Finite Group Databasebeta

← #107221

Subgroups — #107221

Order 512.

885 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
1#34371C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic
3C2×#1966C211yesmaximal, characteristic
4#8093C211yesmaximal, characteristic
5#8074C211yesmaximal, characteristic
6#7973C211yesmaximal, characteristic
7#7960C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×#101—22no—
13C2×C4×M16C2211yescharacteristic
14#2381—22no—
15#2365—22no—
16#2345—22no—
17Q8×C22⋊C4—22no—
18#2331—22no—
19#2327—22no—
20#2313C414yes—
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#1997—22no—
24#1997C414yescomplemented
25#1966—22no—
26#1966C414yescomplemented
27#1821—22no—
28#1303—22no—
29#1300—22no—
30#1288—22no—
31#1286—22no—
32C23×Q8D412yes—
33C22×M16—21no—
34C22×M16—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8D412yes—
46C2×C4⋊Q8Q812yes—
47C2×C4×Q8—24no—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4D412yes—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423C2×C412yes—
64#423C2×C412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×#101—28no—
68C2×#101—48no—
69#407—21no—
70#407—21no—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16C2×C412yes—
76C4×M16C2×C412yes—
77C2×#97—21no—
78C2×C4×C8—21no—
79#391—24no—
80#391—28no—
81#389—24no—
82#387—28no—
83#387—28no—
84#387—28no—
85#383—21no—
86#383D412yes—
87#382—21no—
88#382D412yes—
89#381C2×C412yes—
90#381C2×C412yes—
91#380—24no—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D412yes—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100#376—21no—
101#376D412yes—
102#343—28no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—44no—
108C22×Q8—44no—
109C22×Q8C22⋊C414yes—
110C22×Q8C4⋊C414yes—
111C23×C4C22⋊C414yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C2×M16—24no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—42no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—44no—
124C22×C8—41no—
125C22×C8—41no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4×Q8—84no—
143C4×Q8—84no—
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—28no—
155C2×C4⋊C4C4214yes—
156C2×C4⋊C4C22⋊C414yes—
157C2×C4⋊C4C4⋊C414yes—
158C2×C4⋊C4C4⋊C414yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—28no—
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×Q811yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184#101—416no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189#97—22no—
190#97—22no—
191#97—41no—
192#97—41no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—41no—
196C4×C8—41no—
197#95—28no—
198#95—28no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95C22⋊C414yes—
206#95C22⋊C414yes—
207C24C2×C22⋊C411yescharacteristic
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—48no—
215C2×Q8—44no—
216C2×Q8—44no—
217C2×Q8—84no—
218C2×Q8—84no—
219C2×Q8—82no—
220C2×Q8—82no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—48no—
230C22×C4#9518yes—
231C22×C4C2×C4211yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C4⋊C411yescharacteristic
237C22×C4C2×C4⋊C411yescharacteristic
238C22×C4C2×C4⋊C411yescharacteristic
239C22×C4#11711yescharacteristic
240C22×C4#11711yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249M16—24no—
250M16—44no—
251M16—42no—
252M16—44no—
253M16—44no—
254M16—44no—
255M16—82no—
256C2×C8—44no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—82no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—88no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C42—44no—
277C42—42no—
278C42—42no—
279C42—88no—
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42#12111yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C23—22no—
297C23—22no—
298C23—41no—
299C23C2×#9511yescharacteristic
300C23#37911yescharacteristic
301C23#38111yescharacteristic
302Q8—48no—
303Q8—44no—
304Q8—44no—
305Q8—84no—
306Q8—84no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—28no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—84no—
336C2×C4C4×C22⋊C411yescharacteristic
337C2×C4C4×C22⋊C411yescharacteristic
338C2×C4#38011yescharacteristic
339C2×C4#38011yescharacteristic
340C2×C4#38511yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38611yescharacteristic
345C2×C4#38611yescharacteristic
346C2×C4#38611yescharacteristic
347C2×C4#38611yescharacteristic
348C2×C4#45312yes—
349C2×C4#45512yes—
350C8—44no—
351C8—82no—
352C8—82no—
353C8—82no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—44no—
362C22C22⋊#9511yescharacteristic
363C22#179212yes—
364C22#183012yes—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—48no—
375C4—84no—
376C4—84no—
377C4#185612yes—
378C4#188511yescharacteristic
379C4#188711yescharacteristic
380C4#191212yes—
381C2—22no—
382C2—22no—
383C2—41no—
384C2#1074111yescharacteristic
385C2#1074911yescharacteristic
386C2#1077011yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ 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
24. #1997C4semidirect
26. #1966C4semidirect

Maximal subgroups

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

subgroupindex
#343712
#218292
C2×#19662
#80932
#80742
#79732
#79602