Finite Group Databasebeta

← #496799

Subgroups — #496799

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33432C211yesmaximal, characteristic, complemented
2C2×#2113C211yesmaximal, characteristic, complemented
3#21466C211yesmaximal, characteristic
4#13196C212yesmaximal, complemented
5#13125C212yesmaximal, complemented
6#13122C212yesmaximal, complemented
7#12980C212yesmaximal, complemented
8#12978C211yesmaximal, characteristic, complemented
9#12914C211yesmaximal, characteristic, complemented
10#12683C211yesmaximal, characteristic, complemented
11#12611C211yesmaximal, characteristic, complemented
12C22×#131C2211yescharacteristic
13C2×#504C2211yescharacteristic, complemented
14C22×M32C2211yescharacteristic
15C23×C16C2211yescharacteristic
16#3002C411yescharacteristic
17C2×#443C2212yes—
18#2929C2211yescharacteristic
19#2168—21no—
20#2168—21no—
21D4×C16—21no—
22D4×C16—21no—
23C2×#364C2212yescomplemented
24#2151C2211yescharacteristic, complemented
25C2×#350C2212yes—
26C2×#350C2212yes—
27C2×#350C2212yescomplemented
28#2113—21no—
29#2113—21no—
30#2113C2212yes—
31#2113C2212yes—
32#2113C2211yescharacteristic, complemented
33#2112C2211yescharacteristic
34C2×C22⋊C16—21no—
35C2×C22⋊C16—21no—
36C2×C22⋊C16C2211yescharacteristic
37#2108C2211yescharacteristic, complemented
38C2×#346C2212yescomplemented
39#1876C412yes—
40#1871C411yescharacteristic
41#1866C412yes—
42#1855C411yescharacteristic
43#1789C411yescharacteristic
44#1764C2212yes—
45C8×C22⋊C4C2211yescharacteristic
46#1379C2212yescomplemented
47#1376C2212yescomplemented
48#1374C2212yescomplemented
49C22×#41C2×C411yescharacteristic
50C2×#135C2×C412yes—
51C2×#131—21no—
52C2×#131—21no—
53C2×#131—42no—
54C2×#131—42no—
55C2×#131C2×C412yes—
56C2×#131C2312yes—
57C2×#131C2311yescharacteristic
58C22×M16—21no—
59C22×M16C2×C411yescharacteristic
60C23×C8—21no—
61C23×C8C2311yescharacteristic
62#525C2×C411yescharacteristic
63#504—42no—
64C2×M32—21no—
65C2×M32—21no—
66C2×M32—42no—
67C2×M32C2311yescharacteristic
68C2×M32C2311yescharacteristic
69C2×M32C2312yes—
70C22×C16—21no—
71C22×C16—21no—
72C22×C16—42no—
73C22×C16D411yescharacteristic
74C22×C16D411yescharacteristic
75C22×C16D411yescharacteristic
76C22×C16D411yescharacteristic
77C22×C16C2312yes—
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80#452C812yes—
81#451C811yescharacteristic
82#450C811yescharacteristic
83#449C812yes—
84#448C811yescharacteristic
85#447C811yescharacteristic
86#443—42no—
87#443—42no—
88#435—21no—
89#435—21no—
90C8×D4—21no—
91C8×D4—21no—
92C2×#108C2×C412yes—
93#424C2×C411yescharacteristic
94C2×#104C2×C412yes—
95C2×#104C2×C412yes—
96C2×#104C2×C412yes—
97#416C2×C412yes—
98#408C2×C411yescharacteristic
99#407C2×C411yescharacteristic
100C2×C22⋊C8C2×C411yescharacteristic
101#405C2311yescharacteristic
102C2×C4×C8C2312yes—
103C4×C22⋊C4C2×C411yescharacteristic
104#364—42no—
105C4⋊C16—21no—
106C4⋊C16—21no—
107#350—42no—
108#350—42no—
109#350—42no—
110C22⋊C16—21no—
111C22⋊C16—21no—
112C22⋊C16—21no—
113C22⋊C16—21no—
114C22⋊C16—22no—
115C22⋊C16—22no—
116C22⋊C16—22no—
117C22⋊C16—22no—
118C22⋊C16—21no—
119C22⋊C16—21no—
120#346—21no—
121#346—42no—
122C4×C16—21no—
123#341C2×C412yes—
124#339C2×C412yes—
125#336C2×C412yes—
126C2×#41—21no—
127C2×#41—21no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41C2×C812yes—
131C2×#41C22×C411yescharacteristic
132C2×#41C22×C412yes—
133C22×Q8C2×C811yescharacteristic
134C22×D4—21no—
135C22×D4C2×C811yescharacteristic
136C23×C4—21no—
137C23×C4C22×C411yescharacteristic
138#135—42no—
139#135—42no—
140C2×Q16C2×C812yes—
141C2×SD16C2×C812yes—
142C2×SD16C2×C812yes—
143C2×D8C2×C812yes—
144#131—22no—
145#131—22no—
146#131—22no—
147#131—22no—
148#131—22no—
149#131—22no—
150#131—42no—
151#131—42no—
152#131—42no—
153#131—42no—
154#131—44no—
155C2×M16—21no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—21no—
159C2×M16—44no—
160C2×M16—42no—
161C2×M16C22×C412yes—
162C2×M16C22×C411yescharacteristic
163C2×M16C22×C411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C412yes—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8#4111yescharacteristic
177C22×C8#4111yescharacteristic
178C22×C8C2411yescharacteristic
179#123—21no—
180C22⋊Q8C2×C811yescharacteristic
181C22⋊Q8C2×C811yescharacteristic
182#121C2×C811yescharacteristic
183#121C2×C811yescharacteristic
184#120—21no—
185C4×D4—21no—
186C4×D4—21no—
