Finite Group Databasebeta

← #7623434

Subgroups — #7623434

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60713C211yesmaximal, characteristic
2#60666C211yesmaximal, characteristic, complemented
3#33810C214yesmaximal, complemented
4#33809C214yesmaximal, complemented
5#33799C214yesmaximal, complemented
6#33798C214yesmaximal, complemented
7#33732C211yesmaximal, characteristic, complemented
8#33726C211yesmaximal, characteristic, complemented
9#33706C211yesmaximal, characteristic, complemented
10C2×#2242C211yesmaximal, characteristic, complemented
11#33651C212yesmaximal, complemented
12#33650C212yesmaximal
13#33632C212yesmaximal, complemented
14#33631C212yesmaximal, complemented
15#33420C211yesmaximal, characteristic, complemented
16#3510—21no—
17#3504—21no—
18#3183—22no—
19#3175C2214yes—
20#3173—22no—
21#3168C2214yes—
22#3166—21no—
23#3166—22no—
24#3164C2212yes—
25#3164C2214yescomplemented
26#3163—21no—
27#3163—22no—
28#3160C2211yescharacteristic
29#3160C2211yescharacteristic
30#3160C2212yescomplemented
31C2×#499C2211yescharacteristic
32C2×#499C2212yescomplemented
33C2×#495C2211yescharacteristic
34#3140—21no—
35#3059C2211yescharacteristic
36#3058C2211yescharacteristic
37#3057—21no—
38#2950C2211yescharacteristic
39#2949—21no—
40#2948C2211yescharacteristic
41#2913C2212yes—
42#2912—21no—
43C4×#131C2211yescharacteristic
44#2255C2214yescomplemented
45#2254C2212yescomplemented
46#2253C2212yescomplemented
47#2253C2214yescomplemented
48#2253C2212yescomplemented
49#2252C2212yescomplemented
50#2243C2214yes—
51#2242C2212yes—
52#2242C2214yes—
53#2242C2212yes—
54#2242C2212yes—
55#2242C2212yes—
56#2237C2214yescomplemented
57#2237C2214yescomplemented
58#2236C2214yescomplemented
59#2235C2214yescomplemented
60#2234C2214yes—
61#2234C2214yes—
62#2233C2214yes—
63#2232C2214yes—
64#2229C2214yescomplemented
65#2229C2214yescomplemented
66#2228C2214yes—
67#2228C2214yes—
68#2227C2214yescomplemented
69#2226C2214yes—
70#2225C2214yescomplemented
71#2224C2214yes—
72#2164C2212yescomplemented
73#2164C2212yescomplemented
74#2162C2214yescomplemented
75#2159C2212yes—
76#2159C2212yes—
77#2158C2212yes—
78#2158C2212yes—
79#2151C2212yes—
80#2141C2212yes—
81#2140C2212yes—
82C2×#358C2212yes—
83C2×#357C2212yes—
84#2113C2212yes—
85C4×M32C2211yescharacteristic
86C2×C4×C16C2211yescharacteristic
87C22×Q16C2311yescharacteristic
88C22×SD16—21no—
89C22×D8C2311yescharacteristic
90C2×#131D411yescharacteristic
91C2×#131C2311yescharacteristic
92#559—42no—
93#557—42no—
94#556—42no—
95#554—44no—
96#554—42no—
97#551—44no—
98#544—22no—
99#544—24no—
100#542—24no—
101#536—21no—
102#535—21no—
103#533—21no—
104#533—21no—
105#533—22no—
106#529—24no—
107C2×#125—21no—
108C2×#125—22no—
109C2×#124—21no—
110#520—22no—
111#519—21no—
112#518—21no—
113C4×#41D411yescharacteristic
114C2×M32C2312yes—
115C22×C16C2312yes—
116#501—24no—
117#501—24no—
118#501—24no—
119#500C2312yes—
120#500C2312yes—
121#500C2314yes—
122#499—21no—
123#499—21no—
124#499—22no—
125#499—22no—
126#499—21no—
127#499—21no—
128#499—22no—
129#499C2312yes—
130#499C2312yes—
131#499C2311yescharacteristic
132#499C2314yes—
133#499C2311yescharacteristic
134#499C2312yes—
135#498—21no—
136#498—21no—
137#498—22no—
138#495—21no—
139#495—21no—
140#495C2312yes—
141#495C2312yes—
142#489—24no—
143#488—24no—
144#470—22no—
145#469—22no—
146#468—24no—
147#467C2314yes—
