Finite Group Databasebeta

← #195674

Subgroups — #195674

Order 512.

1196 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3139C211yesmaximal, characteristic
2#34235C211yesmaximal, characteristic
3#21840C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5C2×#1552C211yesmaximal, characteristic
6#9065C214yesmaximal, complemented
7#9065C214yesmaximal, complemented
8#7904C211yesmaximal, characteristic
9#7895C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3139—24no—
13#3139C418yescomplemented
14C2×#491—22no—
15C2×#487C2212yes—
16C2×#487C2212yes—
17C2×#432C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×#419—22no—
20C2×C4×M16C2211yescharacteristic
21#2365C2212yes—
22#2365C2212yes—
23#2360—22no—
24#2356—21no—
25#2351—21no—
26#2343—22no—
27#2336—22no—
28#2320—21no—
29C4×C22⋊Q8—22no—
30#2283C2211yescharacteristic
31#2276—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1993—21no—
34#1993—21no—
35#1987—21no—
36#1987—21no—
37#1987C2214yes—
38#1987C2214yes—
39#1823—22no—
40C2×#340—22no—
41#1552—24no—
42C2×#332C2212yes—
43C2×#332C2212yes—
44#1279C2214yes—
45#1279C2214yes—
46C23×Q8—21no—
47C22×M16—21no—
48C23×C8—21no—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52#533D414yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4⋊Q8C2312yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125D412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×#117D412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#491—216no—
77#487—24no—
78#487—24no—
79#487C2×C418yescomplemented
80#487C2×C418yescomplemented
81#432—22no—
82#432—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88#419—216no—
89C2×#103—24no—
90C2×#103—24no—
91C2×C22⋊C8—21no—
92C4×M16—22no—
93C4×M16C2×C414yes—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8C2312yes—
97#391—22no—
98#391—22no—
99#389—24no—
100#389—22no—
101#389—22no—
102#387—24no—
103#387—24no—
104#387—24no—
105#387—24no—
106#385—24no—
107#383—22no—
108#383—21no—
109#382—22no—
110#382D411yescharacteristic
111#382D411yescharacteristic
112#381—21no—
113#380D412yes—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4C2312yes—
123C4×C4⋊C4C2312yes—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—21no—
126#376—22no—
127#376—21no—
128C2×#95—21no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#332—24no—
134#332—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—24no—
138C22×Q8—22no—
139C22×Q8—48no—
140C22×Q8—41no—
141C22×Q8—42no—
142C23×C4—22no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—44no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—44no—
153C22×C8—42no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—22no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8C22×C414yes—
168C4⋊Q8C22×C412yes—
169C4⋊Q8C22×C412yes—
170C4⋊Q8C2×D412yes—
171C4⋊Q8C2×D412yes—
172#125—24no—
173#125C22⋊C418yes—
174C22⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—44no—
183C4×Q8—44no—
184C4×Q8—84no—
185#117—24no—
186#117C22⋊C418yes—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—82no—
220C2×C42C22×C412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42C2411yescharacteristic
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—42no—
233#103—216no—
234#103—216no—
235C22⋊C8—42no—
236#97—24no—
237#97—42no—
238C4×C8—22no—
239C4×C8C22×C414yes—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—22no—
253C24#12011yescharacteristic
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—24no—
259C2×Q8—416no—
260C2×Q8—48no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—44no—
267C2×Q8—88no—
268C2×Q8—84no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—82no—
285C22×C4—84no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C4×D412yes—
288C22×C4C4×D412yes—
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12112yes—
293C22×C4#12112yes—
294C22×C4#12312yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298M16—48no—
299C2×C8—24no—
300C2×C8—44no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—44no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—84no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—84no—
329C4⋊C4C2×C22⋊C414yes—
330C4⋊C4C2×C22⋊C414yes—
331C4⋊C4#13714yes—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C42—28no—
335C42—48no—
336C42—42no—
337C42—42no—
338C42—44no—
339C42—81no—
340C42—81no—
341C42—84no—
342C42C2×C22⋊C412yes—
343C42C4×D412yes—
344C42C4×D412yes—
345C42#12112yes—
346C42#12112yes—
347C42#12312yes—
348C42C23×C411yescharacteristic
349C42C22×D411yescharacteristic
350C42C2×#4111yescharacteristic
351C42C2×#4111yescharacteristic
352C23—22no—
353C23—21no—
354C23—21no—
355C23#38612yes—
356C23#45414yes—
357C23C2×#12011yescharacteristic
358Q8—48no—
359Q8—44no—
360Q8—44no—
361Q8—42no—
362Q8—42no—
363Q8—44no—
364Q8—84no—
365Q8—88no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—44no—
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—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—88no—
397C2×C4—88no—
398C2×C4—84no—
399C2×C4#38612yes—
400C2×C4#38612yes—
401C2×C4#38612yes—
402C2×C4C22×C22⋊C411yescharacteristic
403C2×C4C2×C4×D411yescharacteristic
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×#12111yescharacteristic
406C2×C4C2×#12111yescharacteristic
407C2×C4C2×#12311yescharacteristic
408C2×C4C2×#13712yes—
409C2×C4#57512yes—
410C2×C4#57712yes—
411C2×C4#57812yes—
412C8—44no—
413C8—82no—
414C8—82no—
415C22—24no—
416C22—24no—
417C22—22no—
418C22—22no—
419C22#188814yes—
420C22C2×#38611yescharacteristic
421C22#300712yes—
422C22#300911yescharacteristic
423C22#301111yescharacteristic
424C22C2×#45412yes—
425C4—22no—
426C4—22no—
427C4—21no—
428C4—21no—
429C4—22no—
430C4—44no—
431C4—44no—
432C4—44no—
433C4—82no—
434C4—82no—
435C4—82no—
436C4—84no—
437C4C4×#13712yes—
438C4#294712yes—
439C4#303911yescharacteristic
440C4#304011yescharacteristic
441C4#304612yes—
442C2—24no—
443C2#1857912yes—
444C2#1858112yes—
445C2#2092711yescharacteristic
446C2#2093111yescharacteristic
447C2C2×#188811yescharacteristic
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. #9065C2semidirect
7. #9065C2semidirect
13. #3139C4semidirect
79. #487C2×C4semidirect
80. #487C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
#342352
#218402
C2×#19872
C2×#15522
#90652
#79042
#78952