Finite Group Databasebeta

← #204134

Subgroups — #204134

Order 512.

1202 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#34955C211yesmaximal, characteristic
2#20755C211yesmaximal, characteristic, complemented
3C8×#117C211yesmaximal, characteristic, complemented
4C2×#1492C211yesmaximal, characteristic
5C2×#1480C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic, complemented
7C22×Q8⋊C8C211yesmaximal, characteristic, complemented
8#9645C214yesmaximal, complemented
9#9639C214yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2592—22no—
17#2591—22no—
18#2568C412yes—
19#2548C412yes—
20#2504C412yes—
21#2436C412yes—
22#2398—22no—
23C4×#124C2214yes—
24#2297—22no—
25C4×#117C2211yescharacteristic
26#1832—22no—
27#1831C2214yes—
28C8×C4⋊C4—22no—
29C8×C22⋊C4—22no—
30#1744C2212yes—
31#1729—22no—
32C42×C8C2212yes—
33#1492—24no—
34#1487—24no—
35#1480—24no—
36C2×Q8⋊C8—24no—
37C2×Q8⋊C8C2212yes—
38C2×Q8⋊C8C2212yes—
39C2×Q8⋊C8C2214yes—
40C2×D4⋊C8C2214yes—
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C22×#41—21no—
44C23×Q8C2×C411yescharacteristic
45C22×M16—21no—
46C23×C8—21no—
47C23×C8—21no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55C2×#124C2×C414yes—
56C2×#121—21no—
57C2×#121C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2312yes—
66C2×C4×D4—21no—
67C2×C4×D4C2312yes—
68C2×#117D412yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#431—22no—
74#431D414yes—
75#423—22no—
76#423—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—22no—
87C2×C4×C8—24no—
88C2×C4×C8D414yes—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2312yes—
92#400—24no—
93#397—24no—
94#394—24no—
95#393—24no—
96#391—24no—
97#391C2×C414yes—
98#390—24no—
99#390C2×C414yes—
100#388—28no—
101#386—24no—
102#386—24no—
103#382C2×C414yes—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—28no—
107C4×C22⋊C4C2×C412yes—
108C4×C22⋊C4D412yes—
109#376D412yes—
110C43C2312yes—
111#336—28no—
112#336—24no—
113Q8⋊C8—24no—
114Q8⋊C8—24no—
115Q8⋊C8—28no—
116D4⋊C8—28no—
117D4⋊C8—24no—
118D4⋊C8—24no—
119C2×#41—42no—
120C2×#41—42no—
121C2×#41—44no—
122C2×#41—44no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—42no—
129C22×Q8C22×C412yes—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4C22×C412yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×M16—42no—
141C2×M16—44no—
142C22×C8—21no—
143C22×C8—24no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—44no—
149C22×C8—42no—
150C22×C8—44no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153#127—22no—
154#127—22no—
155#124—28no—
156#124—44no—
157#124—42no—
158#124—42no—
159#121—24no—
160#121—24no—
161#121—42no—
162#121—42no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—42no—
178C4×D4—42no—
179#117—24no—
180#117—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C22⋊C414yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4C22⋊C414yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C22×C414yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D414yes—
204C2×C42C2×D414yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—24no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—42no—
218C4⋊C8—42no—
219C22⋊C8—24no—
220C4×C8—24no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—22no—
224C4×C8—48no—
225C4×C8C2×D412yes—
226C4×C8C2×D412yes—
227C4×C8C2×D414yes—
228#95—28no—
229#95—28no—
230#95C22⋊C414yes—
231#95C22⋊C414yes—
232C24—22no—
233C24C2×C22⋊C411yescharacteristic
234#41—88no—
235#41—84no—
236#41—84no—
237C2×Q8—22no—
238C2×Q8—28no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—44no—
244C2×Q8—48no—
245C2×Q8—44no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—44no—
249C2×Q8#13114yes—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4—82no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—82no—
274C22×C4C2×C22⋊C418yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4#12112yes—
281C22×C4#12112yes—
282C22×C4#12412yes—
283C22×C4#12712yes—
284C22×C4#13514yes—
285C22×C4#13514yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289M16—84no—
290C2×C8—24no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—24no—
294C2×C8—48no—
295C2×C8—44no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—84no—
302C4⋊C4—28no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C22⋊C4—28no—
313C22⋊C4—416no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—44no—
317C22⋊C4—88no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—44no—
323C42—42no—
324C42—42no—
325C42—44no—
326C42C4×D412yes—
327C42C4×D412yes—
328C42#12112yes—
329C42#12112yes—
330C42#12412yes—
331C42#12712yes—
332C42C22×D411yescharacteristic
333C42C22×D411yescharacteristic
334C42C2×#4111yescharacteristic
335C42C2×#4111yescharacteristic
336C23—22no—
337C23—22no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—42no—
342C23#39012yes—
343C23#41614yes—
344C23C22×C22⋊C411yescharacteristic
345Q8—24no—
346Q8—24no—
347Q8—48no—
348Q8—42no—
349Q8—42no—
350Q8—84no—
351D4—42no—
352D4—42no—
353D4—44no—
354D4—84no—
355D4—88no—
356D4—84no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—48no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4#39012yes—
381C2×C4#39012yes—
382C2×C4#39012yes—
383C2×C4#41614yes—
384C2×C4#48818yes—
385C2×C4#49514yes—
386C2×C4#49514yes—
387C2×C4C2×C4×D411yescharacteristic
388C2×C4C2×C4×D411yescharacteristic
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×#12111yescharacteristic
391C2×C4C2×#12411yescharacteristic
392C2×C4C2×#12711yescharacteristic
393C2×C4C2×#13112yes—
394C2×C4#56812yes—
395C2×C4C2×#13512yes—
396C2×C4C2×#13512yes—
397C8—24no—
398C8—44no—
399C8—42no—
400C8—82no—
401C8—82no—
402C22—28no—
403C22—22no—
404C22—22no—
405C22—88no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22#195814yes—
410C22C2×#39011yescharacteristic
411C22#288012yes—
412C22C2×#41612yes—
413C22#290112yes—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—48no—
419C4—44no—
420C4—82no—
421C4—82no—
422C4#195814yes—
423C4#292911yescharacteristic
424C4#293011yescharacteristic
425C4#293312yes—
426C4C2×#48812yes—
427C4C2×#49511yescharacteristic
428C4C2×#49511yescharacteristic
429C2—24no—
430C2—162no—
431C2#1751014yes—
432C2#2119111yescharacteristic
433C2C2×#195811yescharacteristic
434C2#2121011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20755C2semidirect
3. C8×#117C2semidirect
6. C2×#1487C2semidirect
7. C22×Q8⋊C8C2semidirect
8. #9645C2semidirect
9. #9639C2semidirect

Maximal subgroups

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

subgroupindex
#349552
#207552
C8×#1172
C2×#14922
C2×#14802
C2×#14872
C22×Q8⋊C82
#96452
#96392