no—
418C8D4×D27111yescharacteristic
419C8D4×D27111yescharacteristic
420C22—21no—
421C22—41no—
422C22—81no—
423C22—2711no—
424C22—2711no—
425C22—5421no—
426C22—10841no—
427C22—21681no—
428C22#41×D27111yescharacteristic
429C4—21no—
430C4—41no—
431C4—2711no—
432C4—2711no—
433C4—5421no—
434C4—10841no—
435C4—10841no—
436C4C2×D4×D27111yescharacteristic
437C4C2×D4×D27111yescharacteristic
438C2—21no—
439C2—41no—
440C2—81no—
441C2—5421no—
442C2—5421no—
443C2—10841no—
444C2#121×D27111yescharacteristic
subgroupG/H
centerC2#121×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#471×C271C2
socleC2×C271C2×#121
prime radical#31774693C2
hypercenter#471D271
hypocenterC271#3069
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
2. C2×#22576189C2semidirect
3. #137×D271C2semidirect
4. #31774662C2semidirect
5. #31774173C2semidirect
6. #31774172C2semidirect
7. #31774171C2semidirect
8. #31774170C2semidirect
9. #31774159C2semidirect
10. #31774158C2semidirect
11. #31774157C2semidirect
12. #31773944C2semidirect
13. #31773878C2semidirect
14. #101×D271C2semidirect
15. #471×C271C2semidirect
35. C2×Dic1084C22semidirect
36. C2×#16864644C22semidirect
38. C2×#16864642C22semidirect
51. #22576096C22semidirect
52. #22576095C22semidirect
53. #22576095C22semidirect
54. #22576079C22semidirect
55. #22576078C22semidirect
56. #22576078C22semidirect
57. #22576078C22semidirect
58. #22576078C22semidirect
59. #22576077C22semidirect
62. C2×SD16×C271C22semidirect
64. #108×C271C22semidirect
65. #101×C271C22semidirect
66. #100×C271C22semidirect
222. C271#3069semidirect
235. #471D271semidirect
265. C2×SD16C2×D271semidirect
281. #108C2×D271semidirect
287. #101C2×D271semidirect
289. #100C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746932
C2×#225761892
#137×D2712
#317746622
#317741732
#317741722
#317741712
#317741702
#317741592
#317741582
#317741572
#317739442
#317738782
#101×D2712
#471×C2712
#3069271