Finite Group Databasebeta

← #110894

Subgroups — #110894

Order 512.

582 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17748C211yesmaximal, characteristic
2#17648C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4#17327C211yesmaximal, characteristic
5#8058C211yesmaximal, characteristic
6#8033C211yesmaximal, characteristic
7#7959C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1558C412yescomplemented
16#1556C412yescomplemented
17#1554—22no—
18#1550—22no—
19#1531—22no—
20#1528—21no—
21#1528—21no—
22#1526C414yescomplemented
23#1518—21no—
24#1518—21no—
25#1518C414yes—
26C2×#328—22no—
27#1453—22no—
28#1451—21no—
29#1451—21no—
30C8×M16—21no—
31C8×M16—21no—
32#1298—22no—
33#1295—22no—
34#1286—22no—
35C22×M16—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—
52C22×C4⋊C4—21no—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yes—
61#432C2×C412yes—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431C2×C412yes—
68#431D412yes—
69#431D411yescharacteristic
70#431D411yescharacteristic
71#423—21no—
72#423D412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75#407—21no—
76#407—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8D412yes—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16C2×C412yes—
83C4×M16C2×C412yes—
84C4×M16D411yescharacteristic
85C4×M16D411yescharacteristic
86C2×#97—21no—
87C2×#97D412yes—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#382—21no—
91#382—21no—
92#382—21no—
93#382C2×C412yes—
94C2×#95—21no—
95C2×#95—21no—
96#333—24no—
97#332—24no—
98#330—24no—
99#330—24no—
100#328—24no—
101#328—24no—
102#322—22no—
103#322—22no—
104#322—22no—
105C82—22no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4—41no—
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—21no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—21no—
116C2×M16—22no—
117C2×M16—44no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—41no—
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—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—82no—
162C2×C22⋊C4—41no—
163C2×C42—41no—
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8C22⋊C414yes—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8C22⋊C412yes—
193C22⋊C8C22⋊C412yes—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—41no—
199#97C22⋊C412yes—
200#97C22⋊C412yes—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—22no—
207C4×C8—21no—
208C4×C8—21no—
209#95—22no—
210#95—22no—
211#95—41no—
212#95—41no—
213C24#12411yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—82no—
219C2×Q8—82no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4#11711yescharacteristic
232C22×C4#11711yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12611yescharacteristic
246C22×C4#12611yescharacteristic
247C22×C4#12711yescharacteristic
248M16—24no—
249M16—44no—
250M16—42no—
251M16—42no—
252M16—42no—
253M16—82no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—82no—
263C2×C8—82no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C42—42no—
281C42—42no—
282C42C2×C4211yescharacteristic
283C42C2×C22⋊C411yescharacteristic
284C42C2×C22⋊C411yescharacteristic
285C42#11711yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42#12011yescharacteristic
293C42#12111yescharacteristic
294C42#12111yescharacteristic
295C42#12311yescharacteristic
296C42#12411yescharacteristic
297C42#12411yescharacteristic
298C23—22no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23#38811yescharacteristic
303C23#38811yescharacteristic
304C23#39011yescharacteristic
305Q8—82no—
306Q8—82no—
307Q8—82no—
308Q8—82no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—164no—
335C2×C4C4×C22⋊C411yescharacteristic
336C2×C4C4×C22⋊C411yescharacteristic
337C2×C4C4×C22⋊C411yescharacteristic
338C2×C4#38611yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38611yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#39011yescharacteristic
346C2×C4#39011yescharacteristic
347C2×C4#47112yes—
348C2×C4#47312yes—
349C8—24no—
350C8—42no—
351C8—42no—
352C8—42no—
353C8—44no—
354C8—82no—
355C8—82no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22#143711yescharacteristic
366C22#193212yes—
367C22#197711yescharacteristic
368C22#197911yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—82no—
381C4—162no—
382C4#190812yes—
383C4#193012yes—
384C2—21no—
385C2—22no—
386C2—21no—
387C2—41no—
388C2#1094211yescharacteristic
389C2#1094611yescharacteristic
390C2#1095111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
15. #1558C4semidirect
16. #1556C4semidirect
22. #1526C4semidirect

Maximal subgroups

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

subgroupindex
#177482
#176482
C2×#15182
#173272
#80582
#80332
#79592