Finite Group Databasebeta

← #6345715

Subgroups — #6345715

Order 512.

1573 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3170C211yesmaximal, characteristic
2#60250C212yesmaximal
3#18086C218yesmaximal, complemented
4#18058C218yesmaximal, complemented
5#17980C214yesmaximal, complemented
6#17958C214yesmaximal, complemented
7#17793C211yesmaximal, characteristic
8#17501C212yesmaximal
9C2×C8×M16C211yesmaximal, characteristic
10C2×#533C2212yes—
11#3170C412yes—
12#3170C414yes—
13#3170C412yes—
14#3170C2212yes—
15#3170C2212yes—
16#3170C2214yes—
17C2×#500C411yescharacteristic
18C2×#494C411yescharacteristic
19C2×#487C412yes—
20C2×#467C414yes—
21#2986C2218yes—
22#2985—24no—
23#2984C2218yes—
24#2983—22no—
25#2982—22no—
26#2962—24no—
27#2960C2218yes—
28C2×C4×Q16C2214yes—
29#2924—24no—
30#2920C2212yes—
31C2×#431C2211yescharacteristic
32C2×C4×M16C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34#1591C22116yescomplemented
35#1587C22132yescomplemented
36C8×Q16C22116yescomplemented
37#1568C2214yes—
38#1567C2212yes—
39#1566C2212yes—
40#1564C2214yes—
41#1493C2218yes—
42#1491C2218yes—
43#1485C2218yes—
44C8×M16C2214yes—
45C8×M16C2214yes—
46C2×#322C2212yes—
47C2×C82C2212yes—
48C22×Q16C2×C414yes—
49C22×M16—21no—
50C23×C8C2×C412yes—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55#533C2×C414yes—
56#533C2×C414yes—
57#533C2314yes—
58#533C2314yes—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8C2×C412yes—
61C2×#125—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8C2×C414yes—
64C2×C4×Q8—22no—
65C2×C4×Q8C2314yes—
66C2×#117—22no—
67C22×C42C2311yescharacteristic
68#500C2×C412yes—
69#500C2×C412yes—
70#500C2×C414yes—
71#494C2×C412yes—
72#494C2×C414yes—
73#494C2×C412yes—
74#487C2×C414yes—
75#487C2×C414yes—
76#487C2×C418yes—
77#467C2×C418yes—
78#467C2×C418yes—
79#467C2×C418yes—
80#467C2×C418yes—
81#446—28no—
82#446—216no—
83#446—28no—
84#446C23116yes—
85C8×Q8—28no—
86C8×Q8—28no—
87C8×Q8C23116yes—
88C4×Q16C2×C418yes—
89C4×Q16C2×C418yes—
90C4×Q16C2318yes—
91C4×Q16C2318yes—
92#433—216no—
93#432—22no—
94#432—28no—
95#432C2314yes—
96#431—22no—
97#431—28no—
98#431D411yescharacteristic
99#431D411yescharacteristic
100#431D412yes—
101#431C2314yes—
102#431C2312yes—
103#431C2312yes—
104C2×#107C2×C414yes—
105#424—28no—
106#423C2314yes—
107C2×C4⋊C8D412yes—
108C2×#103C2×C418yes—
109#407—22no—
110C2×C22⋊C8D412yes—
111#405—28no—
112C4×M16—22no—
113C4×M16—21no—
114C4×M16—21no—
115C4×M16C2312yes—
116C4×M16C2312yes—
117C2×#97—21no—
118C2×C4×C8C2×C411yescharacteristic
119C2×C4×C8C2×C411yescharacteristic
120C2×C4×C8C2×C411yescharacteristic
121C2×C4×C8C2×C412yes—
122C2×C4×C8C2×C411yescharacteristic
123C2×C4×C8C2312yes—
124C2×C4×C8C2312yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2312yes—
128C2×C4×C8C2312yes—
129#335C2318yes—
130#334C2318yes—
131Q8⋊C8C23132yes—
132#322C2318yes—
133C82C2318yes—
134C22×Q8—22no—
135C22×Q8C22×C414yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4#4111yescharacteristic
139C2×Q16—48no—
140C2×Q16C22×C418yes—
141C2×Q16C22×C418yes—
142C2×M16—28no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—24no—
146C22×C8—22no—
147C22×C8—44no—
148C22×C8C22×C412yes—
149C22×C8C22×C412yes—
150C22×C8C22×C412yes—
151C22×C8C22×C412yes—
152C22×C8C22×C412yes—
153C22×C8C22×C412yes—
154C22×C8C2×D414yes—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8C22×C414yes—
160C4⋊Q8C22×C414yes—
161C4⋊Q8C22×C414yes—
162C4⋊Q8C22×C414yes—
163#125—24no—
164#125—24no—
165#125—24no—
166#125—24no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C22⋊Q8C22×C418yes—
174C22⋊Q8C22×C418yes—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8C22×C418yes—
182C4×Q8C2418yes—
183#117—24no—
184#117—24no—
185#117—24no—
