Finite Group Databasebeta

← #238095

Subgroups — #238095

Order 512.

911 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic
2#21825C211yesmaximal, characteristic
3#21825C211yesmaximal, characteristic
4#21825C211yesmaximal, characteristic
5#10393C214yesmaximal
6#10391C214yesmaximal
7#8102C211yesmaximal, characteristic
8#8100C211yesmaximal, characteristic
9#8100C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11#2842C2212yes—
12#2840C2212yes—
13#2665—22no—
14#2665—22no—
15#2628—22no—
16#2607—22no—
17#2439—22no—
18#2439—22no—
19#2280C2211yescharacteristic
20#2280C2211yescharacteristic
21#2280C2211yescharacteristic
22#2280C2211yescharacteristic
23#2280C2211yescharacteristic
24#2280C2211yescharacteristic
25#1996—22no—
26#1996—22no—
27#1996—22no—
28#1996—22no—
29#1996—22no—
30#1996—22no—
31#1819C2212yes—
32#1819C2214yes—
33#1819C2214yes—
34#1819C2212yes—
35#1304C2212yes—
36#1304C2214yes—
37#1304C2214yes—
38#1304C2212yes—
39C23×Q8Q811yescharacteristic, complemented
40C24×C4C2311yescharacteristic
41C2×C4⋊Q8—22no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×#117—21no—
55C2×#117—21no—
56C2×#117—21no—
57C2×#117—21no—
58C2×#117—21no—
59C2×#117—21no—
60C22×C4⋊C4Q811yescharacteristic, complemented
61C22×C4⋊C4Q812yescomplemented
62C22×C22⋊C4C2312yes—
63#400—24no—
64#399—24no—
65#396—24no—
66#396—24no—
67#395—24no—
68#393—24no—
69#393—24no—
70#393—24no—
71#391—24no—
72#391—24no—
73#389—22no—
74#389—22no—
75#387—24no—
76#387—24no—
77#382—22no—
78#381—21no—
79#381—21no—
80#381—21no—
81#381—21no—
82#381—21no—
83#381—21no—
84#380—21no—
85#380—21no—
86#380—21no—
87#380—21no—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—21no—
92#380—21no—
93#380D412yes—
94#380D412yes—
95#380D412yes—
96#379C2312yes—
97#379C2312yes—
98#379C2312yes—
99#379C2311yescharacteristic
100#379C2312yes—
101#379C2312yes—
102#379C2311yescharacteristic
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4D412yes—
113C4×C22⋊C4D412yes—
114C4×C22⋊C4D412yes—
115#342—24no—
116#342—28no—
117#342—28no—
118#342—24no—
119C25C2411yescharacteristic
120C22×Q8—24no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—82no—
128C23×C4—82no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×Q812yes—
136C23×C4C2×Q812yes—
137C23×C4C2×Q811yescharacteristic
138C23×C4C2×Q811yescharacteristic
139C23×C4#4111yescharacteristic
140C23×C4#4111yescharacteristic
141C23×C4#4111yescharacteristic
142C23×C4#4111yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C4⋊Q8—84no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156C22⋊Q8—41no—
157C22⋊Q8—41no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C22⋊Q8—41no—
161C22⋊Q8—41no—
162#117—82no—
163#117—82no—
164#117—82no—
165#117—82no—
166#117—82no—
167#117—82no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4C2×D412yes—
213C2×C22⋊C4C2×D414yes—
214C2×C22⋊C4C2×D414yes—
215C2×C22⋊C4C2×D412yes—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—41no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238C24—42no—
239C24—81no—
240C24—81no—
241C24C22×D411yescharacteristic
242C24C22×D411yescharacteristic
243C24C22×D411yescharacteristic
244C24C22×Q811yescharacteristic
245C24C2×#4111yescharacteristic
246C24C2×#4111yescharacteristic
247C24C2×#4111yescharacteristic
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—82no—
253C2×Q8—82no—
254C2×Q8—82no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—82no—
259C2×Q8—82no—
260C2×Q8—82no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4C22⋊Q811yescharacteristic
295C22×C4C22⋊Q811yescharacteristic
296C22×C4C22⋊Q811yescharacteristic
297C22×C4C22⋊Q811yescharacteristic
298C22×C4C22⋊Q811yescharacteristic
299C22×C4C22⋊Q811yescharacteristic
300C22×C4C22⋊Q811yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303C22×C4C22⋊Q811yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4#12411yescharacteristic
307C22×C4#12411yescharacteristic
308C22×C4#12411yescharacteristic
309C22×C4#12411yescharacteristic
310C22×C4#12411yescharacteristic
311C22×C4#12411yescharacteristic
312C22×C4#12411yescharacteristic
313C22×C4#12411yescharacteristic
314C22×C4#12411yescharacteristic
315C22×C4#12411yescharacteristic
316C22×C4#12411yescharacteristic
317C22×C4#12411yescharacteristic
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—82no—
329C4⋊C4—84no—
330C4⋊C4—84no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—81no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—81no—
350C22⋊C4—81no—
351C22⋊C4—81no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C42—44no—
361C42—82no—
362C42—82no—
363C42—82no—
364C42—82no—
365C42—82no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—88no—
388C23C2×#12011yescharacteristic
389C23C2×C22⋊Q811yescharacteristic
390C23C2×C22⋊Q811yescharacteristic
391C23C2×C22⋊Q811yescharacteristic
392C23C2×#12411yescharacteristic
393C23C2×#12411yescharacteristic
394C23C2×#12411yescharacteristic
395Q8—82no—
396Q8—82no—
397Q8—82no—
398Q8—82no—
399Q8—168no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4#39311yescharacteristic
458C2×C4#39311yescharacteristic
459C2×C4#39311yescharacteristic
460C2×C4#39311yescharacteristic
461C2×C4#39311yescharacteristic
462C2×C4#39311yescharacteristic
463C2×C4#39311yescharacteristic
464C2×C4#39311yescharacteristic
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—44no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22C2×#39311yescharacteristic
493C22#302812yes—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—82no—
502C4—84no—
503C4—84no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4—162no—
508C4—162no—
509C4—162no—
510C4#203212yes—
511C2—22no—
512C2—41no—
513C2—41no—
514C2—41no—
515C2—41no—
516C2—41no—
517C2—41no—
518C2#2347011yescharacteristic
519C2C2×#203211yescharacteristic
520C2#2347411yescharacteristic
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
39. C23×Q8Q8semidirect
60. C22×C4⋊C4Q8semidirect
61. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#374802
#218252
#103932
#103912
#81022
#81002