Finite Group Databasebeta

← #206763

Subgroups — #206763

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34966C211yesmaximal, characteristic
2#20809C211yesmaximal, characteristic
3#20550C211yesmaximal, characteristic
4#9868C212yesmaximal, complemented
5#9867C212yesmaximal, complemented
6#9866C212yesmaximal, complemented
7#9865C212yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9#7899C211yesmaximal, characteristic
10#7894C211yesmaximal, characteristic
11#7892C211yesmaximal, characteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2606—21no—
17#2605—21no—
18#2570C412yes—
19#2550C412yes—
20#2442C412yes—
21#2440C412yes—
22#2438—22no—
23#2402—21no—
24#2399—21no—
25C4×C4⋊Q8C2212yes—
26C4×#125C2212yes—
27C4×C22⋊Q8—22no—
28#2283C2211yescharacteristic
29C4×#117C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1843—21no—
32#1832—21no—
33#1798—22no—
34#1770—21no—
35C8×C4⋊C4—21no—
36#1767C2212yes—
37C4×C4⋊C8C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2212yes—
40#1753—22no—
41#1279C2211yescharacteristic
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45#1279C2211yescharacteristic
46#1279C2212yes—
47#1279C2212yes—
48#1279C2211yescharacteristic
49#1279C2212yes—
50#1279C2211yescharacteristic
51C22×M16—21no—
52C2×C4⋊Q8C2×C412yes—
53C2×#125C2×C412yes—
54C2×C22⋊Q8—22no—
55C2×C4×Q8—22no—
56C2×#117D412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#431—22no—
61#431—22no—
62#431—22no—
63#431—22no—
64#431—22no—
65#431Q814yescomplemented
66#423—22no—
67#423Q814yescomplemented
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×#97—22no—
80C2×C4×C8—21no—
81C2×C4×C8—22no—
82C2×C4×C8C2312yes—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#400—22no—
86#400—22no—
87#398—22no—
88#398—22no—
89#398—22no—
90#398—22no—
91#395—22no—
92#395—22no—
93#391—22no—
94#391—22no—
95#389—22no—
96#389—22no—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#384C2×C412yes—
108#383—21no—
109#382C2×C412yes—
110#382C2×C412yes—
111#381—21no—
112#380—22no—
113#380C2×C412yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2312yes—
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4C2×C411yescharacteristic
129C4×C22⋊C4C2×C411yescharacteristic
130C4×C22⋊C4D412yes—
131#376—21no—
132#376D411yescharacteristic
133#376D411yescharacteristic
134C2×#95—21no—
135C43C2312yes—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140C22×Q8—22no—
141C23×C4—22no—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×M16—24no—
146C2×M16—81no—
147C2×M16—81no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C4⋊Q8—24no—
155C4⋊Q8—42no—
156#125—24no—
157#125—42no—
158C22⋊Q8—48no—
159C4×Q8—82no—
160C4×Q8—82no—
161#117—44no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—81no—
189C2×C42—81no—
190C2×C42C22×C412yes—
191C2×C42C22×C412yes—
192C2×C42C22×C411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×Q812yes—
200C2×C42C2×Q812yes—
201C2×C42#4112yes—
202C2×C42#4112yes—
203C2×C42C2411yescharacteristic
204C4⋊C8—24no—
205C4⋊C8—22no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C4⋊C8—42no—
211C4⋊C8—42no—
212C4⋊C8C2×Q814yes—
213C4⋊C8C2×Q812yes—
214C4⋊C8C2×Q812yes—
215C22⋊C8—24no—
216C22⋊C8—42no—
217C22⋊C8—42no—
218#97—82no—
219C4×C8—24no—
220C4×C8—22no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—22no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—82no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95C22⋊C411yescharacteristic
242#95C22⋊C411yescharacteristic
243#95C22⋊C411yescharacteristic
244#95C22⋊C411yescharacteristic
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C24C2×C22⋊C411yescharacteristic
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—82no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—82no—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C4×Q812yes—
273C22×C4C4×Q812yes—
274C22×C4C22⋊Q812yes—
275C22×C4C22⋊Q812yes—
276C22×C4#12412yes—
277C22×C4C4⋊Q812yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281M16—84no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4#13114yes—
318C22⋊C4—24no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—42no—
331C42—42no—
332C42—41no—
333C42—41no—
334C42—42no—
335C42—41no—
336C42—41no—
337C42—81no—
338C42—81no—
339C42C4×Q812yes—
340C42C4×Q812yes—
341C42C22⋊Q812yes—
342C42C22⋊Q812yes—
343C42#12412yes—
344C42C4⋊Q812yes—
345C42C22×Q811yescharacteristic
346C42C22×Q811yescharacteristic
347C42C2×#4111yescharacteristic
348C42C2×#4111yescharacteristic
349C23—22no—
350C23—21no—
351C23—21no—
352C23—22no—
353C23#39112yes—
354C23C22×C22⋊C411yescharacteristic
355Q8—88no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—88no—
394C2×C4#39112yes—
395C2×C4#39112yes—
396C2×C4#39112yes—
397C2×C4#42114yes—
398C2×C4C2×C4×Q811yescharacteristic
399C2×C4C2×C4×Q811yescharacteristic
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4C2×C22⋊Q811yescharacteristic
402C2×C4C2×#12411yescharacteristic
403C2×C4C2×C4⋊Q811yescharacteristic
404C2×C4C2×#13112yes—
405C2×C4#56812yes—
406C2×C4#57611yescharacteristic
407C2×C4#57711yescharacteristic
408C2×C4#57711yescharacteristic
409C2×C4#57811yescharacteristic
410C8—44no—
411C8—42no—
412C8—42no—
413C8—81no—
414C8—82no—
415C8—81no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—42no—
420C22—42no—
421C22C2×#39111yescharacteristic
422C22#288012yes—
423C22#289711yescharacteristic
424C22#289811yescharacteristic
425C22C2×#42112yes—
426C4—22no—
427C4—21no—
428C4—21no—
429C4—22no—
430C4—22no—
431C4—24no—
432C4—42no—
433C4—44no—
434C4—42no—
435C4—42no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4#296011yescharacteristic
440C4#296111yescharacteristic
441C4#296212yes—
442C4#308111yescharacteristic
443C4#308211yescharacteristic
444C4#308311yescharacteristic
445C4#308411yescharacteristic
446C2—42no—
447C2#1752112yes—
448C2#1752512yes—
449C2#2130411yescharacteristic
450C2#2130611yescharacteristic
451C2#2131511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9868C2semidirect
5. #9867C2semidirect
6. #9866C2semidirect
7. #9865C2semidirect
65. #431Q8semidirect
67. #423Q8semidirect

Maximal subgroups

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

subgroupindex
#349662
#208092
#205502
#98682
#98672
#98662
#98652
#79012
#78992
#78942
#78922