Finite Group Databasebeta

← #387067

Subgroups — #387067

Order 512.

1013 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35154C211yesmaximal, characteristic
2#21338C212yesmaximal
3#11352C212yesmaximal
4#11223C212yesmaximal
5#11048C212yesmaximal
6#10940C212yesmaximal
7#10919C212yesmaximal
8#10763C211yesmaximal, characteristic
9#10763C211yesmaximal, characteristic
10C2×#544C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2905C2212yes—
13C22×#101C2212yes—
14C22×#101C2211yescharacteristic
15C22×#101C2211yescharacteristic
16#2873C2212yes—
17#2855C414yes—
18#2627C414yes—
19#2453—24no—
20#2451—24no—
21#2316C2211yescharacteristic
22#2310C2212yes—
23#2296C2212yes—
24#1998—24no—
25#1997—24no—
26C2×#344C2212yes—
27C2×#344C2212yes—
28C2×#344C2212yes—
29C2×#344C2212yes—
30#1742C2212yes—
31#1742C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34#1728C2211yescharacteristic
35#1728C2212yes—
36#1728C2211yescharacteristic
37C23×Q8C2311yescharacteristic
38C22×M16C2312yes—
39C22×M16C2312yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42#544—41no—
43#544—42no—
44#544—41no—
45C2×C4⋊Q8D414yes—
46C2×#125D414yes—
47C2×C22⋊Q8C2×C414yes—
48C2×C4×Q8—24no—
49C2×C4×Q8—41no—
50C2×C4×Q8—41no—
51C2×C4×Q8—42no—
52C2×#117C2312yes—
53C2×#117C2312yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2×C412yes—
57C22×C42C2311yescharacteristic
58#424—42no—
59#424—42no—
60#423—24no—
61C2×#101—24no—
62C2×#101—22no—
63C2×#101—24no—
64C2×#101—41no—
65C2×#101—41no—
66C2×#101—42no—
67C2×#101—41no—
68C2×#101—41no—
69C2×#101Q814yescomplemented
70#405—42no—
71#405—42no—
72C4×M16—24no—
73#400—24no—
74#398—28no—
75#395—24no—
76#391—24no—
77#391—22no—
78#391—22no—
79#391—24no—
80#389—24no—
81#389—24no—
82#389—22no—
83#389—22no—
84#387—28no—
85#385C2×C414yes—
86#385C2×C414yes—
87#381C2×C418yes—
88#380C2×C414yes—
89C4×C4⋊C4—24no—
90#344—24no—
91#344—24no—
92#344—42no—
93#344—42no—
94#343—24no—
95#343—24no—
96#343—22no—
97#343—22no—
98#343—22no—
99#343—22no—
100#336—41no—
101#336—42no—
102#336—41no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8C22⋊C414yes—
112C22×Q8C2×Q812yes—
113C22×Q8#4112yes—
114C23×C4C22×C412yes—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C412yes—
118C23×C4C2411yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16C2×Q814yes—
142C2×M16#4114yes—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149#125—44no—
150#125—44no—
151C22⋊Q8—44no—
152C22⋊Q8—44no—
153C4×Q8—48no—
154C4×Q8—48no—
155#117—22no—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4C22⋊C414yes—
174C2×C4⋊C4C22⋊C418yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4C22×C414yes—
179C2×C22⋊C4C22×C414yes—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42C2×D414yes—
189C4⋊C8—48no—
190#101—44no—
191#101—48no—
192#101—44no—
193#97—44no—
194C4×C8—44no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—28no—
199#95—28no—
200C24C23×C411yescharacteristic
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—44no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—84no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4—82no—
237C22×C4C2×C22⋊C414yes—
238C22×C4C2×C22⋊C414yes—
239C22×C4C2×C22⋊C414yes—
240C22×C4C4×Q814yes—
241C22×C4C22×D412yes—
242C22×C4C22×Q811yescharacteristic
243C22×C4C2×#4111yescharacteristic
244C22×C4C2×#4112yes—
245C22×C4H(5,𝔽2)11yescharacteristic
246C22×C4H(5,𝔽2)11yescharacteristic
247C22×C4H(5,𝔽2)12yes—
248C22×C4H(5,𝔽2)11yescharacteristic
249C22×C42−1+412yes—
250C22×C42−1+411yescharacteristic
251M16—24no—
252M16—24no—
253M16—24no—
254M16—24no—
255M16—42no—
256M16—42no—
257M16—42no—
258M16—42no—
259M16—42no—
260M16—42no—
261M16—42no—
262M16—42no—
263M16—42no—
264M16—42no—
265M16—42no—
266M16—42no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—44no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—84no—
292C4⋊C4—84no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C42—24no—
298C42—24no—
299C42—48no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—42no—
304C23—41no—
305C23—41no—
306C23C22×C22⋊C411yescharacteristic
307C23C2×C4×Q811yescharacteristic
308C23C4×#4111yescharacteristic
309C23#51811yescharacteristic
310C23#51811yescharacteristic
311C23#51911yescharacteristic
312C23#52011yescharacteristic
313Q8—44no—
314Q8—44no—
315Q8—44no—
316Q8—44no—
317Q8—164no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—44no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4#53412yes—
348C2×C4#53412yes—
349C2×C4#53811yescharacteristic
350C2×C4#53911yescharacteristic
351C2×C4#53912yes—
352C2×C4#54212yes—
353C2×C4#54311yescharacteristic
354C2×C4#54411yescharacteristic
355C2×C4#54812yes—
356C2×C4#54812yes—
357C2×C4D4×Q812yes—
358C2×C4#55012yes—
359C2×C4#55112yes—
360C2×C4#55412yes—
361C2×C4#56212yes—
362C2×C4#56312yes—
363C8—44no—
364C8—44no—
365C8—82no—
366C8—82no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22#230811yescharacteristic
378C22#232911yescharacteristic
379C22#233211yescharacteristic
380C22Q8×C22⋊C411yescharacteristic
381C22#234111yescharacteristic
382C22#236811yescharacteristic
383C22#237811yescharacteristic
384C4—44no—
385C4—44no—
386C4—42no—
387C4—44no—
388C4—44no—
389C4—84no—
390C4—81no—
391C4#272812yes—
392C4#272912yes—
393C4#273012yes—
394C4#273212yes—
395C2—21no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2#1444911yescharacteristic
402C2#2177911yescharacteristic
403C2#2178011yescharacteristic
subgroupG/H
center, socleC22#2308
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
69. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
#351542
#213382
#113522
#112232
#110482
#109402
#109192
#107632