Finite Group Databasebeta

← #188135

Subgroups — #188135

Order 512.

2312 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
1C22×#544C211yesmaximal, characteristic
2#21799C212yesmaximal, complemented
3#17394C212yesmaximal, complemented
4#17357C212yesmaximal
5#8551C214yesmaximal, complemented
6#8550C214yesmaximal
7C2×#544—28no—
8C2×#544—28no—
9C2×#544—24no—
10C2×#544—24no—
11C22×C4⋊Q8C411yescharacteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2212yes—
14C22×#125C412yes—
15C22×#125C411yescharacteristic
16C22×#125C2211yescharacteristic
17C22×C22⋊Q8C412yes—
18C22×C22⋊Q8C2212yes—
19C22×#117C411yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C411yescharacteristic
22#2920C2214yes—
23#1966—28no—
24#1784C2218yes—
25#1470—28no—
26#1464—24no—
27#1463C2218yes—
28#1463C2214yes—
29#1462C2214yes—
30#1458—24no—
31C23×Q8C2×C412yes—
32C23×Q8C2312yes—
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C412yes—
35#544—232no—
36#544—232no—
37#544—232no—
38#544—232no—
39C2×C4⋊Q8—28no—
40C2×C4⋊Q8—24no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—28no—
43C2×C4⋊Q8—28no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8D418yes—
50C2×#125—28no—
51C2×#125—28no—
52C2×#125—24no—
53C2×#125—28no—
54C2×#125—24no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—24no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125D418yes—
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—28no—
66C2×C22⋊Q8—28no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—24no—
71C2×#117—24no—
72C2×#117—24no—
73C2×#117D414yes—
74C2×#117D414yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4C2×C412yes—
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C412yes—
82C22×C4⋊C4C2×C412yes—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C4⋊C4C2×C412yes—
89C22×C4⋊C4C2×C412yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4C2×C412yes—
93C22×C22⋊C4C2×C411yescharacteristic
94C22×C22⋊C4C2311yescharacteristic
95C22×C42C2×C412yes—
96C22×C42C2×C411yescharacteristic
97C22×C42C2311yescharacteristic
98#432—24no—
99#432—24no—
100#431—24no—
101#431—24no—
102#423—24no—
103C2×#101—216no—
104#407C2318yes—
105C4×M16—24no—
106#324—28no—
107#324—216no—
108#323—28no—
109C25C22×C411yescharacteristic
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—84no—
116C22×Q8—84no—
117C22×Q8C2×D418yes—
118C23×C4—28no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—84no—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C22×C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2411yescharacteristic
137C2×M16—28no—
138C4⋊Q8—264no—
139C4⋊Q8—216no—
140C4⋊Q8—216no—
141C4⋊Q8—28no—
142C4⋊Q8—28no—
143C4⋊Q8—28no—
144C4⋊Q8—28no—
145#125—264no—
146#125—216no—
147#125—216no—
148#125—28no—
149#125—28no—
150#125—28no—
151#125—28no—
152C22⋊Q8—264no—
153C22⋊Q8—264no—
154#117—216no—
155#117—216no—
156#117—28no—
157#117—28no—
158#117—28no—
159#117—28no—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—216no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4C22⋊C414yes—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C22⋊C414yes—
199C2×C4⋊C4C22⋊C414yes—
200C2×C4⋊C4C22⋊C414yes—
201C2×C4⋊C4C22⋊C418yes—
202C2×C4⋊C4C22⋊C418yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D418yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4C22⋊C414yes—
213C2×C22⋊C4C2×D414yes—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42C22⋊C412yes—
220C2×C42C22⋊C412yes—
221C2×C42C2×D414yes—
222C2×C42C2×D414yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C4⋊C8—44no—
226C4⋊C8—44no—
227#101—816no—
228C22⋊C8—28no—
229C22⋊C8—28no—
230#97—44no—
231C4×C8—44no—
232C24—21no—
233C24—21no—
234C24—24no—
235C24—42no—
236C24—42no—
237C24C2×C22⋊C412yes—
238C24C23×C411yescharacteristic
239C2×Q8—216no—
240C2×Q8—216no—
241C2×Q8—416no—
242C2×Q8—88no—
243C2×Q8—88no—
244C2×Q8—88no—
245C2×Q8—88no—
246C2×Q8—88no—
247C22×C4—216no—
248C22×C4—216no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—216no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—88no—
280C22×C4—88no—
281C22×C4—88no—
282C22×C4#10118yescomplemented
283C22×C4C2×C22⋊C414yes—
284C22×C4C2×C22⋊C414yes—
285C22×C4C2×C22⋊C414yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C414yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C22×D412yes—
298C22×C4C22×D412yes—
299C22×C4C22×D412yes—
300M16—416no—
301C2×C8—48no—
302C4⋊C4—216no—
303C4⋊C4—216no—
304C4⋊C4—216no—
305C4⋊C4—216no—
306C4⋊C4—28no—
307C4⋊C4—216no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—216no—
318C4⋊C4—216no—
319C22⋊C4—216no—
320C22⋊C4—28no—
321C22⋊C4—28no—
322C42—216no—
323C42—24no—
324C42—24no—
325C42#12014yes—
326C42#12014yes—
327C42#12014yes—
328C42#12014yes—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—24no—
333C23—24no—
334C23—22no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—416no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23C2×#10114yes—
344C23C22×C22⋊C411yescharacteristic
345C23C22×C22⋊C411yescharacteristic
346C23C22×C22⋊C411yescharacteristic
347Q8—816no—
348Q8—816no—
349C2×C4—216no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—216no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—816no—
389C2×C4#37912yes—
390C2×C4#37914yes—
391C2×C4#37912yes—
392C2×C4C2×#10114yes—
393C2×C4C2×#10114yes—
394C2×C4C2×#12014yes—
395C8—88no—
396C22—28no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—24no—
404C22—44no—
405C22—48no—
406C22—44no—
407C22—48no—
408C22—42no—
409C22—42no—
410C22C2×#37911yescharacteristic
411C22#288312yes—
412C22C22×#10112yes—
413C22#288912yes—
414C4—24no—
415C4—22no—
416C4—24no—
417C4—24no—
418C4—44no—
419C4—44no—
420C4—44no—
421C4—42no—
422C4—44no—
423C4—42no—
424C4#178414yes—
425C2—22no—
426C2—22no—
427C2—22no—
428C2—44no—
429C2#2058611yescharacteristic
430C2C2×#178411yescharacteristic
431C2#2059311yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21799C2semidirect
3. #17394C2semidirect
5. #8551C2semidirect
282. C22×C4#101semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
#217992
#173942
#173572
#85512
#85502