148#466C2314yes—
149#465—24no—
150#442—22no—
151#441—22no—
152#440—24no—
153C4×Q16C2314yes—
154C4×SD16—24no—
155C4×D8C2314yes—
156#428—22no—
157#428—24no—
158#428—22no—
159#428—24no—
160#427—21no—
161#427—21no—
162#427—22no—
163#427C2312yes—
164#427C2311yescharacteristic
165#427C2311yescharacteristic
166#427C2314yes—
167C2×#107C2312yes—
168C2×#107C2314yes—
169C2×#106—21no—
170C2×#106—22no—
171#419—21no—
172#419—21no—
173#418—21no—
174#418—21no—
175#405D412yes—
176#405D411yescharacteristic
177#405C2311yescharacteristic
178#405C2312yes—
179C4×M16D411yescharacteristic
180C4×M16D412yes—
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2312yes—
183#367C2312yes—
184#367C2312yes—
185#367C2312yes—
186#367C2312yes—
187#367C2314yes—
188#367C2314yes—
189#366C2312yes—
190#366C2312yes—
191#366C2312yes—
192#366C2312yes—
193#366C2314yes—
194#366C2314yes—
195C4⋊C16C2314yes—
196C4⋊C16C2314yes—
197#358C2314yes—
198#358C2314yes—
199#358C2314yes—
200#358C2314yes—
201#357C2314yes—
202#357C2314yes—
203#357C2314yes—
204#357C2314yes—
205C22⋊C16C2314yes—
206C22⋊C16C2314yes—
207#346C2312yes—
208C4×C16C2312yes—
209C4×C16C2311yescharacteristic
210C4×C16C2311yescharacteristic
211C4×C16C2312yes—
212C2×#41C2×D411yescharacteristic
213C22×Q8—21no—
214C22×D4—21no—
215C2×Q16—21no—
216C2×Q16—24no—
217C2×Q16C2412yes—
218C2×SD16—22no—
219C2×SD16—22no—
220C2×SD16—44no—
221C2×D8—21no—
222C2×D8—24no—
223C2×D8C2412yes—
224#131—24no—
225#131—24no—
226C2×M16C2×D411yescharacteristic
227C2×M16C2×D412yes—
228C2×M16C2×D411yescharacteristic
229C2×M16C2×D412yes—
230C22×C8C2×D412yes—
231C22×C8C2×D411yescharacteristic
232C22×C8C2412yes—
233C22×C8C2411yescharacteristic
234C4⋊Q8—22no—
235C4⋊Q8—24no—
236C4⋊Q8—22no—
237C4⋊Q8—41no—
238C4⋊Q8—42no—
239C4⋊Q8—44no—
240#127—41no—
241#126—44no—
242#126—44no—
243#125—22no—
244#125—21no—
245#125—24no—
246#125—22no—
247#125—22no—
248#125—21no—
249#125—42no—
250#125—44no—
251#125—44no—
252#125—42no—
253#124—22no—
254#124—22no—
255#124—41no—
256#124—41no—
257#124—44no—
258#124—42no—
259#123—42no—
260#123—44no—
261#123—44no—
262C22⋊Q8—24no—
263C22⋊Q8—42no—
264C22⋊Q8—42no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270#121—24no—
271#120—42no—
272C4×Q8—24no—
273C4×Q8—42no—
274C4×Q8—42no—
275C4×Q8—44no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8D812yes—
279C4×D4—24no—
280C4×D4—44no—
281C4×D4—42no—
282C4×D4—44no—
283C4×D4—42no—
284C4×D4D814yes—
285C4×D4D812yes—
286#117—21no—
287#117—21no—
288#117—24no—
289#117—22no—
290#117—42no—
291#117—44no—
292#117C2×D411yescharacteristic
293#117C2×D412yes—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D412yes—
302M32—24no—
303C2×C16—24no—
304C2×C16C2412yes—
305C2×C16C2412yes—
306C2×C16C2414yes—
307#107—24no—
308#107—22no—
309#107—24no—
310#107—22no—
311#107—22no—
312#107C2414yes—
313#107C2412yes—
314#107C2412yes—
315#107C2412yes—
316#107C2412yes—
317#106—22no—
318#106—21no—
319#106—24no—
320#106—22no—
321#106—21no—
322#106—22no—
323#106—24no—
324#106—22no—
325#106—21no—
326#106—21no—
327#103—24no—
328#103—24no—
329#102—24no—
330#102—24no—
331#97C2×D412yes—
332#97C2×D412yes—
333#97C2×D412yes—
334C4×C8C2×D412yes—
335C4×C8C2×D411yescharacteristic
336C4×C8C2×D412yes—
337C4×C8C2×D411yescharacteristic
338C4×C8C2411yescharacteristic
339C4×C8C2411yescharacteristic
340C4×C8C2412yes—
341C24—41no—
342#41—22no—
