Finite Group Databasebeta

← #219639

Subgroups — #219639

Order 512.

522 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22258C211yesmaximal, characteristic, complemented
2#22230C211yesmaximal, characteristic
3#19705C211yesmaximal, characteristic
4#9769C211yesmaximal, characteristic
5#9755C211yesmaximal, characteristic, complemented
6#9298C211yesmaximal, characteristic, complemented
7#9297C211yesmaximal, characteristic, complemented
8#9295C211yesmaximal, characteristic
9#9295C211yesmaximal, characteristic
10#9294C211yesmaximal, characteristic
11#9294C211yesmaximal, characteristic
12#9290C211yesmaximal, characteristic, complemented
13#9288C211yesmaximal, characteristic, complemented
14#8795C211yesmaximal, characteristic, complemented
15#8142C211yesmaximal, characteristic, complemented
16#3230C2211yescharacteristic
17#3209C2211yescharacteristic
18#3208C2211yescharacteristic
19#3208C2211yescharacteristic
20#3160C2211yescharacteristic
21C2×#486C2211yescharacteristic
22#2961C2211yescharacteristic
23#2961C2211yescharacteristic
24#2331C2211yescharacteristic
25#2328C2211yescharacteristic
26#1995C2211yescharacteristic
27#1995C2211yescharacteristic
28#1993C2211yescharacteristic
29#1961C2211yescharacteristic
30#1926—21no—
31#1925—21no—
32#1922—21no—
33#1922—21no—
34#1922—21no—
35#1922—21no—
36#1864—21no—
37#1862—21no—
38#1846C2211yescharacteristic
39#1837C2211yescharacteristic
40#1746C2211yescharacteristic
41#1722C412yescomplemented
42#1720C412yescomplemented
43#1719C412yescomplemented
44#1714C412yescomplemented
45#1549C2211yescharacteristic
46#1548C2211yescharacteristic
47#1545C2211yescharacteristic
48#1545C2211yescharacteristic
49#1518C2211yescharacteristic
50#1518C2211yescharacteristic
51#1518C2211yescharacteristic
52#1518C2211yescharacteristic
53#1452C2211yescharacteristic
54#1450C2211yescharacteristic
55#1286C2211yescharacteristic
56#1286C2211yescharacteristic
57#1286C2211yescharacteristic
58#1286C2211yescharacteristic
59#1275C2211yescharacteristic
60#1275C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63#533C2311yescharacteristic
64#533C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×#127C2311yescharacteristic
67C2×C4×D4—21no—
68#500C2×C411yescharacteristic, complemented
69#500C2×C411yescharacteristic, complemented
70#500C2×C412yescomplemented
71#499—21no—
72#499—21no—
73#498C2×C411yescharacteristic, complemented
74#498C2×C411yescharacteristic, complemented
75#494C2×C412yescomplemented
76#492C2×C412yes—
77#492C2×C412yes—
78#487C2×C412yescomplemented
79#487C2×C412yescomplemented
80#486—21no—
81#486—21no—
82#486C2×C412yes—
83#486C2×C412yes—
84#486C2×C412yes—
85#486C2×C412yes—
86#484—41no—
87#477—41no—
88#462—41no—
89#460—41no—
90#460—41no—
91#451—41no—
92#450—41no—
93#450—41no—
94#446—41no—
95#446—41no—
96#446—41no—
97#446—41no—
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#427—21no—
103#427—21no—
104C2×#107—21no—
105C2×#106—21no—
106#424—21no—
107#424—21no—
108#423C2311yescharacteristic
109#423C2311yescharacteristic
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114#405—21no—
115#405—21no—
116C2×#97C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—41no—
119#390—21no—
120#389—21no—
121#387—41no—
122#386—41no—
123#386—41no—
124#382C2311yescharacteristic
125#382C2311yescharacteristic
126C4×C4⋊C4—21no—
127#376C2311yescharacteristic
128#341—21no—
129#341—21no—
130#340—21no—
131#340—21no—
132#339—21no—
133#339—21no—
134#339—21no—
135#339—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138#336D411yescharacteristic
139#336D411yescharacteristic
140#333—41no—
141#332—41no—
142#331—41no—
143#331—41no—
144#328—41no—
145#328—41no—
146#328—41no—
147#328—41no—
148#322—21no—
149#322C2×C412yescomplemented
150#322C2×C412yescomplemented
151C82—21no—
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C23×C4—41no—
157C2×Q16—42no—
158C2×SD16—42no—
159C2×SD16—42no—
160C2×M16—21no—
161C2×M16—21no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C22×C412yes—
170C4⋊Q8C22×C411yescharacteristic
171C4⋊Q8C22×C411yescharacteristic
172C4⋊Q8C22×C411yescharacteristic
