Finite Group Databasebeta

← #118373

Subgroups — #118373

Order 512.

527 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17813C211yesmaximal, characteristic
2#17749C211yesmaximal, characteristic
3C2×#1559C211yesmaximal, characteristic
4#17385C211yesmaximal, characteristic
5#8128C211yesmaximal, characteristic
6#8121C211yesmaximal, characteristic
7#8090C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#1570—22no—
16#1570—22no—
17#1566—22no—
18#1559—22no—
19#1559—22no—
20#1559—22no—
21#1550—22no—
22#1548—21no—
23#1546—21no—
24C2×#333—21no—
25C2×#332—21no—
26#1472C414yes—
27#1470—22no—
28#1461—22no—
29#1309—22no—
30#1307—22no—
31#1302—22no—
32C22×M16—21no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8Q812yes—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432D412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431D412yes—
66#431D412yes—
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⋊C8—21no—
83#407—21no—
84#407C2×C412yes—
85C2×C22⋊C8—21no—
86C2×C22⋊C8C2×C412yes—
87C4×M16—21no—
88C4×M16—21no—
89C2×#97—21no—
90C2×C4×C8—21no—
91#334—28no—
92#333—24no—
93#333—24no—
94#332—24no—
95#332—24no—
96#324—28no—
97C22×Q8—21no—
98C22×Q8C4⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—82no—
109C2×M16—82no—
110C22×C8—22no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122#125—41no—
123#125—41no—
124#125—41no—
125#125—41no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142#117—41no—
143#117—41no—
144#117—41no—
145#117—41no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C4⋊C412yes—
152C2×C22⋊C4—22no—
153C2×C42—22no—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8C22⋊C414yes—
190#97—22no—
191#97—41no—
192#97—41no—
193#97—41no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—41no—
197C4×C8—41no—
198C24#11711yescharacteristic
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—82no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—81no—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C4⋊C411yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12411yescharacteristic
227C22×C4#12411yescharacteristic
228C22×C4#12611yescharacteristic
229C22×C4#12611yescharacteristic
230M16—82no—
231M16—82no—
232M16—82no—
233M16—82no—
234M16—82no—
235M16—82no—
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C2×C8—82no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C42—42no—
261C42—42no—
262C42#12011yescharacteristic
263C42#12011yescharacteristic
264C42#12011yescharacteristic
265C42#12111yescharacteristic
266C42#12111yescharacteristic
267C42#12111yescharacteristic
268C42#12111yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42C22⋊Q811yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42C22⋊Q811yescharacteristic
273C42#12311yescharacteristic
274C42#12311yescharacteristic
275C42#12311yescharacteristic
276C42#12411yescharacteristic
277C42#12711yescharacteristic
278C23—22no—
279C23—41no—
280C23—41no—
281C23#38011yescharacteristic
282C23#38811yescharacteristic
283C23#38811yescharacteristic
284Q8—82no—
285Q8—82no—
286Q8—82no—
287Q8—81no—
288Q8—81no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4#38511yescharacteristic
313C2×C4#38511yescharacteristic
314C2×C4#38611yescharacteristic
315C2×C4#39011yescharacteristic
316C2×C4#39211yescharacteristic
317C2×C4#39311yescharacteristic
318C2×C4#39411yescharacteristic
319C2×C4#39411yescharacteristic
320C2×C4#39611yescharacteristic
321C2×C4#39611yescharacteristic
322C2×C4#39711yescharacteristic
323C2×C4#39911yescharacteristic
324C2×C4#47112yes—
325C2×C4#47112yes—
326C8—44no—
327C8—82no—
328C8—82no—
329C8—82no—
330C8—82no—
331C8—82no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—82no—
341C22#143911yescharacteristic
342C22#181512yes—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—82no—
353C4—82no—
354C4#201911yescharacteristic
355C4#201911yescharacteristic
356C4#204912yes—
357C4#207912yes—
358C2—22no—
359C2—41no—
360C2—41no—
361C2#1109011yescharacteristic
362C2#1110611yescharacteristic
363C2#1114311yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#117
LCSG ▷ C23×C4 ▷ 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
#178132
#177492
C2×#15592
#173852
#81282
#81212
#80902