Finite Group Databasebeta

← #6435218

Subgroups — #6435218

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic
3#29993C218yesmaximal, complemented
4#29989C214yesmaximal, complemented
5#29983C214yesmaximal, complemented
6#23745C211yesmaximal, characteristic
7#23744C211yesmaximal, characteristic
8#23732C211yesmaximal, characteristic
9C2×#2096C211yesmaximal, characteristic
10#23556C212yesmaximal
11#23457C212yesmaximal
12#21174C212yesmaximal
13#20754C211yesmaximal, characteristic
14#20651C212yesmaximal
15C2×#544C2212yes—
16C2×#519C2211yescharacteristic
17C2×#428C2212yes—
18C22×#106C2212yes—
19C2×#424C2212yes—
20C2×#419C2212yes—
21C22×#103C2212yes—
22#2868C2212yes—
23#2859C2212yes—
24#2660C2214yes—
25#2552—22no—
26#2551—22no—
27#2440—24no—
28#2356—22no—
29#2355—22no—
30#2343—24no—
31#2320C2214yes—
32#2317C2211yescharacteristic
33#2317C2212yes—
34#2310C2211yescharacteristic
35#2296C2211yescharacteristic
36#2295C2214yes—
37C2×#391C2211yescharacteristic
38#2285C2211yescharacteristic
39C2×#384C2211yescharacteristic
40#2274C2211yescharacteristic
41C4×#117C2211yescharacteristic
42#2101C2214yes—
43#2100C2212yes—
44#2100C2214yes—
45#2099C2212yes—
46#2099C2214yes—
47#2097C2214yes—
48#2096C2212yes—
49#2096C2214yes—
50#2096C2214yes—
51#2095C2212yes—
52#2059C2218yes—
53#2058C2218yes—
54#2026C2218yes—
55#2024C2218yes—
56#1949C2218yes—
57#1948C2218yes—
58#1832C2214yes—
59#1831C2214yes—
60#1805C2218yes—
61#1803C2218yes—
62#1736C2212yes—
63#1736C2212yes—
64#1735C2212yes—
65#1735C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C23×Q8C2311yescharacteristic
70C22×M16C2312yes—
71C23×C8C2312yes—
72#544—44no—
73#544—44no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8C2314yes—
77C2×#125—22no—
78C2×#125C2314yes—
79C2×C22⋊Q8—24no—
80#519—24no—
81#519Q818yescomplemented
82C2×C4×Q8—22no—
83C2×C4×Q8C2314yes—
84C2×#117—22no—
85C2×#117C2311yescharacteristic
86C2×#117C2312yes—
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C22×C4⋊C4—22no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#428—44no—
95C2×#107C2314yes—
96C2×#106—44no—
97C2×#106C2314yes—
98C2×#106C2318yes—
99#424—28no—
100C2×C4⋊C8C2318yes—
101#419—44no—
102C2×#103—44no—
103C2×#103C2314yes—
104C2×#103C2318yes—
105C2×#103C2314yes—
106#398—28no—
107#395—28no—
108#391—21no—
109#391—21no—
110#391—22no—
111#391C2314yes—
112#391C2312yes—
113#391C2312yes—
114#389—24no—
115#389—24no—
116#389—24no—
117#389C2314yes—
118#389C2312yes—
119#389C2312yes—
120#387—24no—
121#387—24no—
122#387—24no—
123#387—24no—
124#385—22no—
125#385—22no—
126#385—24no—
127#384—21no—
128#384—24no—
129#384—22no—
130#384C2314yes—
131#384C2311yescharacteristic
132#384C2312yes—
133#384C2311yescharacteristic
134#384C2312yes—
135#383—24no—
136#382—21no—
137#382—24no—
138#382C2314yes—
139#382C2312yes—
140#381—22no—
141#380—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4D412yes—
147C4×C4⋊C4D412yes—
148C4×C4⋊C4C2314yes—
149C4×C22⋊C4—22no—
150C4×C22⋊C4D414yes—
151#376—24no—
152#376C2312yes—
153C2×#95—22no—
154C43C2312yes—
155#340C2314yes—
156#340C2318yes—
157#340C2314yes—
158#340C2318yes—
159#340C2314yes—
160#340C2314yes—
161#336C2318yes—
162#336C2312yes—
163#336C2312yes—
164#336C2312yes—
165#336C2312yes—
166C22×Q8—21no—
167C22×Q8—24no—
168C22×Q8—22no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8C2×Q814yes—
172C22×Q8C2412yes—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4C2×D412yes—
177C23×C4C2411yescharacteristic
178C2×M16—24no—
179C2×M16—24no—
180C2×M16—24no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8C2414yes—
185C22×C8C2414yes—
