Finite Group Databasebeta

← #97579

Subgroups — #97579

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2#8707C212yesmaximal
3#8558C211yesmaximal, characteristic
4C4×#324C211yesmaximal, characteristic
5#7575C212yesmaximal
6C22×C4×Q8C411yescharacteristic
7#2355—24no—
8#2355—24no—
9C2×#391C411yescharacteristic
10C2×#391C2211yescharacteristic
11C2×#387—21no—
12C2×#387—21no—
13C2×#384C411yescharacteristic
14C2×#382—21no—
15C2×C4×C4⋊C4—21no—
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#1272—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×Q8C814yescomplemented
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⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C42—21no—
43C22×C42—21no—
44C22×C42—21no—
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42C2311yescharacteristic
48C2×C4⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C4×C8—22no—
52#391—22no—
53#391—22no—
54#391—22no—
55#391D414yes—
56#387—28no—
57#387—28no—
58#384—22no—
59#384—22no—
60#382—28no—
61C4×C4⋊C4—28no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4D414yes—
68C2×#95—21no—
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—
80C22×Q8—21no—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—42no—
84C22×Q8—42no—
85C22×Q8C22⋊C412yes—
86C22×Q8C2×C812yes—
87C22×Q8M1612yes—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C22×C8—42no—
100C22×C8—42no—
101C4×Q8—28no—
102C4×Q8—28no—
103C4×Q8—28no—
104C4×Q8—28no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—42no—
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⋊C4C2×C812yes—
129C2×C4⋊C4M1612yes—
130C2×C4⋊C4SD1612yes—
131C2×C4⋊C4Q1612yes—
132C2×C42—24no—
133C2×C42—24no—
134C2×C42—24no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—24no—
142C2×C42—24no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42—82no—
146C2×C42—82no—
147C2×C42C2×C812yes—
148C2×C42M1612yes—
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C4⋊C8—44no—
152C22⋊C8—24no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C4×C8—44no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—22no—
164#95—22no—
165C24C2×C22⋊C411yescharacteristic
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8C22⋊C814yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4C22⋊C812yes—
224C22×C4C22⋊C812yes—
225C22×C4#10312yes—
226C22×C4#10312yes—
227C22×C4#10412yes—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4#12012yes—
231C22×C4#12012yes—
232C22×C4C22×C811yescharacteristic
233C22×C4C2×M1611yescharacteristic
234C22×C4C2×M1611yescharacteristic
235C22×C4C2×M1611yescharacteristic
236C22×C4C2×SD1611yescharacteristic
237C22×C4C2×Q1611yescharacteristic
238C22×C4#13712yes—
239C2×C8—44no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—216no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—48no—
256C4⋊C4—42no—
257C4⋊C4—84no—
258C4⋊C4C22⋊C814yes—
259C42—216no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—24no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—84no—
269C42—84no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23#37911yescharacteristic
275C23C2×C22⋊C811yescharacteristic
276C23#40711yescharacteristic
277C23C2×#10311yescharacteristic
278C23#41911yescharacteristic
279C23C2×#10411yescharacteristic
280C23#42111yescharacteristic
281Q8—44no—
282Q8—44no—
283Q8—44no—
284Q8—44no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4Q8⋊C814yes—
343C2×C4C2×C22⋊C811yescharacteristic
344C2×C4C2×C22⋊C811yescharacteristic
345C2×C4#40711yescharacteristic
346C2×C4#40711yescharacteristic
347C2×C4#41012yes—
348C2×C4#41312yes—
349C2×C4#44812yes—
350C2×C4#45112yes—
351C8—84no—
352C8—162no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22C2×Q8⋊C811yescharacteristic
366C22#147911yescharacteristic
367C22#148711yescharacteristic
368C22#149111yescharacteristic
369C22#177911yescharacteristic
370C22#178211yescharacteristic
371C22#178511yescharacteristic
372C22#178711yescharacteristic
373C22#178911yescharacteristic
374C22#179511yescharacteristic
375C22#179611yescharacteristic
376C4—21no—
377C4—21no—
378C4—24no—
379C4—22no—
380C4—22no—
381C4—44no—
382C4—42no—
383C4—44no—
384C4—42no—
385C4—42no—
386C4—81no—
387C4—82no—
388C4—82no—
389C4#145812yes—
390C4#146412yes—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2#830011yescharacteristic
396C2#831111yescharacteristic
397C2#832411yescharacteristic
398C2#836511yescharacteristic
399C2#837011yescharacteristic
400C2#838011yescharacteristic
401C2#839211yescharacteristic
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
33. C2×C4×Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
#87072
#85582
C4×#3242
#75752