Finite Group Databasebeta

← #117756

Subgroups — #117756

Order 512.

631 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21879C211yesmaximal, characteristic
2C2×#1848C211yesmaximal, characteristic
3#17749C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic
5#8004C211yesmaximal, characteristic
6#8004C211yesmaximal, characteristic
7#7885C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C22×#108—22no—
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#1997—21no—
17#1966—21no—
18#1901—22no—
19#1848—22no—
20#1848—22no—
21#1801—22no—
22#1559—22no—
23#1550—22no—
24#1548—21no—
25#1546—21no—
26#1493—22no—
27#1493—22no—
28C2×Q8⋊C8C414yes—
29#1292—22no—
30#1292—22no—
31#1278—22no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16D412yes—
35C23×C8—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44C2×C4⋊Q8D411yescharacteristic
45C2×C4⋊Q8D411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8Q812yes—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C412yes—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61C2×#108—28no—
62C2×#108—48no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423—21no—
74#423D412yes—
75#423D412yes—
76#423D412yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2×C412yes—
83C2×#101—24no—
84#407—21no—
85#407—21no—
86C4×M16—21no—
87C4×M16—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2×C412yes—
92C2×C4×C8D412yes—
93C2×C4×C8D412yes—
94#343—24no—
95#333—24no—
96#332—24no—
97Q8⋊C8—28no—
98C22×Q8—21no—
99C22×Q8C4⋊C412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×Q811yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—44no—
112C2×M16—44no—
113C2×M16—82no—
114C22×C8—24no—
115C22×C8—24no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—41no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130#125—41no—
131#125—41no—
132#125—41no—
133#125—41no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—41no—
150#117—41no—
151#117—41no—
152#117—41no—
153#117—41no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C4⋊C412yes—
160C2×C22⋊C4—22no—
161C2×C42—22no—
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174#108—416no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191#101—84no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C22⋊C8—41no—
196#97—41no—
197#97—41no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202C4×C8—41no—
203C4×C8C22⋊C414yes—
204C24C22⋊Q811yescharacteristic
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—42no—
211C2×Q8—42no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4C2×C4⋊C411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4#12011yescharacteristic
223C22×C4#12011yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12611yescharacteristic
237M16—44no—
238M16—44no—
239M16—42no—
240M16—44no—
241M16—42no—
242M16—42no—
243M16—42no—
244M16—82no—
245M16—82no—
246M16—82no—
247C2×C8—24no—
248C2×C8—28no—
249C2×C8—24no—
250C2×C8—28no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—82no—
258C2×C8—82no—
259C2×C8—82no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C42—42no—
275C42—42no—
276C42C2×C22⋊C411yescharacteristic
277C42#11711yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42#12011yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42#12311yescharacteristic
287C42#12311yescharacteristic
288C42#12311yescharacteristic
289C42#12311yescharacteristic
290C42#12611yescharacteristic
291C42#12711yescharacteristic
292C23—22no—
293C23—22no—
294C23—41no—
295C23#38511yescharacteristic
296C23#39311yescharacteristic
297C23#39611yescharacteristic
298Q8—44no—
299Q8—82no—
300Q8—81no—
301Q8—81no—
302Q8—82no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4#38011yescharacteristic
328C2×C4#38511yescharacteristic
329C2×C4#38611yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#39011yescharacteristic
333C2×C4#39211yescharacteristic
334C2×C4#39411yescharacteristic
335C2×C4#39411yescharacteristic
336C2×C4#39611yescharacteristic
337C2×C4#39711yescharacteristic
338C2×C4#39911yescharacteristic
339C2×C4#45312yes—
340C2×C4#45612yes—
341C2×C4#47112yes—
342C2×C4#47312yes—
343C8—28no—
344C8—82no—
345C8—82no—
346C8—82no—
347C8—82no—
348C8—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—44no—
357C22#143911yescharacteristic
358C22#185912yes—
359C22#206412yes—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—82no—
368C4—82no—
369C4#181312yes—
370C4#197711yescharacteristic
371C4#197911yescharacteristic
372C4#200211yescharacteristic
373C4#200511yescharacteristic
374C4#203912yes—
375C4#205212yes—
376C2—22no—
377C2—22no—
378C2—41no—
379C2#1107811yescharacteristic
380C2#1109911yescharacteristic
381C2#1112811yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#218792
C2×#18482
#177492
C2×#14932
#80042
#78852