Finite Group Databasebeta

← #97590

Subgroups — #97590

Order 512.

798 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2370C211yesmaximal, characteristic
2#8706C212yesmaximal
3#8558C211yesmaximal, characteristic
4C4×#324C211yesmaximal, characteristic
5#7576C212yesmaximal
6C22×#125C411yescharacteristic
7#2370—24no—
8#2370—24no—
9C2×#389—21no—
10C2×#389C411yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#387—21no—
13C2×#387—21no—
14C2×C4×C4⋊C4—21no—
15C2×C4×C4⋊C4C411yescharacteristic
16C2×C4×C4⋊C4C411yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×#324—21no—
22C2×#324—21no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26#1271—24no—
27C2×#125—22no—
28C2×#125—22no—
29C2×#125—22no—
30C2×#125—22no—
31C2×#125—22no—
32C2×#125C814yescomplemented
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C42—21no—
44C22×C42—21no—
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C4⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C4×C8—22no—
53#389—28no—
54#389—22no—
55#389—22no—
56#389—22no—
57#389D414yes—
58#387—28no—
59#387—28no—
60C4×C4⋊C4—28no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4D414yes—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95D411yescharacteristic
75C2×#95D411yescharacteristic
76#336—22no—
77#336—22no—
78#324—24no—
79#324—42no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—42no—
92C22×C8—42no—
93#125—28no—
94#125—28no—
95#125—28no—
96#125—28no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—82no—
125C2×C4⋊C4C22⋊C412yes—
126C2×C4⋊C4C22⋊C412yes—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C2×C812yes—
130C2×C4⋊C4C2×C812yes—
131C2×C4⋊C4M1612yes—
132C2×C4⋊C4M1612yes—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—24no—
140C2×C42—24no—
141C2×C42—24no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42—41no—
145C2×C42—41no—
146C2×C42—42no—
147C2×C42—82no—
148C2×C42—82no—
149C2×C42C2×C812yes—
150C2×C42M1612yes—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C4⋊C8—44no—
154C22⋊C8—24no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157C22⋊C8—42no—
158C4×C8—44no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—22no—
165#95—22no—
166C24C2×C22⋊C411yescharacteristic
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
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—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4C22⋊C812yes—
217C22×C4C22⋊C812yes—
218C22×C4#10412yes—
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4#12012yes—
222C22×C4#12012yes—
223C22×C4C22×C811yescharacteristic
224C22×C4C2×M1611yescharacteristic
225C22×C4C2×M1611yescharacteristic
226C22×C4C2×M1611yescharacteristic
227C22×C4#13512yes—
228C22×C4AGL(1,ℤ/8)11yescharacteristic
229C22×C4#13711yescharacteristic
230C2×C8—44no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C4⋊C4—24no—
237C4⋊C4—216no—
238C4⋊C4—216no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—48no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—84no—
253C4⋊C4C22⋊C814yes—
254C4⋊C4C22⋊C814yes—
255C4⋊C4#10114yes—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—24no—
261C42—42no—
262C42—42no—
263C42—42no—
264C42—44no—
265C42—44no—
266C42—84no—
267C42—84no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23#37911yescharacteristic
273C23C2×C22⋊C811yescharacteristic
274C23#40711yescharacteristic
275C23#41611yescharacteristic
276C23#41711yescharacteristic
277C23C2×#10411yescharacteristic
278C23#42111yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
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—24no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4C2×C22⋊C811yescharacteristic
340C2×C4C2×C22⋊C811yescharacteristic
341C2×C4#40711yescharacteristic
342C2×C4#40711yescharacteristic
343C2×C4#41012yes—
344C2×C4C2×#10112yes—
345C2×C4#44912yes—
346C2×C4#45212yes—
347C8—84no—
348C8—162no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22#147611yescharacteristic
362C22#147711yescharacteristic
363C22#148811yescharacteristic
364C22#148911yescharacteristic
365C22#177911yescharacteristic
366C22#178211yescharacteristic
367C22#178411yescharacteristic
368C22#178811yescharacteristic
369C22#178911yescharacteristic
370C22#179511yescharacteristic
371C22#179611yescharacteristic
372C4—21no—
373C4—21no—
374C4—22no—
375C4—22no—
376C4—42no—
377C4—44no—
378C4—42no—
379C4—44no—
380C4—42no—
381C4—42no—
382C4—44no—
383C4—82no—
384C4—81no—
385C4#146112yes—
386C4#146712yes—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2#830311yescharacteristic
392C2#831211yescharacteristic
393C2#832411yescharacteristic
394C2#836211yescharacteristic
395C2#837411yescharacteristic
396C2#838211yescharacteristic
397C2#839811yescharacteristic
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
32. C2×#125C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23702
#87062
#85582
C4×#3242
#75762