Finite Group Databasebeta

← #96753

Subgroups — #96753

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14699C211yesmaximal, characteristic
2#10693C211yesmaximal, characteristic
3C4×#336C211yesmaximal, characteristic
4C2×#1271C212yesmaximal
5#7025C212yesmaximal
6C22×#97—21no—
7C22×C4×C8—21no—
8C2×#389C411yescharacteristic
9C2×#389C2211yescharacteristic
10C2×#387—21no—
11C2×C43C2211yescharacteristic
12C2×#336—21no—
13C2×#336—21no—
14C2×#336C2212yes—
15C2×#336C2212yes—
16#1436—21no—
17#1436—21no—
18C4⋊#95C411yescharacteristic
19C4⋊#95C411yescharacteristic
20C4⋊#95C411yescharacteristic
21C4⋊#95C2211yescharacteristic
22#1434—21no—
23#1271C418yescomplemented
24C23×C8—22no—
25C23×C8—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42C2311yescharacteristic
39C2×#97—24no—
40C2×#97—42no—
41C2×#97—42no—
42C2×C4×C8—24no—
43C2×C4×C8—42no—
44C2×C4×C8—42no—
45#389—22no—
46#389—42no—
47#389C2×C414yes—
48#387—42no—
49#387—42no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C43—42no—
61#336—24no—
62#336—24no—
63#336—22no—
64#336—22no—
65#336C2×C414yes—
66#336C2×C414yes—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—24no—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—24no—
82C22×C8—24no—
83C22×C8—24no—
84C22×C8—42no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8—42no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8—42no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4C4212yes—
112C2×C4⋊C4C4212yes—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42—41no—
132C2×C42—41no—
133C2×C42D812yes—
134C2×C42SD1612yes—
135C2×C42C22×C412yes—
136C2×C42#4112yes—
137#97—48no—
138C4×C8—48no—
139#95—24no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—41no—
151C24C2×C22⋊C411yescharacteristic
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4#9912yes—
213C22×C4#10112yes—
214C22×C4#10212yes—
215C22×C4#10212yes—
216C22×C4#10412yes—
217C22×C4#10412yes—
218C22×C4#10412yes—
219C22×C4#10412yes—
220C22×C4C2×C4211yescharacteristic
221C22×C4#11711yescharacteristic
222C22×C4C4×D412yes—
223C22×C4C4×D412yes—
224C22×C4#13112yes—
225C22×C4#13112yes—
226C22×C4C2×D811yescharacteristic
227C22×C4C2×SD1611yescharacteristic
228C22×C4#13512yes—
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230C22×C4#13712yes—
231C22×C4#13711yescharacteristic
232C2×C8—28no—
233C2×C8—28no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—84no—
255C42—24no—
256C42—24no—
257C42—24no—
258C42—24no—
259C42—42no—
260C42—42no—
261C42—44no—
262C42—44no—
263C42—44no—
264C42—44no—
265C23C4×C22⋊C411yescharacteristic
266C23#40811yescharacteristic
267C23#40811yescharacteristic
268C23C2×#9911yescharacteristic
269C23#41011yescharacteristic
270C23C2×#10111yescharacteristic
271C23#41311yescharacteristic
272C23C2×#10211yescharacteristic
273C23#41611yescharacteristic
274C23#41711yescharacteristic
275C23#41911yescharacteristic
276C23C2×#10411yescharacteristic
277C23C2×#10411yescharacteristic
278C23C2×#10411yescharacteristic
279C23C2×#10411yescharacteristic
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
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—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4#32714yes—
359C2×C4#33014yes—
360C2×C4#40512yes—
361C2×C4#43312yes—
362C2×C4C4×D812yes—
363C2×C4C4×SD1612yes—
364C2×C4#44012yes—
365C2×C4#44112yes—
366C8—48no—
367C8—48no—
368C22#146811yescharacteristic
369C22#147011yescharacteristic
370C22#147211yescharacteristic
371C22#147311yescharacteristic
372C22#148311yescharacteristic
373C22#148311yescharacteristic
374C22#148511yescharacteristic
375C22#148511yescharacteristic
376C22#149211yescharacteristic
377C22#149211yescharacteristic
378C22#149411yescharacteristic
379C22#149411yescharacteristic
380C22C2×#32711yescharacteristic
381C22#150011yescharacteristic
382C22#150111yescharacteristic
383C22#150211yescharacteristic
384C22#150411yescharacteristic
385C22#150611yescharacteristic
386C22#150811yescharacteristic
387C22#150911yescharacteristic
388C22C2×#33011yescharacteristic
389C22#152411yescharacteristic
390C22#152411yescharacteristic
391C22#152611yescharacteristic
392C22#154611yescharacteristic
393C22#154711yescharacteristic
394C22#155211yescharacteristic
395C22#155411yescharacteristic
396C22#175011yescharacteristic
397C22C4×#9911yescharacteristic
398C22C4×#10111yescharacteristic
399C22C4×#10411yescharacteristic
400C22C4×#10411yescharacteristic
401C22C4×#10211yescharacteristic
402C22#176311yescharacteristic
403C4—24no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—24no—
411C4—42no—
412C4—42no—
413C4—82no—
414C4—82no—
415C4—161no—
416C4—161no—
417C2#767511yescharacteristic
418C2#767711yescharacteristic
419C2#767811yescharacteristic
420C2#768011yescharacteristic
421C2#768511yescharacteristic
422C2#768611yescharacteristic
423C2#769111yescharacteristic
424C2#769311yescharacteristic
425C2#815411yescharacteristic
426C2#817711yescharacteristic
427C2#818011yescharacteristic
428C2C4×#32711yescharacteristic
429C2#821911yescharacteristic
430C2C4×#33011yescharacteristic
431C2#823611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1271C4semidirect

Maximal subgroups

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

subgroupindex
#146992
#106932
C4×#3362
C2×#12712
#70252