Finite Group Databasebeta

← #121838

Subgroups — #121838

Order 512.

547 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21939C211yesmaximal, characteristic
2#8102C211yesmaximal, characteristic
3#8100C211yesmaximal, characteristic
4#8049C211yesmaximal, characteristic
5#7982C211yesmaximal, characteristic
6#7946C211yesmaximal, characteristic
7#7927C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10#2280C2211yescharacteristic
11#2280C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14#2280C2211yescharacteristic
15#1854—22no—
16#1854—22no—
17#1808C412yescomplemented
18#1807C412yescomplemented
19#1304—22no—
20#1304—22no—
21#1297—22no—
22#1289—22no—
23#1284—22no—
24#1282—22no—
25C24×C4C2311yescharacteristic
26C22×M16—21no—
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—21no—
32C2×#117—21no—
33C2×#117—21no—
34C2×#107—22no—
35C2×#106—22no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407D412yes—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8D412yes—
47C2×C22⋊C8D412yes—
48#381—21no—
49#381—21no—
50#381—21no—
51#381—21no—
52#381Q812yes—
53#380—21no—
54#380—21no—
55#380—21no—
56#380—21no—
57#380—21no—
58#380—21no—
59#380—21no—
60#380—21no—
61#380C2×C412yes—
62#380C2×C412yes—
63#379—21no—
64#379—21no—
65#379—21no—
66#379—21no—
67#379D412yes—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C4×C22⋊C4—21no—
74C4×C22⋊C4—21no—
75C4×C22⋊C4—21no—
76C4×C22⋊C4—21no—
77C4×C22⋊C4Q812yes—
78#340—22no—
79#340—22no—
80C25C2×D411yescharacteristic
81C23×C4—81no—
82C23×C4—81no—
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×Q811yescharacteristic
88C23×C4C2×Q811yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C2×M16—22no—
98C2×M16—41no—
99C2×M16—81no—
100C2×M16—81no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—41no—
106C22×C8—81no—
107C22×C8—81no—
108C22×C8C22⋊C412yes—
109C22×C8C22⋊C412yes—
110#117—81no—
111#117—81no—
112#117—81no—
113#117—81no—
114#117—81no—
115#117—81no—
116#117—81no—
117#117—81no—
118#117—81no—
119#117—81no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C42—22no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42—41no—
145#107—82no—
146#106—82no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155#95—22no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95—41no—
164#95—41no—
165C24—42no—
166C24—81no—
167C24—81no—
168C24C4×D411yescharacteristic
169C24#12111yescharacteristic
170C24C22⋊Q811yescharacteristic
171C24C22⋊Q811yescharacteristic
172C24#12311yescharacteristic
173C24#12311yescharacteristic
174C24#12711yescharacteristic
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4#11711yescharacteristic
203C22×C4#11711yescharacteristic
204C22×C4C4×D411yescharacteristic
205C22×C4C4×Q811yescharacteristic
206C22×C4C4×Q811yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4C22⋊Q811yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4C22⋊Q811yescharacteristic
211C22×C4C22⋊Q811yescharacteristic
212C22×C4#12311yescharacteristic
213C22×C4#12311yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12311yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4#12411yescharacteristic
219C22×C4#12411yescharacteristic
220C22×C4#12511yescharacteristic
221C22×C4#12511yescharacteristic
222C22×C4#12511yescharacteristic
223C22×C4#12511yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12611yescharacteristic
227C22×C4#12611yescharacteristic
228C22×C4C4⋊Q811yescharacteristic
229M16—82no—
230M16—82no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—81no—
237C2×C8—81no—
238C2×C8—81no—
239C2×C8—81no—
240C2×C8—81no—
241C2×C8—81no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—82no—
251C22⋊C4—44no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C22⋊C4—82no—
266C42—81no—
267C42—81no—
268C42—82no—
269C42—82no—
270C42—82no—
271C42—82no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—88no—
294C23#38711yescharacteristic
295C23#38711yescharacteristic
296C23#39011yescharacteristic
297C23#39711yescharacteristic
298C23#39811yescharacteristic
299C23#39911yescharacteristic
300C23#39911yescharacteristic
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
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—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4#38111yescharacteristic
352C2×C4#38711yescharacteristic
353C2×C4#38711yescharacteristic
354C2×C4#39111yescharacteristic
355C2×C4#39711yescharacteristic
356C2×C4#39811yescharacteristic
357C2×C4#40011yescharacteristic
358C2×C4#40011yescharacteristic
359C8—42no—
360C8—42no—
361C8—82no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—44no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22#144511yescharacteristic
390C22#197612yes—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—162no—
399C4—162no—
400C4—162no—
401C4—162no—
402C4—162no—
403C4#182811yescharacteristic
404C4#182811yescharacteristic
405C2—22no—
406C2—41no—
407C2—41no—
408C2—41no—
409C2—41no—
410C2—41no—
411C2—41no—
412C2#1135111yescharacteristic
413C2#1136111yescharacteristic
414C2#1136411yescharacteristic
subgroupG/H
center, socleC22#1445
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1808C4semidirect
18. #1807C4semidirect

Maximal subgroups

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

subgroupindex
#219392
#81022
#81002
#80492
#79822
#79462
#79272