Finite Group Databasebeta

← #196929

Subgroups — #196929

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35249C211yesmaximal, characteristic
2C2×#2276C211yesmaximal, characteristic, complemented
3#21820C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic
5C2×#1502C211yesmaximal, characteristic
6#9149C214yesmaximal, complemented
7#9148C214yesmaximal, complemented
8#8021C211yesmaximal, characteristic, complemented
9#8013C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×Q8—21no—
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2854C2214yes—
16#2632—22no—
17#2580C414yescomplemented
18#2567C412yescomplemented
19#2548C412yescomplemented
20#2501—21no—
21#2436—21no—
22#2406—22no—
23#2326—22no—
24#2283C2211yescharacteristic
25#2276—24no—
26#2276—24no—
27C2×C4×C4⋊C4—22no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×#376C2212yes—
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—216no—
58C2×C4×Q8—28no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8D412yes—
66C2×C4×Q8D412yes—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D412yes—
72C22×C4⋊C4D411yescharacteristic
73C22×C42—22no—
74C22×C42—21no—
75C22×C42—22no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#432—22no—
80#432—22no—
81#431—22no—
82#431—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87C2×#97—21no—
88C2×C4×C8—21no—
89#399—24no—
90#395—24no—
91#394—24no—
92#393—24no—
93#391—22no—
94#390—22no—
95#390C2×C418yes—
96#389C2×C418yes—
97#388—28no—
98#386—24no—
99#385—28no—
100#384—22no—
101#383—21no—
102#382C2×C412yes—
103#381—21no—
104#380C2312yes—
105C4×C4⋊C4—28no—
106C4×C4⋊C4—216no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4D414yes—
112C4×C22⋊C4—21no—
113#376—21no—
114#376—22no—
115#376—24no—
116#376—24no—
117#376D414yes—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95C2311yescharacteristic
121C2×#95C2312yes—
122C2×#95C2311yescharacteristic
123#336—28no—
124#327—28no—
125#327—24no—
126#327—24no—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41—82no—
130C2×#41—82no—
131C22×Q8—28no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8C22⋊C412yes—
136C22×Q8C22⋊C412yes—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154#127—22no—
155#127—22no—
156#124—48no—
157#124—42no—
158#124—42no—
159#123—84no—
160#123—84no—
161#121—22no—
162#121—22no—
163#121—22no—
164#121—22no—
165#121—42no—
166#121—42no—
167C4×Q8—264no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—42no—
179C4×D4—42no—
180C2×C4⋊C4—216no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C414yes—
192C2×C4⋊C4C22⋊C414yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C414yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—28no—
201C2×C42—28no—
202C2×C42—216no—
203C2×C42—28no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—28no—
211C2×C42—24no—
212C2×C42—82no—
213C2×C42—84no—
214C2×C42C22×C414yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D414yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4112yes—
223C2×C42#4112yes—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230#97—42no—
231C4×C8—42no—
232#95—28no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—28no—
237#95—24no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95C2×D412yes—
243#95C2×D414yes—
244#95C2×D412yes—
245C24—22no—
246C24C22×D411yescharacteristic
247#41—88no—
248#41—84no—
249#41—84no—
250C2×Q8—216no—
251C2×Q8—216no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—28no—
255C2×Q8—24no—
256C2×Q8—42no—
257C2×Q8—82no—
258C2×Q8#10418yes—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—82no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—84no—
297C22×C4C2×C22⋊C414yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C414yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4#12012yes—
304C22×C4#12012yes—
305C22×C4#12312yes—
306C22×C4#12312yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—42no—
316C2×C8—42no—
317C4⋊C4—232no—
318C4⋊C4—216no—
319C4⋊C4—28no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—88no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—416no—
334C22⋊C4—88no—
335C42—232no—
336C42—216no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—44no—
342C42—84no—
343C42—88no—
344C42C4×D412yes—
345C42C4×D412yes—
346C42C4×D412yes—
347C42C4×D412yes—
348C42#12112yes—
349C42#12112yes—
350C42#12112yes—
351C42#12112yes—
352C23—22no—
353C23—22no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—42no—
358C23C22×C22⋊C411yescharacteristic
359C23C2×#12011yescharacteristic
360C23C2×#12311yescharacteristic
361C23#57612yes—
362C23#57712yes—
363Q8—216no—
364Q8—216no—
365Q8—84no—
366D4—82no—
367D4—82no—
368D4—84no—
369D4—84no—
370D4—82no—
371D4—82no—
372D4—84no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—28no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—24no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—416no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—88no—
414C2×C4#38612yes—
415C2×C4#38612yes—
416C2×C4#38612yes—
417C2×C4#38612yes—
418C2×C4C2×#10418yes—
419C2×C4C2×#10414yes—
420C2×C4C2×C4×D411yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×#12111yescharacteristic
423C2×C4C2×#12111yescharacteristic
424C8—82no—
425C8—82no—
426C22—24no—
427C22—24no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—88no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22C2×#38611yescharacteristic
437C22#288412yes—
438C22#289712yes—
439C22C22×#10412yes—
440C22#300811yescharacteristic
441C22#301011yescharacteristic
442C22#309612yes—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—28no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—48no—
452C4—44no—
453C4—44no—
454C4—44no—
455C4—84no—
456C4—82no—
457C4#189614yes—
458C2—21no—
459C2—21no—
460C2—22no—
461C2—162no—
462C2#1755014yes—
463C2#2091411yescharacteristic
464C2C2×#189611yescharacteristic
465C2#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×#2276C2semidirect
3. #21820C2semidirect
6. #9149C2semidirect
7. #9148C2semidirect
8. #8021C2semidirect
9. #8013C2semidirect
17. #2580C4semidirect
18. #2567C4semidirect
19. #2548C4semidirect

Maximal subgroups

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

subgroupindex
#352492
C2×#22762
#218202
C2×#15052
C2×#15022
#91492
#91482
#80212
#80132