Finite Group Databasebeta

← #215574

Subgroups — #215574

Order 512.

864 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#9709C211yesmaximal, characteristic, complemented
5#9708C211yesmaximal, characteristic, complemented
6#9700C211yesmaximal, characteristic, complemented
7#9700C212yesmaximal, complemented
8#9697C211yesmaximal, characteristic, complemented
9#9693C212yesmaximal, complemented
10#8343C212yesmaximal
11#8340C212yesmaximal
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14#2862C2211yescharacteristic
15#2862C2211yescharacteristic
16#2862C2211yescharacteristic
17#2862C2211yescharacteristic
18#2642—21no—
19#2642—21no—
20#2568—22no—
21#2568C414yes—
22#2503—22no—
23#2503C414yes—
24#2310—21no—
25#2310—21no—
26C4×#117C2211yescharacteristic
27#1923—22no—
28#1916—22no—
29#1842C2211yescharacteristic
30#1840—21no—
31#1831C2212yes—
32#1831C2211yescharacteristic
33#1818—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39#1750—21no—
40#1746—21no—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48C2×#332C2212yes—
49C23×Q8—21no—
50#533—42no—
51#533—42no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2×C412yes—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C412yes—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C412yes—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—22no—
82#400—24no—
83#400—24no—
84#400—24no—
85#400—24no—
86#391—24no—
87#391—24no—
88#391—21no—
89#391—21no—
90#391—21no—
91#391—21no—
92#391—21no—
93#391—21no—
94#391—21no—
95#391—21no—
96#391—22no—
97#391—22no—
98#391C2×C414yes—
99#391C2×C414yes—
100#382—22no—
101#382—22no—
102#382C2×C414yes—
103#382C2×C414yes—
104#381—24no—
105#381—24no—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4C2×C412yes—
109C4×C22⋊C4D412yes—
110#376D411yescharacteristic
111#376D411yescharacteristic
112C43C2311yescharacteristic
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336—21no—
118#336—21no—
119#336—22no—
120#336—22no—
121#336—22no—
122#333—42no—
123#333—42no—
124#332—42no—
125#332—42no—
126#332—42no—
127#332—42no—
128#332—42no—
129#332—42no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—44no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—44no—
138C22×Q8C22×C412yes—
139C23×C4—22no—
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—44no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—44no—
159#125—44no—
160C22⋊Q8—44no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C4×Q8—44no—
164C4×Q8—42no—
165C4×Q8—42no—
166#117—24no—
167#117—41no—
168#117—41no—
169#117—44no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2411yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—82no—
214C22⋊C8—44no—
215C22⋊C8—82no—
216#97—42no—
217C4×C8—42no—
218C4×C8—82no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231C24C2×C22⋊C411yescharacteristic
232C2×Q8—24no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—44no—
249C2×Q8—88no—
250C2×Q8—88no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4#13514yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C22⋊C4—24no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C42—22no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—42no—
325C42—42no—
326C42—44no—
327C42H(5,𝔽2)12yes—
328C42H(5,𝔽2)12yes—
329C23—21no—
330C23—22no—
331C23—21no—
332C23—22no—
333C23#41612yes—
334C23C22×C22⋊C411yescharacteristic
335Q8—84no—
336Q8—88no—
337Q8—82no—
338Q8—82no—
339Q8—82no—
340Q8—82no—
341Q8—81no—
342Q8—81no—
343Q8—82no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—44no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—88no—
380C2×C4#41612yes—
381C2×C4#51812yes—
382C2×C4#53512yes—
383C2×C4#53511yescharacteristic
384C2×C4#53511yescharacteristic
385C2×C4C2×#13512yes—
386C2×C4#57511yescharacteristic
387C2×C4#57511yescharacteristic
388C2×C4#57712yes—
389C2×C4#57812yes—
390C8—82no—
391C8—82no—
392C8—82no—
393C8—82no—
394C22—22no—
395C22—24no—
396C22—41no—
397C22—41no—
398C22—42no—
399C22#230911yescharacteristic
400C22#288912yes—
401C22C2×#41611yescharacteristic
402C22#289611yescharacteristic
403C22#289812yes—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—82no—
413C4—82no—
414C4—82no—
415C4—82no—
416C4#321112yes—
417C4#321112yes—
418C4#322111yescharacteristic
419C4#322311yescharacteristic
420C4#322512yes—
421C2—42no—
422C2#1772512yes—
423C2#1776712yes—
424C2#2180211yescharacteristic
425C2#2180611yescharacteristic
426C2#2181411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9709C2semidirect
5. #9708C2semidirect
6. #9700C2semidirect
7. #9700C2semidirect
8. #9697C2semidirect
9. #9693C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#219222
#207862
#97092
#97082
#97002
#96972
#96932
#83432
#83402