Finite Group Databasebeta

← #196850

Subgroups — #196850

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2C2×#2357C211yesmaximal, characteristic
3#21786C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic, complemented
5C2×#1503C211yesmaximal, characteristic, complemented
6#9154C214yesmaximal, complemented
7#9146C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#8014C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2848C2214yes—
15#2626—22no—
16#2580—22no—
17#2566—22no—
18#2563C412yescomplemented
19#2542C412yescomplemented
20#2538C414yescomplemented
21#2357—24no—
22#2357—24no—
23#2317—21no—
24#2309—21no—
25C2×#390—21no—
26C2×#388—22no—
27#2285C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#381C2212yes—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2212yes—
33#1954—22no—
34#1954—22no—
35#1779C2214yes—
36#1518—24no—
37#1503—24no—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C23×D4D411yescharacteristic
45C24×C4C2312yes—
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C418yes—
55C2×C22⋊Q8C2312yes—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C412yes—
64C2×C4×D4D412yes—
65C2×C4×D4D412yes—
66C2×#117—21no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D412yes—
74C22×C42—21no—
75C22×C42—22no—
76C22×C42C2311yescharacteristic
77#431—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C22⋊C8—28no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88#399—24no—
89#396—28no—
90#393—24no—
91#390—28no—
92#390—22no—
93#390—22no—
94#390—24no—
95#389—24no—
96#388—216no—
97#385—28no—
98#385—24no—
99#384—22no—
100#384C2×C412yes—
101#382—22no—
102#382—21no—
103#382—22no—
104#382D414yes—
105#381—24no—
106#381—24no—
107#380—21no—
108#380C2×C418yes—
109#380C2312yes—
110#379—24no—
111C4×C4⋊C4—28no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4D414yes—
116C4×C22⋊C4—24no—
117C4×C22⋊C4—24no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95C2311yescharacteristic
121C2×#95C2312yes—
122C2×#95C2311yescharacteristic
123#328—24no—
124#328—24no—
125#328—28no—
126C25C2×D412yes—
127C22×Q8—21no—
128C22×Q8C22×C412yes—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C22⋊C412yes—
136C22×D4C22⋊C412yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—84no—
145C23×C4C22×C414yes—
146C23×C4C2×D412yes—
147C23×C4C2×D414yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157#125—42no—
158#125—42no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C22⋊Q8—88no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178#117—42no—
179#117—42no—
180#117—42no—
181#117—42no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C414yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—48no—
210C2×C22⋊C4—84no—
211C2×C22⋊C4C22⋊C414yes—
212C2×C22⋊C4C22⋊C414yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—28no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—84no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—88no—
235C4×C8—42no—
236C4×C8—42no—
237#95—28no—
238#95—22no—
239#95—28no—
240#95—24no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95—22no—
245#95C2×D412yes—
246#95C2×D414yes—
247#95C2×D412yes—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—44no—
255C24—44no—
256C24C2×C22⋊C414yes—
257C24C22×D411yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—42no—
262C2×Q8—82no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4C2×C22⋊C414yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4#12012yes—
312C22×C4#12012yes—
313C22×C4#12312yes—
314C22×C4#12312yes—
315C22×C4#13512yes—
316C22×C4#13512yes—
317C22×C4#13712yes—
318C22×C4#13712yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C4⋊C4—48no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—82no—
344C4⋊C4—88no—
345C22⋊C4—28no—
346C22⋊C4—28no—
347C22⋊C4—416no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—88no—
353C22⋊C4—88no—
354C22⋊C4—88no—
355C42—44no—
356C42—44no—
357C42—48no—
358C42—88no—
359C42C4×D412yes—
360C42C4×D412yes—
361C42C4×D412yes—
362C42C4×D412yes—
363C42#12112yes—
364C42#12112yes—
365C42#12112yes—
366C42#12112yes—
367C23—22no—
368C23—22no—
369C23—28no—
370C23—24no—
371C23—24no—
372C23—28no—
373C23—24no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—84no—
383C23—84no—
384C23—84no—
385C23—84no—
386C23C22×C22⋊C411yescharacteristic
387C23C2×#12011yescharacteristic
388C23C2×#12311yescharacteristic
389C23C2×#13511yescharacteristic
390C23C2×#13511yescharacteristic
391C23C2×#13711yescharacteristic
392C23C2×#13711yescharacteristic
393Q8—82no—
394Q8—82no—
395Q8—84no—
396D4—48no—
397D4—48no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—28no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—82no—
435C2×C4—84no—
436C2×C4—82no—
437C2×C4—84no—
438C2×C4#38612yes—
439C2×C4#38612yes—
440C2×C4#38612yes—
441C2×C4#38612yes—
442C2×C4#41612yes—
443C2×C4#41612yes—
444C2×C4#41912yes—
445C2×C4#41912yes—
446C2×C4#45214yes—
447C2×C4#45214yes—
448C2×C4#48514yes—
449C2×C4#48514yes—
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×#12111yescharacteristic
454C8—82no—
455C8—82no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—44no—
465C22—48no—
466C22—48no—
467C22—44no—
468C22—44no—
469C22—84no—
470C22—84no—
471C22—84no—
472C22—84no—
473C22C2×#38611yescharacteristic
474C22#288312yes—
475C22C2×#41611yescharacteristic
476C22C2×#41911yescharacteristic
477C22#290112yes—
478C22C2×#45211yescharacteristic
479C22C2×#45211yescharacteristic
480C22C2×#48511yescharacteristic
481C22C2×#48511yescharacteristic
482C4—22no—
483C4—22no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—44no—
489C4—44no—
490C4—82no—
491C4—84no—
492C4—84no—
493C4—82no—
494C4—82no—
495C4#187314yes—
496C4#187314yes—
497C2—21no—
498C2—21no—
499C2—21no—
500C2—21no—
501C2—84no—
502C2#1761712yes—
503C2#1762712yes—
504C2C2×#187311yescharacteristic
505C2#2098011yescharacteristic
506C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1518C2semidirect
5. C2×#1503C2semidirect
6. #9154C2semidirect
7. #9146C2semidirect
8. #8026C2semidirect
9. #8014C2semidirect
18. #2563C4semidirect
19. #2542C4semidirect
20. #2538C4semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×#23572
#217862
C2×#15182
C2×#15032
#91542
#91462
#80262
#80142