Finite Group Databasebeta

← #194244

Subgroups — #194244

Order 512.

1124 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#34373C211yesmaximal, characteristic
3#21841C211yesmaximal, characteristic
4#21833C211yesmaximal, characteristic
5C2×#1497C211yesmaximal, characteristic
6#8987C214yesmaximal, complemented
7#8986C214yesmaximal, complemented
8#8046C211yesmaximal, characteristic
9#7894C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209C418yescomplemented
14C2×#494—21no—
15C2×#487—21no—
16C2×#462C2214yes—
17C2×#451—22no—
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2378—22no—
22#2362—22no—
23#2355—22no—
24#2355C2212yes—
25Q8×C4⋊C4C2212yes—
26#2336—22no—
27#2331—21no—
28#2327—21no—
29#2314—22no—
30#2285C2211yescharacteristic
31#2283C2211yescharacteristic
32#1993—22no—
33#1990C2214yes—
34#1988—22no—
35#1988C2212yes—
36#1988C2212yes—
37#1824—22no—
38#1823—22no—
39#1497—24no—
40C2×Q8⋊C8C2214yes—
41#1296C2214yes—
42#1279C2212yes—
43#1279C2212yes—
44C23×Q8—21no—
45C22×Q16—24no—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533D414yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8C2312yes—
60C2×#125—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8D412yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8C2312yes—
66C2×#117—21no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C42C2311yescharacteristic
71#494—28no—
72#487—28no—
73#462—28no—
74#462C2×C4116yes—
75#451—216no—
76#432—22no—
77#432—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431C2×C414yes—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×#103—24no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×#97—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2312yes—
92#391—24no—
93#391—22no—
94#389—22no—
95#389—24no—
96#388—24no—
97#388—24no—
98#387—24no—
99#387—24no—
100#387—24no—
101#387—24no—
102#387—24no—
103#384—22no—
104#384C2312yes—
105#383—21no—
106#382—22no—
107#382—21no—
108#382D412yes—
109#381—21no—
110#380—21no—
111#380D412yes—
112#380D412yes—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122#376—21no—
123#340—28no—
124#340—24no—
125#340—24no—
126Q8⋊C8—28no—
127C22×Q8—21no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—41no—
131C22×Q8—42no—
132C22×Q8—82no—
133C22×Q8—82no—
134C22×Q8C2×D412yes—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×Q16—216no—
140C2×Q16—84no—
141C2×Q16—84no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8C22⋊C414yes—
157C4⋊Q8C22⋊C414yes—
158C4⋊Q8C22×C414yes—
159#125—24no—
160#125—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8C22⋊C418yes—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—84no—
170C4×Q8C22×C414yes—
171C4×Q8C2×D414yes—
172#117—24no—
173#117—24no—
174#117—82no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D414yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42C22×C412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42C2411yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—42no—
217C4⋊C8C22×C414yes—
218#103—216no—
219C22⋊C8—24no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222#97—42no—
223C4×C8—24no—
224C4×C8—22no—
225C4×C8—22no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237C24#12011yescharacteristic
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—42no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—82no—
248C2×Q8—88no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×Q8—88no—
252C2×Q8C2×C22⋊C414yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C4×D412yes—
268C22×C4C4×D412yes—
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12112yes—
273C22×C4#12112yes—
274C22×C4#12312yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278Q16—816no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—44no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—44no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4C2×C22⋊C414yes—
307C22⋊C4—28no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C42—24no—
311C42—24no—
312C42—42no—
313C42—42no—
314C42—82no—
315C42—84no—
316C42—84no—
317C42C2×C22⋊C412yes—
318C42C4×D412yes—
319C42C4×D412yes—
320C42#12112yes—
321C42#12112yes—
322C42#12312yes—
323C42C23×C411yescharacteristic
324C42C22×D411yescharacteristic
325C42C2×#4111yescharacteristic
326C42C2×#4111yescharacteristic
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23#38612yes—
332C23C2×#12011yescharacteristic
333Q8—28no—
334Q8—44no—
335Q8—48no—
336Q8—84no—
337Q8—88no—
338Q8—84no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—88no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4#38612yes—
370C2×C4#38612yes—
371C2×C4#38612yes—
372C2×C4C22×C22⋊C411yescharacteristic
373C2×C4C2×C4×D411yescharacteristic
374C2×C4C2×C4×D411yescharacteristic
375C2×C4C2×#12111yescharacteristic
376C2×C4C2×#12111yescharacteristic
377C2×C4C2×#12311yescharacteristic
378C2×C4#57612yes—
379C2×C4#57712yes—
380C2×C4#57814yes—
381C8—44no—
382C8—82no—
383C8—82no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22C2×#38611yescharacteristic
390C22#300811yescharacteristic
391C22#301011yescharacteristic
392C22#301112yes—
393C22#302212yes—
394C4—22no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—44no—
400C4—44no—
401C4—82no—
402C4—84no—
403C4—82no—
404C4—84no—
405C4#289712yes—
406C4#295012yes—
407C4#304412yes—
408C4#309612yes—
409C2—42no—
410C2#1897012yes—
411C2#1897212yes—
412C2#2091411yescharacteristic
413C2#2093511yescharacteristic
414C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8987C2semidirect
7. #8986C2semidirect
13. #3209C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#343732
#218412
#218332
C2×#14972
#89872
#89862
#80462
#78942