186C4⋊Q8—44no—
187C4⋊Q8—44no—
188C4⋊Q8—48no—
189C4⋊Q8—88no—
190#125—44no—
191#125—44no—
192#125—48no—
193C22⋊Q8—416no—
194C4×Q8—216no—
195C4×Q8—28no—
196C4×Q8C2×Q8116yes—
197#117—22no—
198#117—24no—
199#117—216no—
200#117—24no—
201#117—24no—
202#117—24no—
203#117C2×Q814yes—
204#117C2×Q814yes—
205#117#4114yes—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—84no—
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2412yes—
229C2×C4⋊C4C2412yes—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2412yes—
232C2×C4⋊C4C2412yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4C2×D412yes—
237C2×C22⋊C4C2×D412yes—
238C2×C42—24no—
239C2×C42—21no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—24no—
244C2×C42C2×D414yes—
245C2×C42C2×D412yes—
246C2×C42C2×D414yes—
247C2×C42C2412yes—
248C2×C42C2411yescharacteristic
249C2×C42C2411yescharacteristic
250C2×C42C2412yes—
251#107—44no—
252#106—48no—
253#106—44no—
254C4⋊C8—216no—
255#103—48no—
256#103—44no—
257#103—44no—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—28no—
268#95C2×D414yes—
269#95C2×D414yes—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—216no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—48no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—48no—
280C2×Q8—88no—
281C2×Q8C22×Q814yes—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—84no—
298C22×C4—82no—
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C22×D412yes—
302C22×C4C22×D414yes—
303C22×C4C22×D412yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q812yes—
306C22×C4C2×#4112yes—
307C22×C4H(5,𝔽2)14yes—
308C22×C42−1+414yes—
309C22×C4C2511yescharacteristic
310M16—48no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—48no—
318C4⋊C4—216no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—44no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C4⋊C4—44no—
339C4⋊C4—84no—
340C4⋊C4—82no—
341C4⋊C4—84no—
342C4⋊C4—88no—
343C4⋊C4C22⋊Q818yes—
344C4⋊C4C22×Q814yes—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4C22⋊Q818yes—
350C42—216no—
351C42—24no—
352C42—24no—
353C42—28no—
354C42—24no—
355C42—22no—
356C42—22no—
357C42—44no—
358C42—44no—
359C42—42no—
360C42—84no—
361C42C22×Q814yes—
362C42C2×#4114yes—
363C23—22no—
364C23—22no—
365C23C2×C22⋊Q812yes—
366C23#53512yes—
367C23#53612yes—
368C23C23×D411yescharacteristic
369Q8—28no—
370Q8—48no—
371Q8—44no—
372Q8—168no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—164no—
404C2×C4—162no—
405C2×C4C2×C22⋊Q818yes—
406C2×C4C2×C22⋊Q812yes—
407C2×C4#53512yes—
408C2×C4#53612yes—
409C2×C4#54218yes—
410C2×C4#54418yes—
411C2×C4#57714yes—
412C2×C4C23×Q811yescharacteristic
413C2×C4C22×#4111yescharacteristic
414C2×C4C2×H(5,𝔽2)12yes—
415C2×C4C2×2−1+412yes—
416C8—44no—
417C8—44no—
418C22—24no—
419C22—22no—
420C22—22no—
421C22—24no—
422C22#262814yes—
423C22C22×C22⋊Q811yescharacteristic
424C22C2×#53511yescharacteristic
425C22C2×#53611yescharacteristic
426C22C2×#57712yes—
427C22#358612yes—
428C4—28no—
429C4—24no—
430C4—44no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—84no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4#262814yes—
441C4C2×#54212yes—
442C4C2×#54412yes—
443C2—24no—
444C2C2×#262811yescharacteristic
445C2#6049712yes—
446C2#6081811yescharacteristic
447C2#6082811yescharacteristic
448C2#6087212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29993C2semidirect
4. #29989C2semidirect
5. #29983C2semidirect
81. #519Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#342542
#299932
#299892
#299832
#237452
#237442
#237322
C2×#20962
#235562
#234572
#211742
#207542
#206512