186#117—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4C22×C414yes—
193C2×C22⋊C4—24no—
194C2×C42—24no—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42C2411yescharacteristic
205C2×C42C2411yescharacteristic
206C2×C42C2411yescharacteristic
207#107C22×C418yes—
208#107C22×C418yes—
209C4⋊C8—216no—
210C4⋊C8—216no—
211C4⋊C8—24no—
212C4⋊C8C2×D412yes—
213C4⋊C8C2×D412yes—
214C4⋊C8C2×D414yes—
215C4⋊C8C2418yes—
216#103C22×C4116yes—
217#103C22×C4116yes—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8C2×D412yes—
221C22⋊C8C2×D412yes—
222C22⋊C8C2×D414yes—
223#97—216no—
224#97—24no—
225#97—22no—
226#97—22no—
227C4×C8—216no—
228C4×C8—24no—
229C4×C8—22no—
230C4×C8C22×C412yes—
231C4×C8C22×C412yes—
232C4×C8C22×C412yes—
233C4×C8C22×C411yescharacteristic
234C4×C8C22×C411yescharacteristic
235C4×C8C22×C411yescharacteristic
236C4×C8C22×C412yes—
237C4×C8C22×C411yescharacteristic
238C4×C8C2×D412yes—
239C4×C8C2×D412yes—
240C4×C8C2414yes—
241C4×C8C2412yes—
242C4×C8C2412yes—
243C4×C8C2414yes—
244C24C4×D411yescharacteristic
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—48no—
249C2×Q8—48no—
250C2×Q8C23×C418yes—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—42no—
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4C23×C411yescharacteristic
266C22×C4C23×C411yescharacteristic
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274Q16—216no—
275Q16—48no—
276M16—28no—
277M16—28no—
278M16—48no—
279C2×C8—28no—
280C2×C8—28no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—28no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8C23×C412yes—
290C2×C8C23×C412yes—
291C2×C8C23×C412yes—
292C2×C8C23×C412yes—
293C2×C8C22×D414yes—
294C2×C8C22×D414yes—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4C23×C418yes—
307C22⋊C4—28no—
308C22⋊C4—28no—
309C42—28no—
310C42—28no—
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42C23×C411yescharacteristic
318C42C23×C411yescharacteristic
319C42C23×C411yescharacteristic
320C42C22×D411yescharacteristic
321C42C22×D411yescharacteristic
322C42C22×D411yescharacteristic
323C42C2×#4111yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C42C2511yescharacteristic
327C23—22no—
328C23—22no—
329C23—41no—
330C23C2×C4×D411yescharacteristic
331C23C2×C4×D411yescharacteristic
332C23C2×C4×D411yescharacteristic
333Q8—216no—
334Q8—48no—
335Q8—48no—
336Q8—84no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—48no—
356C2×C4—84no—
357C2×C4#44314yes—
358C2×C4C2×C4×D411yescharacteristic
359C2×C4C2×C4×D411yescharacteristic
360C2×C4C2×C4×D411yescharacteristic
361C2×C4C2×C4×D411yescharacteristic
362C2×C4C2×C4×D411yescharacteristic
363C2×C4C2×C4×D411yescharacteristic
364C2×C4C2×C4×D411yescharacteristic
365C2×C4C2×C4×D411yescharacteristic
366C2×C4C2×C4×D411yescharacteristic
367C2×C4#57814yes—
368C2×C4C24×C411yescharacteristic
369C2×C4C23×D411yescharacteristic
370C2×C4C22×#4111yescharacteristic
371C8—28no—
372C8—22no—
373C8—22no—
374C8—22no—
375C8—24no—
376C8—24no—
377C8—48no—
378C8#56814yes—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22#295012yes—
388C22C2×#44312yes—
389C22C22×C4×D411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—48no—
398C4—48no—
399C4#295012yes—
400C4C2×#44312yes—
401C4C2×#44312yes—
402C4C2×#56812yes—
403C4C2×#57812yes—
404C2—22no—
405C2—22no—
406C2—41no—
407C2#6014411yescharacteristic
408C2C2×#295011yescharacteristic
409C2C22×#44311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalC24C4×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #18086C2semidirect
4. #18058C2semidirect
5. #17980C2semidirect
6. #17958C2semidirect
34. #1591C22semidirect
35. #1587C22semidirect
36. C8×Q16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
#602502
#180862
#180582
#179802
#179582
#177932
#175012
C2×C8×M162