Finite Group Databasebeta

← #110896

Subgroups — #110896

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20545C211yesmaximal, characteristic
2C2×#1552C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5#8089C211yesmaximal, characteristic
6#8044C211yesmaximal, characteristic
7#7984C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1770—21no—
16C8×C4⋊C4—21no—
17#1768—22no—
18#1752—21no—
19C8×C22⋊C4—21no—
20#1552—22no—
21#1552—21no—
22#1552—21no—
23C2×#332C412yescomplemented
24C2×#332C412yescomplemented
25#1520—21no—
26#1520—21no—
27#1520—22no—
28#1518C414yescomplemented
29#1515C414yescomplemented
30#1506—22no—
31#1503—22no—
32#1302—22no—
33#1296—22no—
34#1289—22no—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432C2×C412yes—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431D412yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2×C412yes—
72C2×C4⋊C8C2×C412yes—
73C2×C4⋊C8D412yes—
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8D411yescharacteristic
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8D411yescharacteristic
79C2×C22⋊C8D411yescharacteristic
80C2×#97—21no—
81C2×#97—21no—
82C2×#97—22no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85#384—21no—
86#383C2×C412yes—
87#382—21no—
88#382C2×C412yes—
89#381—21no—
90#380—21no—
91#380—21no—
92#380D412yes—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C22⋊C4—21no—
97#376D412yes—
98#336—22no—
99#336—22no—
100#332—22no—
101#332—22no—
102#332—22no—
103#332—22no—
104#328—24no—
105#328—24no—
106#328—28no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—41no—
120C22×C8—82no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C4212yes—
128C4⋊Q8C4212yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117—41no—
153#117—82no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—22no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—41no—
192C4⋊C8C22⋊C412yes—
193C4⋊C8C22⋊C412yes—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8C22⋊C412yes—
201C22⋊C8C22⋊C412yes—
202#97—22no—
203#97—41no—
204#97—41no—
205#97—41no—
206#97—82no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—82no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95C22⋊C414yes—
216C24#12111yescharacteristic
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—81no—
222C2×Q8—81no—
223C2×Q8—82no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12411yescharacteristic
248C22×C4#12411yescharacteristic
249C22×C4#12611yescharacteristic
250C22×C4#12711yescharacteristic
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—82no—
264C2×C8—82no—
265C2×C8—82no—
266C2×C8—82no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—44no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—82no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C4⋊C4—82no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—82no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42—82no—
294C42C2×C4211yescharacteristic
295C42C2×C22⋊C411yescharacteristic
296C42#11711yescharacteristic
297C42#11711yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42C4×D411yescharacteristic
303C42C4×D411yescharacteristic
304C42#12111yescharacteristic
305C42#12111yescharacteristic
306C42#12111yescharacteristic
307C42#12311yescharacteristic
308C42#12411yescharacteristic
309C42#12611yescharacteristic
310C23—41no—
311C23—41no—
312C23#38611yescharacteristic
313C23#38811yescharacteristic
314C23#39011yescharacteristic
315C23#47112yes—
316C23#47312yes—
317Q8—82no—
318Q8—82no—
319Q8—81no—
320Q8—81no—
321Q8—82no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4C4×C22⋊C411yescharacteristic
356C2×C4C4×C22⋊C411yescharacteristic
357C2×C4C4×C22⋊C411yescharacteristic
358C2×C4#38611yescharacteristic
359C2×C4#38611yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#38811yescharacteristic
362C2×C4#38811yescharacteristic
363C2×C4#38811yescharacteristic
364C2×C4#38811yescharacteristic
365C2×C4#39011yescharacteristic
366C2×C4#39011yescharacteristic
367C8—42no—
368C8—42no—
369C8—42no—
370C8—42no—
371C8—82no—
372C8—82no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—42no—
380C22—42no—
381C22#143711yescharacteristic
382C22#190812yes—
383C22#193012yes—
384C22#197711yescharacteristic
385C22#197911yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—84no—
397C4—82no—
398C4—82no—
399C4—162no—
400C4#190411yescharacteristic
401C4#190511yescharacteristic
402C4#197612yes—
403C2—22no—
404C2—21no—
405C2—22no—
406C2—21no—
407C2#1094211yescharacteristic
408C2#1094411yescharacteristic
409C2#1095411yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
23. C2×#332C4semidirect
24. C2×#332C4semidirect
28. #1518C4semidirect
29. #1515C4semidirect

Maximal subgroups

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

subgroupindex
#205452
C2×#15522
#175962
#175752
#80892
#80442
#79842