Finite Group Databasebeta

← #239198

Subgroups — #239198

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic
3#21394C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6C2×#1552C211yesmaximal, characteristic
7#10542C214yesmaximal, complemented
8#10538C214yesmaximal, complemented
9#7879C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#462—22no—
17C2×#451—22no—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1988—21no—
24#1988—21no—
25#1988C2212yes—
26#1988C2214yes—
27#1981C2212yes—
28#1940—22no—
29#1939C2214yes—
30#1872—22no—
31#1797—22no—
32#1763—21no—
33#1763—21no—
34C2×#340—22no—
35#1556—24no—
36#1552—24no—
37C2×#332C2212yes—
38C2×#332C2214yes—
39C2×#332C2212yes—
40#1277C2214yes—
41C23×Q8—21no—
42C22×Q16—24no—
43C23×C8—21no—
44#533—22no—
45#533—22no—
46#533—22no—
47#533D414yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8D414yes—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2312yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×#125—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#494—24no—
70#494—24no—
71#487—24no—
72#487—24no—
73#462—216no—
74#451—216no—
75#432—22no—
76#432—22no—
77#432—22no—
78#432—22no—
79#431—22no—
80#431—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8D412yes—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—24no—
90C2×#103—22no—
91C2×#103—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8D412yes—
95C2×#97—21no—
96C2×#97—21no—
97C2×C4×C8C2312yes—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4D411yescharacteristic
101C4×C4⋊C4D411yescharacteristic
102C4×C4⋊C4C2312yes—
103C2×#95—21no—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—24no—
108#336—24no—
109#332—24no—
110#332—24no—
111#332—28no—
112C22×Q8—24no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—41no—
117C22×Q8—42no—
118C22×Q8—82no—
119C22×Q8—82no—
120C23×C4—22no—
121C23×C4—41no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4#4111yescharacteristic
125C2×Q16—216no—
126C2×Q16—84no—
127C2×Q16—84no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—44no—
132C22×C8—42no—
133C22×C8C2×D414yes—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137C4⋊Q8—21no—
138C4⋊Q8—22no—
139C4⋊Q8—21no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8C2×D414yes—
150C4⋊Q8C2×D412yes—
151C4⋊Q8C2×D412yes—
152#125—24no—
153#125—42no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—42no—
161C4×Q8—42no—
162#117—24no—
163#117—42no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C22⋊C4—41no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—41no—
188C2×C42—82no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42#4112yes—
194C2×C42#4112yes—
195C2×C42#4111yescharacteristic
196C2×C42C2411yescharacteristic
197C4⋊C8—24no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—42no—
205C4⋊C8—84no—
206#103—216no—
207#103—44no—
208#103—44no—
209C22⋊C8—24no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212#97—42no—
213#97—42no—
214C4×C8—22no—
215C4×C8—22no—
216#95—21no—
217#95—21no—
218#95—22no—
219C24#12111yescharacteristic
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—42no—
226C2×Q8—44no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—42no—
232C2×Q8—44no—
233C2×Q8—88no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—88no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4#12012yes—
254C22×C4#12112yes—
255C22×C4#12112yes—
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12312yes—
260C22×C4#12312yes—
261C22×C4#12412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4111yescharacteristic
265Q16—816no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—84no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4#13714yes—
299C22⋊C4—44no—
300C42—44no—
301C42—44no—
302C42—81no—
303C42—81no—
304C42#12012yes—
305C42#12112yes—
306C42#12112yes—
307C42#12312yes—
308C42#12312yes—
309C42#12412yes—
310C42C22×D411yescharacteristic
311C42C22×D411yescharacteristic
312C42C2×#4111yescharacteristic
313C42C2×#4111yescharacteristic
314C23—21no—
315C23—21no—
316C23—22no—
317C23#39412yes—
318C23#47314yes—
319C23C2×#12111yescharacteristic
320Q8—44no—
321Q8—48no—
322Q8—42no—
323Q8—42no—
324Q8—88no—
325Q8—82no—
326Q8—84no—
327Q8—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—88no—
350C2×C4—88no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4#39412yes—
357C2×C4#39412yes—
358C2×C4#39412yes—
359C2×C4C2×#12011yescharacteristic
360C2×C4C2×#12111yescharacteristic
361C2×C4C2×#12111yescharacteristic
362C2×C4C2×#12311yescharacteristic
363C2×C4C2×#12311yescharacteristic
364C2×C4C2×#12411yescharacteristic
365C2×C4C2×#13712yes—
366C2×C4#57512yes—
367C2×C4#57612yes—
368C2×C4#57712yes—
369C8—44no—
370C8—82no—
371C8—82no—
372C8—82no—
373C22—24no—
374C22—24no—
375C22—22no—
376C22—22no—
377C22#204714yes—
378C22C2×#39411yescharacteristic
379C22#304111yescharacteristic
380C22#304311yescharacteristic
381C22#305612yes—
382C22C2×#47312yes—
383C4—22no—
384C4—21no—
385C4—21no—
386C4—22no—
387C4—22no—
388C4—44no—
389C4—82no—
390C4—84no—
391C4—82no—
392C4—82no—
393C4#301212yes—
394C4#303712yes—
395C4#303911yescharacteristic
396C4#304011yescharacteristic
397C4#313612yes—
398C2—24no—
399C2#1945912yes—
400C2#1947112yes—
401C2#2351111yescharacteristic
402C2C2×#204711yescharacteristic
403C2#2354711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
7. #10542C2semidirect
8. #10538C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#219532
#213942
C2×#19882
C2×#15562
C2×#15522
#105422
#105382
#78792