Finite Group Databasebeta

← #195272

Subgroups — #195272

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic
2#34398C211yesmaximal, characteristic
3#21840C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic, complemented
6#9043C218yesmaximal, complemented
7#8013C211yesmaximal, characteristic, complemented
8#7914C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3226—24no—
12#3226C418yescomplemented
13C2×#500—21no—
14C2×#486—21no—
15C2×#480C2214yes—
16C2×#474—22no—
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2379—22no—
20#2361—22no—
21#2357—21no—
22#2349—21no—
23#2339—22no—
24#2331—21no—
25#2328—21no—
26#2325C2214yes—
27#2318—22no—
28#2283C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1993—22no—
32#1987—21no—
33#1987—21no—
34#1823C2214yes—
35#1505—24no—
36C2×#327C2214yes—
37#1446—22no—
38#1445C2214yes—
39C4⋊#95—22no—
40#1294C2214yes—
41#1280C2214yes—
42C22×#41—21no—
43C23×Q8—21no—
44#532—22no—
45#532—22no—
46#532C2×C414yes—
47#532D414yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yescomplemented
56C2×C4⋊Q8D411yescharacteristic
57C2×#127D411yescharacteristic
58C2×#124—21no—
59C2×#121—21no—
60C2×#121C2312yes—
61C2×C4×D4—22no—
62C2×C4×D4—24no—
63C2×C4×D4—21no—
64C2×C4×D4D412yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#500—28no—
71#486—28no—
72#480—28no—
73#480C2×C4116yescomplemented
74#474—216no—
75#432—22no—
76#432—22no—
77#431—22no—
78#431C2×C414yescomplemented
79C2×#107—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×#102—24no—
83C2×C22⋊C8C2312yes—
84C2×C22⋊C8C2312yes—
85C2×#97—21no—
86C2×C4×C8—21no—
87#391—24no—
88#390—22no—
89#389—22no—
90#388—24no—
91#388—24no—
92#387—24no—
93#386—24no—
94#386—24no—
95#386—24no—
96#385—24no—
97#385—24no—
98#383—21no—
99#382—21no—
100#382—21no—
101#382—21no—
102#382—21no—
103#382D412yes—
104#381—22no—
105#381—21no—
106#380—22no—
107#380C2312yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4D411yescharacteristic
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113#376—21no—
114#376—22no—
115#376—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—28no—
119C2×#95D412yes—
120C2×#95C2312yes—
121C2×#95C2312yes—
122#340—24no—
123#340—24no—
124#327—28no—
125C2×#41—24no—
126C2×#41—24no—
127C2×#41—84no—
128C22×Q8—24no—
129C22×Q8—21no—
130C22×Q8—41no—
131C22×Q8—41no—
132C22×Q8—42no—
133C22×Q8—84no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C2×D412yes—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D414yes—
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—22no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8C22⋊C414yes—
158#127—22no—
159#127C22⋊C414yes—
160#124—24no—
161#124—24no—
162#121—24no—
163#121—24no—
164#121—22no—
165#121—22no—
166#121C22×C414yes—
167#121C2×D414yes—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—84no—
172C4×D4—88no—
173C4×D4C22⋊C418yes—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4C22×C414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C42—24no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—24no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42C22×C412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212#107—216no—
213C4⋊C8—24no—
214C4⋊C8—42no—
215#102—216no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8C22×C414yes—
220#97—42no—
221C4×C8—24no—
222#95—24no—
223#95—24no—
224#95—21no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—42no—
236#95—88no—
237C24—22no—
238C24C22×D411yescharacteristic
239#41—88no—
240#41—88no—
241C2×Q8—24no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—88no—
245C2×Q8—82no—
246C2×Q8—84no—
247C2×Q8—84no—
248C2×Q8—88no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—82no—
259C2×D4C2×C22⋊C414yes—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—84no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12112yes—
292C22×C4#12112yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C2×C8—24no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—42no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—82no—
315C4⋊C4—84no—
316C4⋊C4C2×C22⋊C414yes—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—28no—
320C22⋊C4—44no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C42—48no—
325C42—42no—
326C42—42no—
327C42—84no—
328C42C2×C22⋊C412yes—
329C42C2×C22⋊C412yes—
330C42C4×D412yes—
331C42C4×D412yes—
332C42#12112yes—
333C42#12112yes—
334C42#12312yes—
335C42#12312yes—
336C23—21no—
337C23—21no—
338C23—22no—
339C23—24no—
340C23—44no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23C2×C4×D411yescharacteristic
345C23C2×#12011yescharacteristic
346C23C2×#12111yescharacteristic
347C23#57614yes—
348Q8—84no—
349Q8—88no—
350Q8—88no—
351D4—44no—
352D4—48no—
353D4—48no—
354D4—48no—
355D4—82no—
356D4—84no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—88no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—88no—
393C2×C4—168no—
394C2×C4#38612yes—
395C2×C4#38612yes—
396C2×C4#38612yes—
397C2×C4#38612yes—
398C2×C4#45512yes—
399C2×C4#45612yes—
400C2×C4C22×C22⋊C411yescharacteristic
401C2×C4C2×C4×D411yescharacteristic
402C2×C4C2×#12111yescharacteristic
403C2×C4C2×#12311yescharacteristic
404C8—44no—
405C8—82no—
406C22—24no—
407C22—24no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—44no—
413C22—44no—
414C22—44no—
415C22—84no—
416C22—84no—
417C22C2×#38611yescharacteristic
418C22#294812yes—
419C22#300812yes—
420C22C2×#45511yescharacteristic
421C22C2×#45611yescharacteristic
422C22#302812yes—
423C22#304112yes—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—44no—
432C4—84no—
433C4—84no—
434C4—82no—
435C4—82no—
436C4—84no—
437C4#188214yes—
438C2—22no—
439C2—22no—
440C2—84no—
441C2#1843612yes—
442C2#1844012yes—
443C2#2093411yescharacteristic
444C2C2×#188211yescharacteristic
445C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21840C2semidirect
5. #14282C2semidirect
6. #9043C2semidirect
7. #8013C2semidirect
8. #7914C2semidirect
12. #3226C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
73. #480C2×C4semidirect
78. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#343982
#218402
C2×#15052
#142822
#90432
#80132
#79142