Finite Group Databasebeta

← #496121

Subgroups — #496121

Order 512.

780 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2251C211yesmaximal, characteristic
2C2×#2176C211yesmaximal, characteristic, complemented
3C2×#2175C211yesmaximal, characteristic
4C2×C4×Q32C211yesmaximal, characteristic, complemented
5C2×C4×SD32C211yesmaximal, characteristic
6#21698C211yesmaximal, characteristic, complemented
7#21696C211yesmaximal, characteristic
8#13240C211yesmaximal, characteristic, complemented
9#13238C211yesmaximal, characteristic
10#13237C211yesmaximal, characteristic
11#13236C211yesmaximal, characteristic, complemented
12#13204C211yesmaximal, characteristic
13#13203C211yesmaximal, characteristic, complemented
14#12908C211yesmaximal, characteristic, complemented
15#12539C211yesmaximal, characteristic, complemented
16C22×Q32C2211yescharacteristic
17C22×Q32C2211yescharacteristic
18C22×SD32C2211yescharacteristic
19C22×SD32C2211yescharacteristic
20C2×#495C2211yescharacteristic
21C2×#494C2211yescharacteristic
22C2×#441—21no—
23C2×#440—21no—
24C2×C4×Q16C2211yescharacteristic
25C2×C4×Q16C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×D8C2211yescharacteristic
28#2327—21no—
29#2326—21no—
30#2251C418yescomplemented
31#2176—24no—
32#2175—24no—
33C4×Q32—24no—
34C4×SD32—24no—
35C2×#367C2211yescharacteristic
36C2×#367C2211yescharacteristic
37C2×#366C2211yescharacteristic
38C2×#366C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#358C2211yescharacteristic
42C2×#358C2211yescharacteristic
43C2×#358C2211yescharacteristic
44C2×#358C2211yescharacteristic
45C2×#357C2211yescharacteristic
46C2×#357C2211yescharacteristic
47C2×#346C2211yescharacteristic
48C2×#346C2211yescharacteristic
49C2×#346C2211yescharacteristic
50C2×C4×C16C2211yescharacteristic
51C2×C4×C16C2211yescharacteristic
52C2×C4×C16C2211yescharacteristic
53#1973—21no—
54#1971—21no—
55#1970C2211yescharacteristic
56#1969C2211yescharacteristic
57#1969C2211yescharacteristic
58#1969C2211yescharacteristic
59#1960—21no—
60#1959—21no—
61#1846C2211yescharacteristic
62#1845C2211yescharacteristic
63#1726C2211yescharacteristic
64C22×Q16—21no—
65C22×Q16C2311yescharacteristic
66C22×Q16C2311yescharacteristic
67C22×Q16C2311yescharacteristic
68C22×SD16—21no—
69C22×D8C2311yescharacteristic
70C2×C4⋊Q8—21no—
71C2×#124—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4—21no—
76C2×Q32—22no—
77C2×Q32—22no—
78C2×Q32—22no—
79C2×Q32—22no—
80C2×Q32C2×C414yescomplemented
81C2×Q32C2×C414yescomplemented
82C2×SD32—22no—
83C2×SD32—22no—
84C2×SD32—22no—
85C2×SD32—22no—
86C2×SD32C2×C414yes—
87C2×SD32C2×C414yes—
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92#495—22no—
93#495C2×C414yes—
94#494—22no—
95#494C2×C414yescomplemented
96#441—44no—
97#440—44no—
98C4×Q16—22no—
99C4×Q16—22no—
100C4×Q16—22no—
101C4×Q16—22no—
102C4×Q16—22no—
103C4×Q16—22no—
104C4×D8—22no—
105C4×D8—22no—
106C2×#107—21no—
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#106—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#103—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#97D411yescharacteristic
121C2×#97D411yescharacteristic
122C2×#97D411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391—21no—
127#391—21no—
128#391—21no—
129#390—21no—
130#388—42no—
131#387—42no—
132#387—42no—
133#385—42no—
134#384—21no—
135#382—21no—
136#376D411yescharacteristic
137#367—22no—
138#367—22no—
139#366—22no—
140#366—22no—
141#358—22no—
142#358—22no—
143#358—22no—
144#358—22no—
145#358—22no—
146#358—22no—
147#357—22no—
148#357—22no—
149#346—22no—
150#346C2×C414yescomplemented
151#346D414yes—
152C4×C16—22no—
153C4×C16—22no—
154C4×C16D414yes—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×D4—21no—
159C23×C4—41no—
160C2×Q16—24no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×Q16—42no—
177C2×Q16—42no—
178C2×Q16C22×C412yes—
179C2×Q16C22×C412yes—
180C2×Q16C22×C412yes—
181C2×SD16—24no—
182C2×SD16—42no—
183C2×SD16—42no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8C22×C412yes—
