Finite Group Databasebeta

← #116707

Subgroups — #116707

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17810C211yesmaximal, characteristic
2#17449C211yesmaximal, characteristic
3C2×#1487C211yesmaximal, characteristic
4C2×#1467C211yesmaximal, characteristic
5#8058C211yesmaximal, characteristic
6#8003C211yesmaximal, characteristic
7#7990C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#1573—22no—
16#1572—22no—
17#1565—22no—
18#1491C414yes—
19#1487—22no—
20#1487—22no—
21#1487—22no—
22#1479—22no—
23C2×Q8⋊C8—22no—
24#1467—22no—
25#1467—22no—
26C2×#324—22no—
27#1298—22no—
28#1292—22no—
29#1290—22no—
30C23×Q8Q811yescharacteristic
31C22×M16—21no—
32C2×C4×Q8—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8D412yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4Q811yescharacteristic
46C22×C42—21no—
47C22×C42C2311yescharacteristic
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432C2×C412yes—
59#432D412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431D412yes—
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⋊C8—21no—
78C2×C4⋊C8D412yes—
79#407—21no—
80#407—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8D412yes—
84C4×M16—21no—
85C4×M16C2×C412yes—
86C2×#97—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89#335—24no—
90#334—24no—
91Q8⋊C8—28no—
92Q8⋊C8—28no—
93#324—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—44no—
100C22×Q8C4⋊C412yes—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×Q811yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—82no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114C22×C8—41no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—41no—
124C4×Q8—41no—
125C4×Q8—41no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4C4⋊C412yes—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—41no—
146C2×C42—41no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42C22×C411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8C22⋊C414yes—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—41no—
189#97—41no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—41no—
195C24C22⋊Q811yescharacteristic
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—44no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8—41no—
207C2×Q8—41no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—81no—
220C22×C4—82no—
221C22×C4C2×C4⋊C411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12711yescharacteristic
239M16—42no—
240M16—42no—
241M16—82no—
242M16—82no—
243M16—82no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C42—42no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42C2×C22⋊C411yescharacteristic
267C42#11711yescharacteristic
268C42C4×D411yescharacteristic
269C42C4×D411yescharacteristic
270C42#12011yescharacteristic
271C42#12011yescharacteristic
272C42#12111yescharacteristic
273C42#12111yescharacteristic
274C42#12111yescharacteristic
275C42#12111yescharacteristic
276C42#12111yescharacteristic
277C42#12311yescharacteristic
278C42#12411yescharacteristic
279C42#12411yescharacteristic
280C42#12411yescharacteristic
281C42#12611yescharacteristic
282C23—22no—
283C23—22no—
284C23—41no—
285C23#38511yescharacteristic
286C23#39611yescharacteristic
287C23#39911yescharacteristic
288Q8—24no—
289Q8—24no—
290Q8—42no—
291Q8—42no—
292Q8—44no—
293Q8—44no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—24no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4#38011yescharacteristic
323C2×C4#38511yescharacteristic
324C2×C4#38611yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#38811yescharacteristic
327C2×C4#39011yescharacteristic
328C2×C4#39211yescharacteristic
329C2×C4#39311yescharacteristic
330C2×C4#39411yescharacteristic
331C2×C4#39411yescharacteristic
332C2×C4#39611yescharacteristic
333C2×C4#39711yescharacteristic
334C2×C4#47112yes—
335C2×C4#47312yes—
336C8—44no—
337C8—82no—
338C8—82no—
339C8—82no—
340C8—82no—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—44no—
350C22#143911yescharacteristic
351C22#206412yes—
352C22#209112yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—44no—
364C4—82no—
365C4#190812yes—
366C4#204611yescharacteristic
367C4#204711yescharacteristic
368C2—22no—
369C2—22no—
370C2—41no—
371C2#1106211yescharacteristic
372C2#1110111yescharacteristic
373C2#1111511yescharacteristic
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
#178102
#174492
C2×#14872
C2×#14672
#80582
#80032
#79902