Finite Group Databasebeta

← #387069

Subgroups — #387069

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35090C211yesmaximal, characteristic
2#21333C212yesmaximal
3#11351C212yesmaximal
4#11205C212yesmaximal
5#11070C212yesmaximal
6#10936C212yesmaximal
7#10915C212yesmaximal
8#10726C211yesmaximal, characteristic
9#10726C211yesmaximal, characteristic
10C2×#543C2211yescharacteristic
11C22×#117C2211yescharacteristic
12#2855C414yescomplemented
13#2627C414yescomplemented
14#2438—28no—
15#2315C2212yes—
16#2315C2212yes—
17#2315C2211yescharacteristic
18C2×#381C2212yes—
19C2×#381C2212yes—
20C2×#381C2211yescharacteristic
21C2×#381C2211yescharacteristic
22#2273C2212yes—
23#1996—28no—
24C2×#342C2212yes—
25C2×#342C2212yes—
26C2×#342C2212yes—
27C2×#342C2212yes—
28C2×#342C2212yes—
29C2×#342C2212yes—
30C2×#342C2212yes—
31C2×#342C2212yes—
32C2×#342C2211yescharacteristic
33C2×#342C2212yes—
34C2×#342C2211yescharacteristic
35C24×C4C2311yescharacteristic
36#543—41no—
37#543—42no—
38#543—41no—
39C2×C22⋊Q8C2×C414yes—
40C2×C22⋊Q8D418yes—
41C2×#117—24no—
42C2×#117—41no—
43C2×#117—41no—
44C2×#117—42no—
45C22×C4⋊C4C2×C412yes—
46C22×C22⋊C4C2312yes—
47C22×C22⋊C4C2312yes—
48C22×C22⋊C4C2311yescharacteristic
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2312yes—
51C22×C22⋊C4C2312yes—
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2311yescharacteristic
54C22×C42C2×C412yes—
55#400—24no—
56#398—28no—
57#395—24no—
58#391—24no—
59#389—24no—
60#387—28no—
61#385—24no—
62#385—24no—
63#385—42no—
64#385—42no—
65#385C2×C418yes—
66#381—24no—
67#381—24no—
68#381—22no—
69#381—24no—
70#381—41no—
71#381—41no—
72#381C2×C414yes—
73#381C2×C414yes—
74#381Q814yescomplemented
75#380—24no—
76#380—24no—
77#380C2×C414yes—
78C4×C22⋊C4—24no—
79C4×C22⋊C4—42no—
80C4×C22⋊C4—42no—
81C2×#95—24no—
82C2×#95—41no—
83C2×#95—42no—
84C2×#95—41no—
85#342—24no—
86#342—24no—
87#342—24no—
88#342—24no—
89#342—24no—
90#342—24no—
91#342—42no—
92#342—42no—
93#342—41no—
94#342—42no—
95#342—41no—
96C25C2411yescharacteristic
97C22×Q8C22⋊C414yes—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C2×D414yes—
116C23×C4C2×Q812yes—
117C23×C4#4112yes—
118C22⋊Q8—44no—
119C22⋊Q8—44no—
120C22⋊Q8—48no—
121C22⋊Q8—48no—
122#117—48no—
123#117—48no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4C22⋊C418yes—
131C2×C4⋊C4C22⋊C414yes—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4C22×C414yes—
171C2×C22⋊C4C22×C414yes—
172C2×C22⋊C4C2×D414yes—
173C2×C22⋊C4C2×D414yes—
174C2×C22⋊C4C2×Q814yes—
175C2×C22⋊C4#4114yes—
176C2×C42—24no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—44no—
180#95—28no—
181#95—28no—
182#95—28no—
183#95—24no—
184#95—44no—
185#95—48no—
186#95—44no—
187#95—44no—
188#95—44no—
189C24—82no—
190C24C23×C411yescharacteristic
191C24C22×D412yes—
192C24C22×Q811yescharacteristic
193C24C2×#4111yescharacteristic
194C24C2×#4112yes—
195C24H(5,𝔽2)11yescharacteristic
196C24H(5,𝔽2)11yescharacteristic
197C24H(5,𝔽2)12yes—
198C24H(5,𝔽2)11yescharacteristic
199C242−1+412yes—
200C242−1+411yescharacteristic
201C2×Q8—44no—
202C2×Q8—84no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4C2×C22⋊C418yes—
262C22×C4C2×C22⋊C414yes—
263C22×C4C4×Q812yes—
264C22×C4C4×Q812yes—
265C4⋊C4—48no—
266C4⋊C4—44no—
267C4⋊C4—48no—
268C4⋊C4—88no—
269C4⋊C4—84no—
270C4⋊C4—164no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C42—48no—
300C42—164no—
301C23—42no—
302C23—44no—
303C23—42no—
304C23—42no—
305C23—44no—
306C23—81no—
307C23—81no—
308C23—82no—
309C23—81no—
310C23—82no—
311C23—81no—
312C23C22×C22⋊C411yescharacteristic
313C23C2×C4×Q811yescharacteristic
314C23C4×#4111yescharacteristic
315C23#51811yescharacteristic
316C23#51811yescharacteristic
317C23#51911yescharacteristic
318C23#52011yescharacteristic
319C23#53412yes—
320C23#53412yes—
321C23#53811yescharacteristic
322C23#53911yescharacteristic
323C23#53912yes—
324C23#54212yes—
325C23#54311yescharacteristic
326C23#54411yescharacteristic
327C23#54812yes—
328C23#54812yes—
329C23D4×Q812yes—
330C23#55012yes—
331C23#55112yes—
332C23#55412yes—
333C23#56212yes—
334C23#56312yes—
335Q8—164no—
336C2×C4—24no—
337C2×C4—28no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—162no—
381C2×C4—162no—
382C2×C4—162no—
383C2×C4—162no—
384C2×C4—162no—
385C2×C4—162no—
386C22—22no—
387C22—24no—
388C22—24no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22#230811yescharacteristic
410C22#232911yescharacteristic
411C22#233211yescharacteristic
412C22Q8×C22⋊C411yescharacteristic
413C22#234111yescharacteristic
414C22#236811yescharacteristic
415C22#237811yescharacteristic
416C22#272812yes—
417C22#272912yes—
418C22#273012yes—
419C22#273212yes—
420C4—44no—
421C4—48no—
422C4—42no—
423C4—42no—
424C4—84no—
425C4—84no—
426C4—84no—
427C4—82no—
428C4—162no—
429C4—162no—
430C2—21no—
431C2—22no—
432C2—22no—
433C2—22no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—21no—
438C2—21no—
439C2—22no—
440C2#1444911yescharacteristic
441C2#2177712yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2855C4semidirect
13. #2627C4semidirect
74. #381Q8semidirect

Maximal subgroups

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

subgroupindex
#350902
#213332
#113512
#112052
#110702
#109362
#109152
#107262