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2411yescharacteristic
194C4⋊Q8—24no—
195C4⋊Q8—42no—
196#124—24no—
197#124—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×D4—42no—
205C4×D4—42no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—42no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224Q32—28no—
225Q32—28no—
226SD32—28no—
227SD32—28no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16C22×C412yes—
233C2×C16C22×C412yes—
234C2×C16C2×D412yes—
235C2×C16C2×D412yes—
236C2×C16C2×D412yes—
237C2×C16C2×D412yes—
238C2×C16#4112yes—
239C2×C16#4112yes—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—42no—
249#106—42no—
250#103—42no—
251#103—42no—
252#103—42no—
253#103—42no—
254#103—42no—
255#103—42no—
256#102—42no—
257#102—42no—
258#97—22no—
259#97—22no—
260#97D814yes—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8C22×C412yes—
264C4×C8C2×D412yes—
265C4×C8C2×D412yes—
266C4×C8#4112yes—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95D812yes—
272#95D812yes—
273C24—41no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—162no—
307C22×C4C2×D812yes—
308C22×C4C2×D812yes—
309C22×C4C2×D811yescharacteristic
310C22×C4C2×D811yescharacteristic
311C22×C4C22×D411yescharacteristic
312Q16—22no—
313Q16—22no—
314Q16—22no—
315Q16—22no—
316Q16—22no—
317Q16—22no—
318Q16—41no—
319Q16—41no—
320Q16—41no—
321Q16—41no—
322Q16—41no—
323Q16—41no—
324Q16—48no—
325SD16—48no—
326D8—22no—
327D8—22no—
328D8—41no—
329D8—41no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8C4×D412yes—
335C2×C8#12712yes—
336C2×C8C2×D812yes—
337C2×C8C2×D812yes—
338C2×C8#13512yes—
339C2×C8C23×C411yescharacteristic
340C2×C8C22×D411yescharacteristic
341C2×C8C22×D411yescharacteristic
342C2×C8C2×#4111yescharacteristic
343C2×C8C2×#4111yescharacteristic
344C2×C8C2×#4111yescharacteristic
345C4⋊C4—48no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—162no—
358C22⋊C4—48no—
359C22⋊C4—82no—
360C22⋊C4—162no—
361C42—21no—
362C42—21no—
363C42—82no—
364C42—82no—
365C42—82no—
366C42C4×D412yes—
367C42#12712yes—
368C42C2×D812yes—
369C42#13512yes—
370C16—22no—
371C16—22no—
372C16C4×D414yes—
373C16C4×D414yes—
374C23—42no—
375C23—81no—
376C23—82no—
377C23—82no—
378C23—81no—
379C23C22×D811yescharacteristic
380Q8—42no—
381Q8—42no—
382Q8—42no—
383Q8—42no—
384Q8—42no—
385Q8—42no—
386Q8—81no—
387Q8—81no—
388Q8—81no—
389Q8—81no—
390Q8—81no—
391Q8—81no—
392D4—42no—
393D4—42no—
394D4—81no—
395D4—81no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—164no—
421C2×C4C4×D814yes—
422C2×C4#49312yes—
423C2×C4#49312yes—
424C2×C4C2×C4×D411yescharacteristic
425C2×C4C2×#12711yescharacteristic
426C2×C4#53211yescharacteristic
427C2×C4C22×D811yescharacteristic
428C2×C4C2×#13511yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C8—22no—
431C8—22no—
432C8C4×D814yes—
433C8C2×C4×D411yescharacteristic
434C8C2×C4×D411yescharacteristic
435C8#53211yescharacteristic
436C8#53211yescharacteristic
437C22—84no—
438C22—82no—
439C22—161no—
440C22—161no—
441C22—162no—
442C22C2×C4×D811yescharacteristic
443C22C2×#49311yescharacteristic
444C22#316911yescharacteristic
445C22#341612yes—
446C22#341712yes—
447C4—24no—
448C4—22no—
449C4—22no—
450C4—42no—
451C4—82no—
452C4—82no—
453C4—82no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4C4×#12711yescharacteristic
459C4C2×C4×D811yescharacteristic
460C4#314711yescharacteristic
461C4#316911yescharacteristic
462C2—162no—
463C2C4×#49311yescharacteristic
464C2#3345512yes—
465C2#3372111yescharacteristic
466C2#3372211yescharacteristic
467C2#3374512yes—
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2176C2semidirect
4. C2×C4×Q32C2semidirect
6. #21698C2semidirect
8. #13240C2semidirect
11. #13236C2semidirect
13. #13203C2semidirect
14. #12908C2semidirect
15. #12539C2semidirect
30. #2251C4semidirect
80. C2×Q32C2×C4semidirect
81. C2×Q32C2×C4semidirect
95. #494C2×C4semidirect
150. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22512
C2×#21762
C2×#21752
C2×C4×Q322
C2×C4×SD322
#216982
#216962
#132402
#132382
#132372
#132362
#132042
#132032
#129082
#125392