Finite Group Databasebeta

← #378944

Subgroups — #378944

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2090C211yesmaximal, characteristic
2#21159C212yesmaximal
3C2×#1944C211yesmaximal, characteristic
4#20436C211yesmaximal, characteristic
5#11366C211yesmaximal, characteristic
6#11331C211yesmaximal, characteristic
7#11239C211yesmaximal, characteristic
8#10913C212yesmaximal
9#10896C212yesmaximal
10#10896C212yesmaximal
11#10705C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C2×C4×M16C2212yes—
20C22×#97C2211yescharacteristic
21#2296C2211yescharacteristic
22C2×#389C2212yes—
23#2090C418yescomplemented
24#1944—22no—
25#1944—22no—
26#1944—22no—
27#1943—22no—
28#1942—22no—
29#1941—22no—
30#1771—22no—
31#1770—22no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38C2×#341C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41C2×#340C2212yes—
42#1729C2211yescharacteristic
43#1728C2211yescharacteristic
44C22×M16C2311yescharacteristic
45C22×M16C2312yes—
46C22×M16C2311yescharacteristic
47C22×M16C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56#428—24no—
57#428—24no—
58#428—42no—
59#428C2×C418yes—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#107—42no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—22no—
67C2×#106—22no—
68C2×#106C2×C414yes—
69#424—22no—
70#424C2×C414yes—
71C2×C4⋊C8—22no—
72#405—22no—
73#405—22no—
74#405—42no—
75C4×M16—24no—
76C4×M16—42no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—22no—
80C2×#97—42no—
81C2×#97D412yes—
82C2×#97D412yes—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389D412yes—
91#389D412yes—
92#385—21no—
93#385—21no—
94C4×C4⋊C4—22no—
95C2×#95—22no—
96#343—24no—
97#343—24no—
98#343—42no—
99#343C2×C418yes—
100#341—22no—
101#341C2×C414yes—
102#341Q818yes—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#336—22no—
110#336—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4C2411yescharacteristic
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—21no—
120C2×M16—21no—
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×M16C22×C412yes—
131C2×M16C22×C414yes—
132C2×M16C2×Q814yes—
133C2×M16C2×Q814yes—
134C2×M16#4112yes—
135C2×M16#4114yes—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8C22×C412yes—
146C22×C8C2×D412yes—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×Q814yes—
150C22×C8#4112yes—
151#117—21no—
152#117—21no—
153#117—22no—
154#117—22no—
155#117C22×C412yes—
156#117#4112yes—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—84no—
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D412yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185#107—28no—
186#107—44no—
187#107—44no—
188#106—28no—
189#106—28no—
190#106—44no—
191C4⋊C8—28no—
192#97—24no—
193#97—42no—
194#97—42no—
195#97—44no—
196#97—42no—
197C4×C8—24no—
198C4×C8—44no—
199C4×C8—42no—
200#95—22no—
201#95—22no—
202#95—42no—
203#95—42no—
204C24C2×#4111yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4#12112yes—
223C22×C4#12112yes—
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D411yescharacteristic
227C22×C4C22×Q812yes—
228C22×C4C2×#4111yescharacteristic
229C22×C4C2×#4111yescharacteristic
230C22×C4C2×#4112yes—
231C22×C4C2×#4111yescharacteristic
232C22×C4H(5,𝔽2)12yes—
233C22×C42−1+412yes—
234M16—24no—
235M16—24no—
236M16—24no—
237M16—24no—
238M16—24no—
239M16—24no—
240M16—24no—
241M16—42no—
242M16—42no—
243M16—42no—
244M16—42no—
245M16C4×Q818yes—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—24no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8C4×D414yes—
269C2×C8C4⋊Q812yes—
270C2×C8C4⋊Q812yes—
271C4⋊C4—22no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C4⋊C4—84no—
286C4⋊C4C4×D414yes—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—42no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—41no—
297C42—41no—
298C23—22no—
299C23—22no—
300C23C4×#4111yescharacteristic
301C23C2×#12111yescharacteristic
302C23#52911yescharacteristic
303C23#55312yes—
304C23#55712yes—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—84no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—84no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—162no—
329C2×C4—162no—
330C2×C4C2×C4×D411yescharacteristic
331C2×C4C2×C4×D411yescharacteristic
332C2×C4C2×C4×Q812yes—
333C2×C4#52012yes—
334C2×C4#52511yescharacteristic
335C2×C4C2×C4⋊Q811yescharacteristic
336C2×C4#53211yescharacteristic
337C2×C4#53211yescharacteristic
338C2×C4#53312yes—
339C2×C4#53411yescharacteristic
340C2×C4#53611yescharacteristic
341C2×C4#53912yes—
342C2×C4#54612yes—
343C2×C4D4×Q812yes—
344C2×C4#55212yes—
345C2×C4#55212yes—
346C2×C4#55412yes—
347C2×C4#55412yes—
348C8—24no—
349C8—22no—
350C8—22no—
351C8—28no—
352C8—42no—
353C8—42no—
354C8—44no—
355C22—24no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22C4×#12111yescharacteristic
361C22#237012yes—
362C22#254311yescharacteristic
363C22#254611yescharacteristic
364C22#258512yes—
365C22#306812yes—
366C22#306912yes—
367C4—44no—
368C4—42no—
369C4—42no—
370C4—84no—
371C4—81no—
372C4—81no—
373C4—162no—
374C4C4×C4⋊Q811yescharacteristic
375C4#232211yescharacteristic
376C4#236012yes—
377C4#255612yes—
378C4#258311yescharacteristic
379C4#261111yescharacteristic
380C2—22no—
381C2—22no—
382C2#1422611yescharacteristic
383C2#2155712yes—
384C2#2859411yescharacteristic
385C2#2859511yescharacteristic
386C2#2870312yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #2090C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20902
#211592
C2×#19442
#204362
#113662
#113312
#112392
#109132
#108962
#107052