Finite Group Databasebeta

← #117876

Subgroups — #117876

Order 512.

541 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1569C211yesmaximal, characteristic
2#17648C211yesmaximal, characteristic
3#17496C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#8061C211yesmaximal, characteristic
6#8060C211yesmaximal, characteristic
7#7922C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1569—22no—
16#1569—22no—
17C2×#335—21no—
18C2×#334—21no—
19#1531C412yes—
20#1531C412yes—
21#1528—22no—
22#1526—22no—
23#1497—22no—
24#1493—22no—
25#1491—22no—
26#1487—22no—
27#1481—22no—
28#1481—21no—
29#1481—21no—
30#1299—22no—
31#1299—22no—
32#1281—22no—
33C22×M16—21no—
34C22×M16—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533D412yes—
43C2×C4⋊Q8Q811yescharacteristic
44C2×C4⋊Q8Q811yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
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#432C2×C412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431D412yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423D412yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8D411yescharacteristic
78C2×C4⋊C8D411yescharacteristic
79#407—21no—
80#407—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16D412yes—
87C4×M16D412yes—
88C4×M16D412yes—
89C2×#97—21no—
90C2×#97C2×C411yescharacteristic
91C2×#97C2×C411yescharacteristic
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#335—24no—
95#334—24no—
96#330—24no—
97#330—24no—
98Q8⋊C8—28no—
99Q8⋊C8—24no—
100Q8⋊C8—24no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—44no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—82no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—41no—
119C22×C8—41no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C4⋊C412yes—
127C4⋊Q8C4⋊C412yes—
128#125—22no—
129#125—22no—
130#125—41no—
131#125—41no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148#117—22no—
149#117—22no—
150#117—41no—
151#117—41no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C42—41no—
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×Q811yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—22no—
178C4⋊C8—21no—
179C4⋊C8—22no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#97—22no—
195#97—21no—
196#97—21no—
197#97—22no—
198#97—41no—
199#97—41no—
200#97C22⋊C412yes—
201#97C22⋊C412yes—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—41no—
207C4×C8—41no—
208C24#12611yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—82no—
214C2×Q8—82no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—81no—
221C22×C4#11711yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4#12611yescharacteristic
239M16—44no—
240M16—44no—
241M16—82no—
242M16—82no—
243M16—82no—
244M16—82no—
245C2×C8—24no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—82no—
258C2×C8—82no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C42—42no—
274C42—42no—
275C42C2×C22⋊C411yescharacteristic
276C42C2×C4⋊C411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42#12011yescharacteristic
280C42#12011yescharacteristic
281C42#12011yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42C22⋊Q811yescharacteristic
286C42C22⋊Q811yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42#12311yescharacteristic
289C42#12411yescharacteristic
290C42#12711yescharacteristic
291C23—22no—
292C23—41no—
293C23—41no—
294C23#38811yescharacteristic
295C23#39611yescharacteristic
296C23#39711yescharacteristic
297Q8—44no—
298Q8—44no—
299Q8—82no—
300Q8—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4#38011yescharacteristic
325C2×C4#38511yescharacteristic
326C2×C4#38511yescharacteristic
327C2×C4#38611yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#39011yescharacteristic
330C2×C4#39211yescharacteristic
331C2×C4#39311yescharacteristic
332C2×C4#39411yescharacteristic
333C2×C4#39411yescharacteristic
334C2×C4#39611yescharacteristic
335C2×C4#39911yescharacteristic
336C2×C4#47112yes—
337C2×C4#47312yes—
338C8—24no—
339C8—42no—
340C8—82no—
341C8—82no—
342C8—82no—
343C8—81no—
344C8—82no—
345C8—81no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—82no—
355C22#143911yescharacteristic
356C22#193012yes—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—82no—
368C4#186112yes—
369C4#201911yescharacteristic
370C4#202011yescharacteristic
371C4#206412yes—
372C2—22no—
373C2—41no—
374C2—41no—
375C2#1108011yescharacteristic
376C2#1111811yescharacteristic
377C2#1114511yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
C2×#15692
#176482
#174962
#174862
#80612
#80602
#79222