Finite Group Databasebeta

← #196938

Subgroups — #196938

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35249C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic, complemented
3#21820C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic
5C2×#1502C211yesmaximal, characteristic
6#9157C214yesmaximal, complemented
7#9156C214yesmaximal, complemented
8#8021C211yesmaximal, characteristic, complemented
9#8013C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2854C2214yes—
15#2632—22no—
16#2580C414yescomplemented
17#2567C412yescomplemented
18#2548C412yescomplemented
19#2501—21no—
20#2436—21no—
21#2406—22no—
22#2326—22no—
23#2320—24no—
24#2320—24no—
25C2×#391—21no—
26C2×#389—21no—
27C2×#387—22no—
28#2283C2211yescharacteristic
29C2×#383C2212yes—
30C2×#382C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1985—22no—
34#1984—22no—
35#1817C2214yes—
36#1505—24no—
37#1502—24no—
38C2×#327C2214yes—
39C2×#327C2212yes—
40C2×#327C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C22×#41—21no—
45C23×Q8D411yescharacteristic
46#532—22no—
47#532—22no—
48#532—22no—
49#532—22no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#124—22no—
53C2×#124—21no—
54C2×#123—24no—
55C2×#121—21no—
56C2×#121C2312yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8D412yes—
64C2×C4×Q8D412yes—
65C2×C4×D4—21no—
66C2×C4×D4C2×C412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#431—22no—
78#431—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C22⋊C8C2312yes—
82C2×C22⋊C8C2312yes—
83C2×#97—21no—
84C2×C4×C8—21no—
85#399—24no—
86#395—24no—
87#394—24no—
88#393—24no—
89#391—28no—
90#391—22no—
91#390—22no—
92#390C2×C418yes—
93#389—28no—
94#389C2×C418yes—
95#388—28no—
96#387—216no—
97#386—24no—
98#385—28no—
99#384—22no—
100#383—21no—
101#383—24no—
102#383—24no—
103#382—24no—
104#382—24no—
105#382C2×C412yes—
106#381—21no—
107#380C2312yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D414yes—
111C4×C22⋊C4—21no—
112#376—22no—
113#376—21no—
114#376D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95C2311yescharacteristic
120C2×#95C2311yescharacteristic
121C2×#95C2312yes—
122#336—28no—
123#327—28no—
124#327—24no—
125#327—24no—
126C2×#41—42no—
127C2×#41—42no—
128C2×#41—82no—
129C2×#41—82no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—44no—
134C22×Q8C22⋊C412yes—
135C22×Q8C22⋊C412yes—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C22×C412yes—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153#127—22no—
154#127—22no—
155#124—48no—
156#124—42no—
157#124—42no—
158#123—84no—
159#123—84no—
160#121—22no—
161#121—22no—
162#121—22no—
163#121—22no—
164#121—42no—
165#121—42no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—42no—
177C4×D4—42no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—84no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C414yes—
190C2×C4⋊C4C22⋊C414yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C414yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—28no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—44no—
207C2×C42—82no—
208C2×C42C22×C414yes—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D414yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4112yes—
217C2×C42#4112yes—
218C4⋊C8—42no—
219C4⋊C8—42no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224#97—42no—
225C4×C8—42no—
226#95—28no—
227#95—28no—
228#95—28no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—28no—
233#95—24no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95C2×D412yes—
239#95C2×D412yes—
240#95C2×D414yes—
241C24—22no—
242C24C22×D411yescharacteristic
243#41—88no—
244#41—84no—
245#41—84no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—44no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8—42no—
254C2×Q8—82no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—44no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—82no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—84no—
293C22×C4C2×C22⋊C414yes—
294C22×C4C2×C22⋊C414yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4#12012yes—
300C22×C4#12012yes—
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—44no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—88no—
323C4⋊C4—88no—
324C22⋊C4—416no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—88no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—48no—
336C42—44no—
337C42—84no—
338C42C4×D412yes—
339C42C4×D412yes—
340C42C4×D412yes—
341C42C4×D412yes—
342C42#12112yes—
343C42#12112yes—
344C42#12112yes—
345C42#12112yes—
346C23—22no—
347C23—22no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23—42no—
352C23C22×C22⋊C411yescharacteristic
353C23C2×#12011yescharacteristic
354C23C2×#12311yescharacteristic
355C23#57612yes—
356C23#57712yes—
357Q8—48no—
358Q8—48no—
359Q8—84no—
360D4—82no—
361D4—82no—
362D4—84no—
363D4—84no—
364D4—82no—
365D4—82no—
366D4—84no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—28no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—88no—
406C2×C4—84no—
407C2×C4—88no—
408C2×C4—84no—
409C2×C4#38612yes—
410C2×C4#38612yes—
411C2×C4#38612yes—
412C2×C4#38612yes—
413C2×C4#42114yes—
414C2×C4C2×C4×D411yescharacteristic
415C2×C4C2×C4×D411yescharacteristic
416C2×C4C2×#12111yescharacteristic
417C2×C4C2×#12111yescharacteristic
418C8—82no—
419C8—82no—
420C22—24no—
421C22—24no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—82no—
427C22—88no—
428C22—82no—
429C22—82no—
430C22C2×#38611yescharacteristic
431C22#288412yes—
432C22#289712yes—
433C22C2×#42112yes—
434C22#300811yescharacteristic
435C22#301011yescharacteristic
436C22#309612yes—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—44no—
445C4—44no—
446C4—82no—
447C4—84no—
448C4—82no—
449C4—82no—
450C4—84no—
451C4#189714yes—
452C2—21no—
453C2—21no—
454C2—22no—
455C2—162no—
456C2#1760214yes—
457C2#2091411yescharacteristic
458C2C2×#189711yescharacteristic
459C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2320C2semidirect
3. #21820C2semidirect
6. #9157C2semidirect
7. #9156C2semidirect
8. #8021C2semidirect
9. #8013C2semidirect
16. #2580C4semidirect
17. #2567C4semidirect
18. #2548C4semidirect

Maximal subgroups

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

subgroupindex
#352492
C2×#23202
#218202
C2×#15052
C2×#15022
#91572
#91562
#80212
#80132