173C4⋊Q8C22×C411yescharacteristic
174#127—21no—
175#127—21no—
176#127—21no—
177#127C22×C412yes—
178#125—21no—
179#125—21no—
180C22⋊Q8—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×D4—81no—
189C4×D4—81no—
190#117—21no—
191#117—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—41no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2411yescharacteristic
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—42no—
209#107—81no—
210#106—21no—
211#106—21no—
212#106—21no—
213#106—21no—
214#106—81no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225#103—42no—
226#103—42no—
227#102—22no—
228#102—22no—
229#102—22no—
230#102—22no—
231#102—42no—
232#102—42no—
233#102—81no—
234#102—81no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239#97—21no—
240#97—21no—
241#97—41no—
242#97—41no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—41no—
246C4×C8—41no—
247C4×C8C22×C412yes—
248C4×C8C22×C412yes—
249C4×C8C22×C411yescharacteristic
250C4×C8C22×C411yescharacteristic
251C4×C8#4111yescharacteristic
252C4×C8#4111yescharacteristic
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261C24—41no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—161no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—161no—
280C2×D4—161no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—161no—
288C22×C4—161no—
289C22×C4C22×D411yescharacteristic
290C22×C4H(5,𝔽2)11yescharacteristic
291C22×C4H(5,𝔽2)11yescharacteristic
292Q16—162no—
293SD16—162no—
294SD16—162no—
295M16—41no—
296M16—41no—
297M16—41no—
298M16—41no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8C4×D411yescharacteristic
316C2×C8C4×D411yescharacteristic
317C2×C8C4×D411yescharacteristic
318C2×C8C4×D411yescharacteristic
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—161no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—82no—
343C22⋊C4—161no—
344C22⋊C4—161no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—41no—
350C42—41no—
351C42—161no—
352C42C23×C411yescharacteristic
353C42C2×#4111yescharacteristic
354C42H(5,𝔽2)11yescharacteristic
355C42H(5,𝔽2)11yescharacteristic
356C23—42no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23#53511yescharacteristic
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—82no—
368Q8—81no—
369Q8—161no—
370D4—42no—
371D4—42no—
372D4—82no—
373D4—82no—
374D4—81no—
375D4—81no—
376D4—161no—
377D4—161no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—164no—
400C2×C4C2×C4×D411yescharacteristic
401C2×C4#51811yescharacteristic
402C2×C4#51811yescharacteristic
403C2×C4#53511yescharacteristic
404C2×C4#54011yescharacteristic
405C2×C4#54011yescharacteristic
406C2×C4#57711yescharacteristic
407C2×C4#57711yescharacteristic
408C2×C4#57811yescharacteristic
409C2×C4#57811yescharacteristic
410C8—21no—
411C8—21no—
412C8—21no—
413C8—21no—
414C8—42no—
415C8—42no—
416C8—41no—
417C8—41no—
418C8—81no—
419C8—81no—
420C8—81no—
421C8—81no—
422C22—21no—
423C22—21no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22#232811yescharacteristic
432C22#322211yescharacteristic
433C22#322311yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—82no—
443C4—81no—
444C4—81no—
445C4—161no—
446C4—161no—
447C4#294711yescharacteristic
448C4#294711yescharacteristic
449C4#322511yescharacteristic
450C4#322511yescharacteristic
451C2—21no—
452C2—21no—
453C2—162no—
454C2#2227811yescharacteristic
455C2#2228411yescharacteristic
456C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22258C2semidirect
5. #9755C2semidirect
6. #9298C2semidirect
7. #9297C2semidirect
12. #9290C2semidirect
13. #9288C2semidirect
14. #8795C2semidirect
15. #8142C2semidirect
41. #1722C4semidirect
42. #1720C4semidirect
43. #1719C4semidirect
44. #1714C4semidirect
68. #500C2×C4semidirect
69. #500C2×C4semidirect
70. #500C2×C4semidirect
73. #498C2×C4semidirect
74. #498C2×C4semidirect
75. #494C2×C4semidirect
78. #487C2×C4semidirect
79. #487C2×C4semidirect
149. #322C2×C4semidirect
150. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222582
#222302
#197052
#97692
#97552
#92982
#92972
#92952
#92942
#92902
#92882
#87952
#81422