187#117C22×C411yescharacteristic
188C2×C22⋊C4—21no—
189C2×C42C22×C412yes—
190M32—22no—
191M32—22no—
192M32—22no—
193M32—22no—
194M32—42no—
195M32—42no—
196C2×C16—22no—
197C2×C16—22no—
198C2×C16—21no—
199C2×C16—22no—
200C2×C16—21no—
201C2×C16—22no—
202C2×C16—22no—
203C2×C16—22no—
204C2×C16—41no—
205C2×C16—41no—
206C2×C16—41no—
207C2×C16—41no—
208C2×C16—42no—
209C2×C16—42no—
210C2×C16C2×D411yescharacteristic
211C2×C16C2×D411yescharacteristic
212C2×C16C2×D411yescharacteristic
213C2×C16C2×D411yescharacteristic
214#108—42no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217#104—42no—
218#104—42no—
219#104—42no—
220#103C2×C812yes—
221#103C2×C812yes—
222#102C2×C812yes—
223#102C2×C812yes—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8C2×C811yescharacteristic
229C22⋊C8C2×C811yescharacteristic
230C22⋊C8C2×C812yes—
231#97—21no—
232C4×C8—21no—
233C4×C8—22no—
234C4×C8—22no—
235#95—21no—
236C24—21no—
237C24#13111yescharacteristic
238#41—22no—
239#41—22no—
240#41—22no—
241#41—22no—
242#41—22no—
243#41—22no—
244#41—42no—
245#41—42no—
246#41—42no—
247#41—44no—
248#41—42no—
249C2×Q8—21no—
250C2×Q8—42no—
251C2×Q8C22×C812yes—
252C2×Q8C22×C811yescharacteristic
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—22no—
258C2×D4—21no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4C22×C812yes—
262C2×D4C22×C811yescharacteristic
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C4×D411yescharacteristic
273C22×C4C4×D411yescharacteristic
274C22×C4C22×C811yescharacteristic
275C22×C4C22×C811yescharacteristic
276C22×C4#13111yescharacteristic
277C22×C4C23×C411yescharacteristic
278Q16—42no—
279SD16—42no—
280SD16—42no—
281D8—42no—
282M16—22no—
283M16—22no—
284M16—22no—
285M16—22no—
286M16—44no—
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—44no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8C4×D411yescharacteristic
310C2×C8C4×D411yescharacteristic
311C2×C8C22×C812yes—
312C2×C8C22×C812yes—
313C2×C8C22×D411yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8H(5,𝔽2)12yes—
316C2×C8H(5,𝔽2)12yes—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4C22×C811yescharacteristic
320C4⋊C4C22×C811yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—42no—
327C42—21no—
328C42—22no—
329C42—22no—
330C16—21no—
331C16—21no—
332C16—21no—
333C16—21no—
334C16—41no—
335C16—42no—
336C16—41no—
337C16—42no—
338C23—22no—
339C23—22no—
340C23—21no—
341C23—22no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23C8×D411yescharacteristic
347C23C8×D411yescharacteristic
348C23C2×#13111yescharacteristic
349Q8—22no—
350Q8—22no—
351Q8—22no—
352Q8—42no—
353D4—22no—
354D4—22no—
355D4—22no—
356D4—42no—
357D4—42no—
358D4—44no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—42no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4C8×D411yescharacteristic
384C2×C4C8×D411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4#51812yes—
387C2×C4C23×C811yescharacteristic
388C2×C4#56812yes—
389C8—21no—
390C8—21no—
391C8—22no—
392C8—22no—
393C8—42no—
394C8—41no—
395C8—41no—
396C8#53411yescharacteristic
397C8#53511yescharacteristic
398C8#53812yes—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—44no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—42no—
413C22—84no—
414C22#217112yes—
415C22C2×C8×D411yescharacteristic
416C22#297612yes—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—82no—
424C4#232111yescharacteristic
425C4#293011yescharacteristic
426C4#297212yes—
427C2—22no—
428C2—22no—
429C2—41no—
430C2—81no—
431C2—82no—
432C2#2202811yescharacteristic
433C2C2×#217111yescharacteristic
434C2#3344111yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radical#3002C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33432C2semidirect
2. C2×#2113C2semidirect
4. #13196C2semidirect
5. #13125C2semidirect
6. #13122C2semidirect
7. #12980C2semidirect
8. #12978C2semidirect
9. #12914C2semidirect
10. #12683C2semidirect
11. #12611C2semidirect
13. C2×#504C22semidirect
23. C2×#364C22semidirect
24. #2151C22semidirect
27. C2×#350C22semidirect
32. #2113C22semidirect
37. #2108C22semidirect
38. C2×#346C22semidirect
46. #1379C22semidirect
47. #1376C22semidirect
48. #1374C22semidirect

Maximal subgroups

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

subgroupindex
#334322
C2×#21132
#214662
#131962
#131252
#131222
#129802
#129782
#129142
#126832
#126112