343#41—22no—
344C2×Q8—22no—
345C2×Q8—41no—
346C2×Q8—42no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8C2×D811yescharacteristic
352C2×D4—22no—
353C2×D4—44no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4C2×D812yes—
358C2×D4C2×D811yescharacteristic
359C22×C4—41no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4C2×D812yes—
364C22×C4C2×D814yes—
365C22×C4C22×D412yes—
366C22×C4C22×D411yescharacteristic
367Q16—24no—
368Q16—42no—
369SD16—44no—
370SD16—44no—
371D8—24no—
372D8—42no—
373M16—22no—
374M16—22no—
375M16—24no—
376M16—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8#13714yes—
380C2×C8C22×D412yes—
381C2×C8C22×D412yes—
382C2×C8C22×D411yescharacteristic
383C2×C8C22×D411yescharacteristic
384C2×C8C22×D411yescharacteristic
385C2×C82−1+414yes—
386C2×C8C2511yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4C2×D812yes—
408C4⋊C4C2×D814yes—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4C2×D812yes—
419C22⋊C4C2×D814yes—
420C42—41no—
421C42—44no—
422C42—42no—
423C42—41no—
424C42—44no—
425C42C2×D814yes—
426C42C2×D812yes—
427C42C22×D411yescharacteristic
428C42C22×D412yes—
429C42C22×D411yescharacteristic
430C16—22no—
431C16—22no—
432C16—24no—
433C23—42no—
434C23—82no—
435C23—162no—
436C23C22×D812yes—
437C23C22×D811yescharacteristic
438Q8—22no—
439Q8—42no—
440Q8—42no—
441Q8—82no—
442Q8—164no—
443Q8—162no—
444Q8—162no—
445D4—24no—
446D4—22no—
447D4—42no—
448D4—42no—
449D4—82no—
450D4—164no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—164no—
463C2×C4—162no—
464C2×C4—164no—
465C2×C4#53612yes—
466C2×C4C22×D812yes—
467C2×C4C22×D814yes—
468C2×C4C22×D812yes—
469C2×C4C22×D812yes—
470C2×C4C22×D811yescharacteristic
471C2×C4C2×#13712yes—
472C2×C4C23×D411yescharacteristic
473C8—22no—
474C8—22no—
475C8#53612yes—
476C8C2×#13712yes—
477C8C2×#13712yes—
478C8C2×2−1+412yes—
479C22—22no—
480C22—22no—
481C22—24no—
482C22—82no—
483C22—84no—
484C22—322no—
485C22#322612yes—
486C22C23×D811yescharacteristic
487C4—22no—
488C4—22no—
489C4—24no—
490C4—22no—
491C4—82no—
492C4—82no—
493C4—84no—
494C4—82no—
495C4—82no—
496C4—82no—
497C4#322612yes—
498C4C2×#53611yescharacteristic
499C4C22×#13711yescharacteristic
500C2—22no—
501C2—42no—
502C2—162no—
503C2C2×#322611yescharacteristic
504C2#6263512yes—
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#3058C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60666C2semidirect
3. #33810C2semidirect
4. #33809C2semidirect
5. #33799C2semidirect
6. #33798C2semidirect
7. #33732C2semidirect
8. #33726C2semidirect
9. #33706C2semidirect
10. C2×#2242C2semidirect
11. #33651C2semidirect
13. #33632C2semidirect
14. #33631C2semidirect
15. #33420C2semidirect
25. #3164C22semidirect
30. #3160C22semidirect
32. C2×#499C22semidirect
44. #2255C22semidirect
45. #2254C22semidirect
46. #2253C22semidirect
47. #2253C22semidirect
48. #2253C22semidirect
49. #2252C22semidirect
56. #2237C22semidirect
57. #2237C22semidirect
58. #2236C22semidirect
59. #2235C22semidirect
64. #2229C22semidirect
65. #2229C22semidirect
68. #2227C22semidirect
70. #2225C22semidirect
72. #2164C22semidirect
73. #2164C22semidirect
74. #2162C22semidirect

Maximal subgroups

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

subgroupindex
#607132
#606662
#338102
#338092
#337992
#337982
#337322
#337262
#337062
C2×#22422
#336512
#336502
#336322
#336312
#334202