Finite Group Databasebeta

← #117378

Subgroups — #117378

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21834C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3C2×#1574C211yesmaximal, characteristic
4C2×#1546C211yesmaximal, characteristic
5C2×#1299C211yesmaximal, characteristic
6#8041C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#1997—22no—
16#1997—22no—
17#1990—22no—
18#1987—21no—
19#1987—21no—
20#1824—22no—
21C2×#343—22no—
22#1574—22no—
23#1574—22no—
24C2×#335—21no—
25C2×#334—21no—
26#1546—22no—
27#1546C414yescomplemented
28C2×#332—22no—
29#1299—22no—
30#1296—22no—
31#1279—22no—
32C23×Q8—21no—
33C22×M16—21no—
34C22×M16—21no—
35C22×M16—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⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C42C2311yescharacteristic
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431C2×C412yes—
59#431D412yes—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423C2×C412yes—
67#423D412yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8D412yes—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16D412yes—
81C2×#97—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#384D412yes—
86#382—21no—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91#343—28no—
92#340—24no—
93#340—24no—
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—24no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—42no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—44no—
118C2×M16—82no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141#117—82no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4C4⋊C414yes—
157C2×C22⋊C4—41no—
158C2×C42—41no—
159C2×C42C22×C411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×Q811yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—21no—
174C4⋊C8—22no—
175C4⋊C8—21no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8C22⋊C414yes—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189#97—22no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—41no—
195C4×C8—41no—
196C4×C8—41no—
197#95—41no—
198#95—41no—
199C24#12311yescharacteristic
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—84no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—84no—
208C2×Q8—84no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8—82no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—84no—
224C22×C4C2×C4⋊C411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4#12611yescharacteristic
242M16—24no—
243M16—44no—
244M16—42no—
245M16—44no—
246M16—82no—
247M16—82no—
248M16—82no—
249C2×C8—24no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—82no—
261C2×C8—82no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—82no—
275C22⋊C4—82no—
276C42—82no—
277C42C2×C22⋊C411yescharacteristic
278C42#11711yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42#12011yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42C22⋊Q811yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42#12311yescharacteristic
291C42#12411yescharacteristic
292C42#12711yescharacteristic
293C23—22no—
294C23—22no—
295C23—21no—
296C23—21no—
297C23#38511yescharacteristic
298C23#39411yescharacteristic
299C23#39711yescharacteristic
300Q8—84no—
301Q8—84no—
302Q8—84no—
303Q8—82no—
304Q8—82no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—44no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4#38011yescharacteristic
333C2×C4#38511yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#39011yescharacteristic
338C2×C4#39211yescharacteristic
339C2×C4#39311yescharacteristic
340C2×C4#39411yescharacteristic
341C2×C4#39611yescharacteristic
342C2×C4#39611yescharacteristic
343C2×C4#39911yescharacteristic
344C2×C4#45412yes—
345C2×C4#45612yes—
346C2×C4#47112yes—
347C2×C4#47112yes—
348C2×C4#47212yes—
349C2×C4#47312yes—
350C8—44no—
351C8—44no—
352C8—82no—
353C8—81no—
354C8—82no—
355C8—81no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—42no—
363C22—42no—
364C22#143911yescharacteristic
365C22#203812yes—
366C22#204611yescharacteristic
367C22#204711yescharacteristic
368C22#205112yes—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—84no—
376C4—162no—
377C4#185712yes—
378C4#190812yes—
379C4#197711yescharacteristic
380C4#197811yescharacteristic
381C4#200311yescharacteristic
382C4#200511yescharacteristic
383C4#206312yes—
384C4#206412yes—
385C2—41no—
386C2—41no—
387C2#1049112yes—
388C2#1049512yes—
389C2#1106211yescharacteristic
390C2#1112711yescharacteristic
391C2#1113611yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
27. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#218342
C2×#19972
C2×#15742
C2×#15462
C2×#12992
#80412
#78962