Finite Group Databasebeta

← #97573

Subgroups — #97573

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic
3#8311C212yesmaximal
4C4×#324C211yesmaximal, characteristic
5#7573C212yesmaximal
6C22×C4×Q8C411yescharacteristic
7C22×C4×Q8C411yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2351—24no—
10#2351—24no—
11C2×#389—21no—
12C2×#389—21no—
13C2×#389C411yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#382—21no—
16C2×#382C411yescharacteristic
17C2×C4×C4⋊C4—21no—
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×Q8⋊C8—24no—
22C2×#324—21no—
23C2×#324—21no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C2×C4×Q8—22no—
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—44no—
37C2×C4×Q8—44no—
38C2×C4×Q8C814yescomplemented
39C2×C4×Q8D414yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C42—21no—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2311yescharacteristic
56C2×C4⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C4×C8—22no—
60#389—28no—
61#389—28no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389D414yes—
66#382—28no—
67#382—22no—
68#382—22no—
69C4×C4⋊C4—28no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95D411yescharacteristic
77C2×#95D411yescharacteristic
78#336—22no—
79#336—22no—
80Q8⋊C8—48no—
81#324—24no—
82#324—42no—
83C22×Q8—24no—
84C22×Q8—24no—
85C22×Q8—21no—
86C22×Q8—21no—
87C22×Q8—21no—
88C22×Q8C22⋊C412yes—
89C22×Q8C2×C812yes—
90C22×Q8M1612yes—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C22×C8—42no—
103C22×C8—42no—
104C4×Q8—28no—
105C4×Q8—28no—
106C4×Q8—28no—
107C4×Q8—28no—
108C4×Q8—28no—
109C4×Q8—44no—
110C4×Q8—48no—
111C4×Q8—48no—
112C4×Q8—44no—
113C4×Q8—44no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C2×C812yes—
146C2×C4⋊C4M1612yes—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—24no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—24no—
158C2×C42—42no—
159C2×C42—44no—
160C2×C42—44no—
161C2×C42—42no—
162C2×C42C2×C812yes—
163C2×C42M1612yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C4⋊C8—44no—
167C22⋊C8—24no—
168C22⋊C8—42no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171C4×C8—44no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—24no—
176#95—22no—
177#95—22no—
178C24C2×C22⋊C411yescharacteristic
179C2×Q8—24no—
180C2×Q8—28no—
181C2×Q8—28no—
182C2×Q8—24no—
183C2×Q8—28no—
184C2×Q8—28no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8C22⋊C814yes—
191C2×Q8#9914yes—
192C2×Q8#10114yes—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4C22⋊C812yes—
243C22×C4C22⋊C812yes—
244C22×C4#10412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4#12012yes—
248C22×C4#12012yes—
249C22×C4C22×C811yescharacteristic
250C22×C4C2×M1611yescharacteristic
251C22×C4C2×M1611yescharacteristic
252C22×C4C2×M1611yescharacteristic
253C22×C4#13512yes—
254C22×C4AGL(1,ℤ/8)11yescharacteristic
255C22×C4#13711yescharacteristic
256C2×C8—44no—
257C2×C8—82no—
258C2×C8—82no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C4⋊C4—216no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—44no—
268C4⋊C4—48no—
269C4⋊C4—42no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4C22⋊C814yes—
279C42—216no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—24no—
285C42—42no—
286C42—48no—
287C42—48no—
288C42—42no—
289C42—42no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#37911yescharacteristic
295C23C2×C22⋊C811yescharacteristic
296C23#40711yescharacteristic
297C23#41611yescharacteristic
298C23#41711yescharacteristic
299C23C2×#10411yescharacteristic
300C23#42111yescharacteristic
301Q8—28no—
302Q8—28no—
303Q8—24no—
304Q8—24no—
305Q8—24no—
306Q8#32418yescomplemented
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4C2×C22⋊C811yescharacteristic
368C2×C4C2×C22⋊C811yescharacteristic
369C2×C4#40711yescharacteristic
370C2×C4#40711yescharacteristic
371C2×C4C2×#9912yes—
372C2×C4C2×#10112yes—
373C2×C4#44912yes—
374C2×C4#45212yes—
375C8—84no—
376C8—162no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22#147611yescharacteristic
390C22#147711yescharacteristic
391C22#148811yescharacteristic
392C22#148911yescharacteristic
393C22#177911yescharacteristic
394C22#178111yescharacteristic
395C22#178411yescharacteristic
396C22#178811yescharacteristic
397C22#178911yescharacteristic
398C22#179611yescharacteristic
399C22#179611yescharacteristic
400C4—28no—
401C4—21no—
402C4—21no—
403C4—22no—
404C4—22no—
405C4—42no—
406C4—44no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—82no—
411C4—81no—
412C4—82no—
413C4C2×#32412yes—
414C4#146312yes—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2#829911yescharacteristic
420C2#831211yescharacteristic
421C2#832311yescharacteristic
422C2#836611yescharacteristic
423C2#837411yescharacteristic
424C2#837711yescharacteristic
425C2#839011yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
38. C2×C4×Q8C8semidirect
306. Q8#324semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
#85582
#83112
C4×#3242
#75732