Finite Group Databasebeta

← #6854520

Subgroups — #6854520

Order 512.

1424 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36574C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic
4#31541C218yesmaximal
5#30307C212yesmaximal
6#29193C212yesmaximal
7#29144C211yesmaximal, characteristic
8#29144C212yesmaximal
9#28511C211yesmaximal, characteristic
10#24832C214yesmaximal
11#24344C214yesmaximal
12#22270C212yesmaximal
13#21691C211yesmaximal, characteristic
14#21340C211yesmaximal, characteristic
15C2×#557C2212yes—
16C2×#494C2211yescharacteristic
17C2×#477C2214yes—
18C2×#441C2214yes—
19#2868C2211yescharacteristic
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2859C2212yes—
23#2584—22no—
24#2581C2212yes—
25#2568C2212yes—
26#2565C2214yes—
27#2546—22no—
28#2509—22no—
29#2508—22no—
30#2506—22no—
31#2506—22no—
32#2453C2214yes—
33#2370—22no—
34#2365—22no—
35#2351C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2212yes—
40C42×Q8C2211yescharacteristic
41#2095C2214yes—
42#2085C2218yes—
43#2068C2218yes—
44#2059C2218yes—
45#2026C2218yes—
46#2026C2214yes—
47#1993C2214yes—
48#1992C2214yes—
49#1985C2212yes—
50#1971C2214yes—
51#1956C2211yescharacteristic
52#1956C2212yes—
53#1956C2212yes—
54#1956C2211yescharacteristic
55#1940C2212yes—
56#1917C2214yes—
57#1878C2218yes—
58#1878C2218yes—
59#1863C2218yes—
60#1845C2212yes—
61#1844C2211yescharacteristic
62#1834C2212yes—
63#1834C2212yes—
64#1834C2212yes—
65#1805C2214yes—
66#1805C2218yes—
67C4×C4⋊C8C2212yes—
68C4×C4⋊C8C2211yescharacteristic
69#1744C2212yes—
70C4×#97C2211yescharacteristic
71C22×Q16C2314yes—
72#557—48no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2312yes—
78C2×#125—24no—
79C2×#125—22no—
80C2×#125C2314yes—
81C2×#125C2312yes—
82C2×C4×Q8—22no—
83C2×C4×Q8D412yes—
84C2×C4×Q8C2312yes—
85C2×C4×Q8C2312yes—
86C2×C4×Q8C2314yes—
87C2×C4×Q8C2314yes—
88C2×C4×Q8C2311yescharacteristic
89#494Q818yescomplemented
90#477—48no—
91#441—48no—
92C2×#107C2318yes—
93C2×#106C2314yes—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2314yes—
96C2×#103C2318yes—
97C2×#103C2314yes—
98C2×#103C2318yes—
99C2×#103C2314yes—
100C2×#97C2312yes—
101C2×#97C2312yes—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2312yes—
104C2×C4×C8C2311yescharacteristic
105#400—22no—
106#400—22no—
107#400—22no—
108#400—24no—
109#398—24no—
110#398—24no—
111#398—24no—
112#395—22no—
113#391—24no—
114#391C2312yes—
115#391C2314yes—
116#391C2314yes—
117#391C2312yes—
118#389—24no—
119#389—24no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389C2314yes—
135#389C2314yes—
136#387—24no—
137#387—24no—
138#387—24no—
139#387—24no—
140#384—22no—
141#384C2312yes—
142#384C2311yescharacteristic
143#384C2312yes—
144#384C2311yescharacteristic
145#384C2312yes—
146#382—22no—
147#382—22no—
148#382C2314yes—
149#382C2312yes—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4D412yes—
158C4×C4⋊C4D412yes—
159C4×C4⋊C4D412yes—
160C4×C4⋊C4C2312yes—
161C4×C4⋊C4C2311yescharacteristic
162C43C2312yes—
163C43C2311yescharacteristic
164#340C2318yes—
165#340C2318yes—
166#340C2318yes—
167#336C2314yes—
168#336C2314yes—
169#336C2312yes—
170#336C2314yes—
171C22×Q8—22no—
172C22×Q8C2×D411yescharacteristic
173C22×Q8C2412yes—
174C22×Q8C2412yes—
175C2×Q16—44no—
176C2×Q16—44no—
177C2×Q16C2×Q8116yes—
178C22×C8C2414yes—
179C22×C8C2412yes—
180C22×C8C2412yes—
181C4⋊Q8—28no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—48no—
185C4⋊Q8—48no—
186C4⋊Q8C2×Q814yes—
187C4⋊Q8C2×Q814yes—
188#125—48no—
189#125—44no—
190#125—48no—
191#125—416no—
192C4×Q8—24no—
193C4×Q8—28no—
194C4×Q8—28no—
195C4×Q8—44no—
196C4×Q8—48no—
197C4×Q8—48no—
198C4×Q8—48no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2412yes—
224C2×C4⋊C4C2414yes—
225C2×C4⋊C4C2414yes—
226C2×C4⋊C4C2412yes—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2412yes—
238C2×C42C2412yes—
239C2×C42C2411yescharacteristic
240C2×C42C2411yescharacteristic
241C2×C42C2411yescharacteristic
242#107—48no—
243#106—44no—
244C4⋊C8—24no—
245C4⋊C8—44no—
246#103—48no—
247#103—44no—
248#103—44no—
249#103—48no—
250#97—24no—
251#97—42no—
252C4×C8—22no—
253C4×C8—24no—
254C4×C8C2×Q814yes—
255#95—22no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95C2×D411yescharacteristic
268#95C2×D411yescharacteristic
269#95C2×D412yes—
270#95C2×D412yes—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—42no—
279C2×Q8—48no—
280C2×Q8—44no—
281C2×Q8C22×Q814yes—
282C2×Q8C22×Q814yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4C22×D412yes—
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4C2511yescharacteristic
298Q16—416no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8C22×Q814yes—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—22no—
344C42—24no—
345C42—48no—
346C42—44no—
347C42—44no—
348C42—84no—
349C42C22×Q812yes—
350C42H(5,𝔽2)14yes—
351C422−1+412yes—
352C23C23×D411yescharacteristic
353Q8—28no—
354Q8—28no—
355Q8—44no—
356Q8—44no—
357Q8—44no—
358Q8—88no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4#53512yes—
393C2×C4#53512yes—
394C2×C4D4×Q814yes—
395C2×C4#57514yes—
396C2×C4#57814yes—
397C2×C4C23×Q811yescharacteristic
398C2×C4C2×H(5,𝔽2)12yes—
399C2×C4C2×2−1+411yescharacteristic
400C2×C4#58512yes—
401C8—24no—
402C8—42no—
403C8—44no—
404C22C2×#53511yescharacteristic
405C22C2×D4×Q811yescharacteristic
406C22#350011yescharacteristic
407C22C2×#57512yes—
408C22C2×#57812yes—
409C4—22no—
410C4—22no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—44no—
417C4—44no—
418C4—84no—
419C4—82no—
420C4—84no—
421C4—82no—
422C4#339018yes—
423C4#350112yes—
424C4#354712yes—
425C2#3961811yescharacteristic
426C2#6080111yescharacteristic
427C2#6082111yescharacteristic
428C2C2×#339012yes—
429C2#6254512yes—
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
89. #494Q8semidirect

Maximal subgroups

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

subgroupindex
#365742
#365602
#332972
#315412
#303072
#291932
#291442
#285112
#248322
#243442
#222702
#216912
#213402