Finite Group Databasebeta

← #110553

Subgroups — #110553

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1554C211yesmaximal, characteristic
2#17590C211yesmaximal, characteristic
3#17385C211yesmaximal, characteristic
4#17326C211yesmaximal, characteristic
5#8096C211yesmaximal, characteristic
6#7997C211yesmaximal, characteristic
7#7926C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2283C2211yescharacteristic
15#1554—22no—
16#1554—21no—
17#1554—21no—
18C2×#333C412yescomplemented
19C2×#332C412yescomplemented
20#1518—22no—
21#1511C414yes—
22#1506—22no—
23#1472—22no—
24#1470—21no—
25#1470—21no—
26#1461C414yescomplemented
27#1454—22no—
28#1452—21no—
29#1451—21no—
30#1450—21no—
31C8×M16—21no—
32#1303—22no—
33#1291—22no—
34#1282—22no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432D412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431C2×C412yes—
67#431C2×C412yes—
68#431D412yes—
69#431D411yescharacteristic
70#431D411yescharacteristic
71#431D411yescharacteristic
72#431D411yescharacteristic
73#431D412yes—
74#423—21no—
75#423—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2×C412yes—
78C2×C4⋊C8C2×C412yes—
79#407—21no—
80#407C2×C412yes—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8D412yes—
84C4×M16—21no—
85C4×M16—21no—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89#383—21no—
90#382—21no—
91#381C2×C412yes—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C22⋊C4—21no—
95#376—21no—
96#333—22no—
97#333—22no—
98#332—22no—
99#332—22no—
100#328—28no—
101#324—24no—
102#324—24no—
103#322—22no—
104#322—22no—
105#322—22no—
106C82—22no—
107C22×Q8—21no—
108C22×Q8C4212yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—41no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125#125—41no—
126#125—41no—
127#125—41no—
128#125—41no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145#117—41no—
146#117—41no—
147#117—41no—
148#117—41no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C4212yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C42—22no—
159C2×C42—41no—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—22no—
181C4⋊C8—41no—
182C4⋊C8C22⋊C412yes—
183C4⋊C8C22⋊C412yes—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8C22⋊C412yes—
194C22⋊C8C22⋊C412yes—
195C22⋊C8C22⋊C414yes—
196#97—22no—
197#97—41no—
198#97—41no—
199#97—41no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—22no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—22no—
208#95—22no—
209#95—22no—
210#95—41no—
211#95—41no—
212C24C4×D411yescharacteristic
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—44no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—82no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4C2×C4211yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4#11711yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12411yescharacteristic
241C22×C4#12611yescharacteristic
242C22×C4#12711yescharacteristic
243M16—44no—
244M16—82no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—24no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—82no—
254C2×C8—82no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—82no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—44no—
271C42—42no—
272C42—42no—
273C42—82no—
274C42C2×C22⋊C411yescharacteristic
275C42#11711yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42#12011yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42#12311yescharacteristic
287C42#12311yescharacteristic
288C42#12411yescharacteristic
289C42#12611yescharacteristic
290C23—22no—
291C23—41no—
292C23—41no—
293C23C4×C22⋊C411yescharacteristic
294C23#38811yescharacteristic
295C23#39011yescharacteristic
296Q8—82no—
297Q8—82no—
298Q8—82no—
299Q8—82no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4C4×C22⋊C411yescharacteristic
326C2×C4C4×C22⋊C411yescharacteristic
327C2×C4#38611yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#38611yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#39011yescharacteristic
336C2×C4#39011yescharacteristic
337C8—42no—
338C8—42no—
339C8—44no—
340C8—42no—
341C8—42no—
342C8—82no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—82no—
353C22#143711yescharacteristic
354C22#175712yes—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—82no—
367C4—162no—
368C4#190212yes—
369C4#197612yes—
370C2—22no—
371C2—41no—
372C2—41no—
373C2#1094411yescharacteristic
374C2#1095211yescharacteristic
375C2#1096111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
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
18. C2×#333C4semidirect
19. C2×#332C4semidirect
26. #1461C4semidirect

Maximal subgroups

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

subgroupindex
C2×#15542
#175902
#173852
#173262
#80962
#79972
#79262