Finite Group Databasebeta

← #107495

Subgroups — #107495

Order 512.

893 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34371C211yesmaximal, characteristic
2#21259C211yesmaximal, characteristic
3#17746C211yesmaximal, characteristic
4#8093C211yesmaximal, characteristic
5#8052C211yesmaximal, characteristic
6#7962C211yesmaximal, characteristic
7#7941C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#419—22no—
13#2381C414yescomplemented
14#2365—22no—
15#2345—22no—
16Q8×C22⋊C4—22no—
17#2331—22no—
18#2327—22no—
19#2313—22no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1981—22no—
24#1981C414yescomplemented
25#1558—22no—
26#1548—22no—
27#1548C414yescomplemented
28#1303—22no—
29#1297—22no—
30#1286—22no—
31#1284—22no—
32C23×Q8D412yes—
33C22×M16—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8C2×C412yes—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8D412yes—
44C2×C4⋊Q8D412yes—
45C2×C4⋊Q8Q812yes—
46C2×C4×Q8—24no—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431C2×C412yes—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423C2×C412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2×C412yes—
67#419—48no—
68C2×#103—28no—
69#407—21no—
70#407—21no—
71C4×M16—21no—
72C4×M16—21no—
73#391—24no—
74#391—28no—
75#389—24no—
76#387—28no—
77#387—28no—
78#387—28no—
79#383C2×C412yes—
80#383C2×C412yes—
81#382—21no—
82#382D412yes—
83#381—21no—
84#381D412yes—
85#380—24no—
86#380—21no—
87#380—21no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4D412yes—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4D412yes—
98#376—21no—
99#376—21no—
100#376—21no—
101C43—21no—
102#333—28no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—44no—
107C22×Q8—44no—
108C22×Q8C22⋊C414yes—
109C22×Q8C22⋊C414yes—
110C22×Q8C4⋊C414yes—
111C23×C4—22no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C2×M16—22no—
116C2×M16—44no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C22×C8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8C4214yes—
134C4⋊Q8C22⋊C414yes—
135C4⋊Q8C4⋊C414yes—
136C4⋊Q8C4⋊C414yes—
137C4×Q8—84no—
138C4×Q8—84no—
139#117—24no—
140#117—42no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—28no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—41no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—28no—
163C2×C42—22no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×Q811yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186#103—416no—
187C22⋊C8—22no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191#97—41no—
192#97—41no—
193C4×C8—22no—
194C4×C8—41no—
195#95—28no—
196#95—28no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—41no—
205#95C22⋊C414yes—
206#95C22⋊C414yes—
207C24#12011yescharacteristic
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—48no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—48no—
215C2×Q8—44no—
216C2×Q8—44no—
217C2×Q8—84no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×Q8#9518yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—84no—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12311yescharacteristic
250M16—82no—
251M16—82no—
252M16—82no—
253M16—82no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—88no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C42—44no—
276C42—44no—
277C42—42no—
278C42—42no—
279C42—82no—
280C42—82no—
281C42—88no—
282C42C2×C4211yescharacteristic
283C42C2×C22⋊C411yescharacteristic
284C42C2×C4⋊C411yescharacteristic
285C42C2×C4⋊C411yescharacteristic
286C42#11711yescharacteristic
287C42#11711yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42#12111yescharacteristic
293C42#12111yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42#12311yescharacteristic
297C42#12311yescharacteristic
298C23—22no—
299C23—22no—
300C23—41no—
301C23#37911yescharacteristic
302C23#38511yescharacteristic
303C23#38611yescharacteristic
304Q8—48no—
305Q8—48no—
306Q8—44no—
307Q8—44no—
308Q8—84no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—28no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—48no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4C2×#9511yescharacteristic
342C2×C4C4×C22⋊C411yescharacteristic
343C2×C4C4×C22⋊C411yescharacteristic
344C2×C4#38011yescharacteristic
345C2×C4#38011yescharacteristic
346C2×C4#38111yescharacteristic
347C2×C4#38511yescharacteristic
348C2×C4#38511yescharacteristic
349C2×C4#38511yescharacteristic
350C2×C4#38611yescharacteristic
351C2×C4#38611yescharacteristic
352C2×C4#38611yescharacteristic
353C2×C4#45312yes—
354C2×C4#45512yes—
355C8—82no—
356C8—82no—
357C8—82no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—44no—
366C22C22⋊#9511yescharacteristic
367C22#179212yes—
368C22#190512yes—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—48no—
379C4—84no—
380C4—84no—
381C4—84no—
382C4#185612yes—
383C4#188511yescharacteristic
384C4#188711yescharacteristic
385C4#190512yes—
386C2—22no—
387C2—22no—
388C2—41no—
389C2#1074911yescharacteristic
390C2#1078611yescharacteristic
391C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2381C4semidirect
24. #1981C4semidirect
27. #1548C4semidirect

Maximal subgroups

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

subgroupindex
#343712
#212592
#177462
#80932
#80522
#79622
#79412