Finite Group Databasebeta

← #238340

Subgroups — #238340

Order 512.

843 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35344C211yesmaximal, characteristic, complemented
2#21815C211yesmaximal, characteristic
3#17595C211yesmaximal, characteristic
4#17561C211yesmaximal, characteristic
5#10378C218yesmaximal, complemented
6#8070C211yesmaximal, characteristic, complemented
7#7977C211yesmaximal, characteristic, complemented
8#7949C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2864—21no—
14#2862—21no—
15#2628—22no—
16#2607—22no—
17#2607—22no—
18#2605—22no—
19#2605C2214yescomplemented
20#2331—22no—
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1980C2214yes—
25#1962—21no—
26#1962—21no—
27#1789—22no—
28#1519—22no—
29#1514—22no—
30#1514C2214yes—
31#1510—22no—
32#1505C2214yes—
33#1502—22no—
34#1300C2214yescomplemented
35#1288C2214yescomplemented
36#1285C2214yescomplemented
37C22×#41—21no—
38#532—21no—
39#532—21no—
40#532—21no—
41#532—21no—
42#532—21no—
43#532—21no—
44#532Q812yescomplemented
45#532C2312yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8Q811yescharacteristic, complemented
48C2×#127Q811yescharacteristic, complemented
49C2×#124—21no—
50C2×#121—21no—
51C2×#121—21no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×#117—21no—
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2312yes—
63#432C2312yes—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#423—21no—
69#423—21no—
70C2×C4⋊C8—21no—
71C2×#104—24no—
72C2×#104—24no—
73#407—21no—
74#407—21no—
75C2×C22⋊C8—21no—
76C4×M16—21no—
77C4×M16C2312yes—
78C2×#97—21no—
79#400—24no—
80#399—24no—
81#395—24no—
82#393—24no—
83#391—24no—
84#391—24no—
85#391—24no—
86#391—24no—
87#389—22no—
88#389—22no—
89#387—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#383D412yes—
94#383C2312yes—
95#382—22no—
96#382—21no—
97#382—21no—
98#381—21no—
99#381D412yes—
100#380—21no—
101#380D412yes—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4C2312yes—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4D412yes—
110#376—21no—
111#376—21no—
112#376C2312yes—
113C43D411yescharacteristic
114C43D411yescharacteristic
115#327—28no—
116#327—28no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—41no—
120C2×#41—41no—
121C2×#41—84no—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—41no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—22no—
133C2×M16—41no—
134C22×C8—41no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—48no—
139C4⋊Q8C2×Q812yes—
140#127—21no—
141#127—21no—
142#127—21no—
143#127C2×Q812yes—
144#124—22no—
145#124—22no—
146#124—41no—
147#124—41no—
148#121—22no—
149#121—22no—
150#121—22no—
151#121—22no—
152#121—41no—
153#121—41no—
154#121—41no—
155#121—41no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—41no—
161C4×D4—41no—
162C4×D4—41no—
163C4×D4—41no—
164#117—44no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×Q811yescharacteristic
190C2×C42C2×Q811yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42C2411yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204#104—84no—
205#104—84no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212#97—22no—
213#97—22no—
214#97—22no—
215#97—41no—
216C4×C8—22no—
217C4×C8—41no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95C2×D414yes—
231#95C2×D414yes—
232C24—41no—
233C24#12011yescharacteristic
234#41—44no—
235#41—44no—
236#41—84no—
237#41—84no—
238#41—84no—
239C2×Q8—22no—
240C2×Q8—48no—
241C2×Q8—41no—
242C2×Q8—84no—
243C2×Q8—84no—
244C2×Q8—82no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—82no—
254C2×D4—82no—
255C2×D4—82no—
256C2×D4—84no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—81no—
267C22×C4—84no—
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4C22⋊Q811yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4#12411yescharacteristic
278C22×C4#12411yescharacteristic
279C22×C4#12411yescharacteristic
280C22×C4#12411yescharacteristic
281C22×C4#12411yescharacteristic
282C22×C4#12411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286M16—44no—
287M16—82no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—82no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—48no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C22⋊C4—44no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C42—24no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—82no—
314C42—81no—
315C42—81no—
316C42—82no—
317C42C22⋊Q811yescharacteristic
318C42C22⋊Q811yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42C22⋊Q811yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42#12411yescharacteristic
324C42#12411yescharacteristic
325C42#12411yescharacteristic
326C42#12411yescharacteristic
327C42#12411yescharacteristic
328C42#12411yescharacteristic
329C42C22×Q811yescharacteristic
330C42C2×#4111yescharacteristic
331C42C2×#4111yescharacteristic
332C42C2×#4111yescharacteristic
333C23—22no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—82no—
338C23—82no—
339C23—82no—
340C23#39311yescharacteristic
341C23#39311yescharacteristic
342C23C2×#12011yescharacteristic
343Q8—44no—
344Q8—82no—
345Q8—168no—
346D4—44no—
347D4—82no—
348D4—82no—
349D4—82no—
350D4—82no—
351D4—84no—
352D4—84no—
353D4—82no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4#39311yescharacteristic
390C2×C4#39311yescharacteristic
391C2×C4#39311yescharacteristic
392C2×C4#39311yescharacteristic
393C2×C4#39311yescharacteristic
394C2×C4#39311yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×C22⋊Q811yescharacteristic
398C2×C4C2×#12411yescharacteristic
399C2×C4C2×#12411yescharacteristic
400C2×C4C2×#12411yescharacteristic
401C8—82no—
402C8—82no—
403C8—82no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—164no—
417C22C2×#39311yescharacteristic
418C22#302112yes—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—44no—
431C4—82no—
432C4—84no—
433C4—84no—
434C4—82no—
435C4#203112yes—
436C2—22no—
437C2—41no—
438C2—41no—
439C2—162no—
440C2#2346311yescharacteristic
441C2C2×#203111yescharacteristic
442C2#2347211yescharacteristic
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35344C2semidirect
5. #10378C2semidirect
6. #8070C2semidirect
7. #7977C2semidirect
8. #7949C2semidirect
19. #2605C22semidirect
34. #1300C22semidirect
35. #1288C22semidirect
36. #1285C22semidirect
44. #532Q8semidirect
47. C2×C4⋊Q8Q8semidirect
48. C2×#127Q8semidirect

Maximal subgroups

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

subgroupindex
#353442
#218152
#175952
#175612
#103782
#80702
#79772
#79492