Finite Group Databasebeta

← #117276

Subgroups — #117276

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21829C211yesmaximal, characteristic
2#21313C211yesmaximal, characteristic
3#17817C211yesmaximal, characteristic
4C2×#1546C211yesmaximal, characteristic
5#8097C211yesmaximal, characteristic
6#8004C211yesmaximal, characteristic
7#7870C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1997—22no—
16#1997—21no—
17#1997—21no—
18#1988—22no—
19#1987—22no—
20#1821—22no—
21#1736—22no—
22#1574—22no—
23#1573—22no—
24#1567—21no—
25#1566—21no—
26#1546—22no—
27#1546—22no—
28C2×#332C414yescomplemented
29#1303—22no—
30#1292—22no—
31#1276—22no—
32C23×Q8—21no—
33C22×M16—21no—
34C22×M16—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⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8Q812yes—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D412yes—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431D412yes—
62#431D412yes—
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#423D412yes—
72#423D411yescharacteristic
73#423D411yescharacteristic
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2×C412yes—
77C2×C4⋊C8C2×C412yes—
78C2×C4⋊C8D412yes—
79#407—21no—
80#407—21no—
81C2×C22⋊C8—21no—
82C4×M16—21no—
83C4×M16—21no—
84C2×C4×C8—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4D412yes—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89#376—21no—
90C43—21no—
91#343—24no—
92#343—24no—
93#340—28no—
94#335—24no—
95#334—24no—
96#332—28no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—82no—
104C22×Q8—82no—
105C23×C4—22no—
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—22no—
112C2×M16—44no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—82no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137#117—24no—
138#117—42no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C4⋊C414yes—
152C2×C22⋊C4—41no—
153C2×C42—22no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42C22×C411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×Q811yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—21no—
172C4⋊C8—22no—
173C4⋊C8—21no—
174C4⋊C8—22no—
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8C22⋊C414yes—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190#97—41no—
191#97—41no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—41no—
195C4×C8—41no—
196#95—22no—
197#95—41no—
198C24#12311yescharacteristic
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—84no—
204C2×Q8—82no—
205C2×Q8—82no—
206C2×Q8—84no—
207C2×Q8—84no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8—82no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—84no—
222C22×C4C2×C4⋊C411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4#12611yescharacteristic
240M16—42no—
241M16—42no—
242M16—42no—
243M16—42no—
244M16—82no—
245M16—82no—
246M16—82no—
247M16—82no—
248C2×C8—24no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—82no—
256C2×C8—82no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C22⋊C4—44no—
268C42—44no—
269C42—42no—
270C42—42no—
271C42—82no—
272C42C2×C22⋊C411yescharacteristic
273C42#11711yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42#12011yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42C22⋊Q811yescharacteristic
285C42#12311yescharacteristic
286C42#12411yescharacteristic
287C42#12711yescharacteristic
288C23—22no—
289C23—22no—
290C23—41no—
291C23#38511yescharacteristic
292C23#39411yescharacteristic
293C23#39711yescharacteristic
294Q8—84no—
295Q8—84no—
296Q8—84no—
297Q8—82no—
298Q8—82no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—82no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4#38011yescharacteristic
327C2×C4#38511yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#39011yescharacteristic
332C2×C4#39211yescharacteristic
333C2×C4#39311yescharacteristic
334C2×C4#39411yescharacteristic
335C2×C4#39611yescharacteristic
336C2×C4#39611yescharacteristic
337C2×C4#39911yescharacteristic
338C2×C4#45412yes—
339C2×C4#45612yes—
340C2×C4#47112yes—
341C2×C4#47212yes—
342C8—44no—
343C8—82no—
344C8—82no—
345C8—81no—
346C8—81no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22#143911yescharacteristic
357C22#185712yes—
358C22#205112yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—84no—
367C4—84no—
368C4#190212yes—
369C4#197711yescharacteristic
370C4#197811yescharacteristic
371C4#200311yescharacteristic
372C4#200511yescharacteristic
373C4#203812yes—
374C4#206312yes—
375C2—22no—
376C2—22no—
377C2—41no—
378C2#1105911yescharacteristic
379C2#1112711yescharacteristic
380C2#1113611yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ 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
28. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#218292
#213132
#178172
C2×#15462
#80972
#80042
#78702