Finite Group Databasebeta

← #195162

Subgroups — #195162

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2#34359C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic
4C2×#1511C211yesmaximal, characteristic
5#14342C211yesmaximal, characteristic
6#9046C218yesmaximal, complemented
7#8024C211yesmaximal, characteristic
8#7913C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#484—24no—
12C2×#484—22no—
13C2×#484C418yescomplemented
14C2×#484C2214yes—
15C22×#107—21no—
16C22×#103—21no—
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2378—22no—
20#2368—22no—
21#2344—22no—
22#2342—22no—
23#2332—22no—
24#2329C2214yes—
25#2313—21no—
26#2310—21no—
27#2280C2211yescharacteristic
28C2×#381C2211yescharacteristic
29C2×#381C2211yescharacteristic
30#1824C2214yes—
31#1823—22no—
32#1736—22no—
33#1511—24no—
34C2×#328C2214yes—
35#1445—22no—
36#1445C2214yes—
37C22⋊#95—22no—
38#1294C2214yes—
39#1280C2214yes—
40C23×Q8D411yescharacteristic
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×M16—21no—
44C23×C8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8C2×C414yes—
50C2×C22⋊Q8D414yes—
51C2×C22⋊Q8D412yes—
52C2×C22⋊Q8C2312yes—
53C2×#117—22no—
54C2×#117D412yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C414yes—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D412yes—
70#484—216no—
71#484—28no—
72#484C2×C4116yes—
73C2×#107—28no—
74C2×#107—24no—
75C2×#103—28no—
76C2×#103—24no—
77#407—22no—
78#407—22no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8C2×C414yes—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85#391—24no—
86#391—24no—
87#389—24no—
88#388—24no—
89#388—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#387—24no—
94#385—24no—
95#381—22no—
96#381—21no—
97#381—21no—
98#381—21no—
99#381—21no—
100#381—21no—
101#381—22no—
102#381—22no—
103#380—22no—
104#380D412yes—
105#380C2312yes—
106#379—21no—
107C4×C4⋊C4—24no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C2×#95—21no—
113C2×#95—28no—
114C2×#95D412yes—
115C2×#95C2312yes—
116C2×#95C2312yes—
117#340—28no—
118#328—28no—
119C25C2×D411yescharacteristic
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8C22⋊C414yes—
124C22×Q8C2×D412yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—44no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4—42no—
134C23×C4—84no—
135C23×C4C22×C412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D414yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C2×M16—44no—
147C2×M16—42no—
148C22×C8—24no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—44no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8C22⋊C418yes—
160C22⋊Q8C22×C414yes—
161C22⋊Q8C2×D414yes—
162#117—44no—
163#117—84no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C22⋊C414yes—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D414yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—22no—
208#107—216no—
209#103—216no—
210C22⋊C8—24no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—42no—
215C22⋊C8C22×C414yes—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—42no—
229#95—88no—
230C24—24no—
231C24—24no—
232C24—42no—
233C24C4×D412yes—
234C24#12011yescharacteristic
235C24#12011yescharacteristic
236C24#12112yes—
237C24C22×D411yescharacteristic
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—48no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—84no—
244C2×Q8—82no—
245C2×Q8C2×C22⋊C414yes—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—48no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—82no—
277C22×C4—88no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—88no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4#12312yes—
292C22×C4#12312yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299M16—84no—
300C2×C8—24no—
301C2×C8—48no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—42no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4C2×C22⋊C414yes—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—28no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—84no—
329C42—82no—
330C42—84no—
331C42—84no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—24no—
337C23—24no—
338C23—24no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—42no—
344C23—42no—
345C23—416no—
346C23#38612yes—
347C23#38612yes—
348C23#45814yes—
349C23C2×C4×D411yescharacteristic
350C23C2×#12011yescharacteristic
351C23C2×#12111yescharacteristic
352C23#57814yes—
353Q8—44no—
354Q8—82no—
355Q8—84no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—48no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—88no—
391C2×C4—88no—
392C2×C4—84no—
393C2×C4—88no—
394C2×C4—82no—
395C2×C4—88no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4#38612yes—
399C2×C4#38612yes—
400C2×C4C22×C22⋊C411yescharacteristic
401C2×C4C2×C4×D411yescharacteristic
402C2×C4C2×#12111yescharacteristic
403C2×C4C2×#12311yescharacteristic
404C8—44no—
405C8—82no—
406C22—28no—
407C22—28no—
408C22—22no—
409C22—21no—
410C22—21no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—48no—
418C22—48no—
419C22—44no—
420C22#189014yes—
421C22C2×#38611yescharacteristic
422C22#295012yes—
423C22#301112yes—
424C22#302112yes—
425C22C2×#45812yes—
426C22#304412yes—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—44no—
435C4—84no—
436C4—84no—
437C4—84no—
438C4—162no—
439C2—22no—
440C2—24no—
441C2—22no—
442C2—42no—
443C2#1823514yes—
444C2#2093511yescharacteristic
445C2#2094511yescharacteristic
446C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9046C2semidirect
13. C2×#484C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
#343592
#207002
C2×#15112
#143422
#90462
#80242
#79132