Finite Group Databasebeta

← #115871

Subgroups — #115871

Order 512.

710 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21312C211yesmaximal, characteristic
2C2×#1518C211yesmaximal, characteristic
3C2×#1503C211yesmaximal, characteristic
4#13955C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#382C2211yescharacteristic
13C2×C4×C4⋊C4C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1989—22no—
16#1988C412yescomplemented
17#1987C412yescomplemented
18#1736—22no—
19#1518—22no—
20#1518—22no—
21#1503—22no—
22#1503—22no—
23C2×#328—22no—
24C2×#328—22no—
25#1442—22no—
26C4⋊#95—22no—
27#1434—22no—
28#1303—22no—
29#1286—22no—
30#1279—22no—
31C22×M16—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533D412yes—
39#533Q812yes—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431D412yes—
61#431D412yes—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8C2×C412yes—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74#382—21no—
75#382—21no—
76#382—21no—
77#382—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4C2×C412yes—
85C4×C4⋊C4D412yes—
86C2×#95—21no—
87C2×#95—24no—
88C2×#95—21no—
89C2×#95—24no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#328—28no—
95#328—28no—
96C22×Q8—21no—
97C22×Q8—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—42no—
107C2×M16—42no—
108C22×C8—22no—
109C22×C8—41no—
110C22×C8—41no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8C4⋊C412yes—
118C4⋊Q8C4⋊C412yes—
119#125—22no—
120#125—22no—
121#125—41no—
122#125—41no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139#117—22no—
140#117—22no—
141#117—41no—
142#117—41no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4C22⋊C414yes—
162C2×C22⋊C4—41no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42—82no—
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×Q811yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—41no—
198C4×C8—41no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—48no—
206#95—41no—
207#95—41no—
208#95—84no—
209C24#12011yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—81no—
215C2×Q8—81no—
216C2×Q8—82no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12411yescharacteristic
254C22×C4#12411yescharacteristic
255C22×C4#12411yescharacteristic
256C22×C4#12711yescharacteristic
257C22×C4#13512yes—
258C22×C4#13512yes—
259C22×C4#13712yes—
260C22×C4#13712yes—
261M16—82no—
262M16—81no—
263M16—81no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—44no—
268C2×C8—81no—
269C2×C8—81no—
270C2×C8—82no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—82no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C42—42no—
295C42—42no—
296C42—82no—
297C42—82no—
298C42—82no—
299C42—81no—
300C42—81no—
301C42C2×C4⋊C411yescharacteristic
302C42#11711yescharacteristic
303C42C4×D411yescharacteristic
304C42C4×D411yescharacteristic
305C42#12111yescharacteristic
306C42#12111yescharacteristic
307C42#12111yescharacteristic
308C42C22⋊Q811yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42#12311yescharacteristic
312C42#12311yescharacteristic
313C42#12311yescharacteristic
314C42#12311yescharacteristic
315C42#12611yescharacteristic
316C42#12611yescharacteristic
317C23—21no—
318C23—21no—
319C23—22no—
320C23#38611yescharacteristic
321C23#39211yescharacteristic
322C23#39311yescharacteristic
323C23#45212yes—
324C23#45212yes—
325Q8—82no—
326Q8—82no—
327Q8—82no—
328Q8—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4#38011yescharacteristic
366C2×C4#38511yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#38811yescharacteristic
370C2×C4#39011yescharacteristic
371C2×C4#39411yescharacteristic
372C2×C4#39411yescharacteristic
373C2×C4#39611yescharacteristic
374C2×C4#39611yescharacteristic
375C2×C4#39711yescharacteristic
376C2×C4#39911yescharacteristic
377C2×C4#41612yes—
378C2×C4#41912yes—
379C2×C4#46412yes—
380C2×C4#46412yes—
381C2×C4#47912yes—
382C2×C4#47912yes—
383C2×C4#48512yes—
384C2×C4#48512yes—
385C2×C4#48812yes—
386C2×C4#49112yes—
387C2×C4#49511yescharacteristic
388C2×C4#49511yescharacteristic
389C2×C4#49711yescharacteristic
390C2×C4#49711yescharacteristic
391C8—82no—
392C8—82no—
393C8—82no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22#143911yescharacteristic
403C22#187312yes—
404C22#190411yescharacteristic
405C22#190511yescharacteristic
406C22#190612yes—
407C22#201511yescharacteristic
408C22#201511yescharacteristic
409C22#202612yes—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—84no—
419C4—82no—
420C4—82no—
421C4—84no—
422C4—162no—
423C4#180512yes—
424C4#195811yescharacteristic
425C4#196011yescharacteristic
426C4#207412yes—
427C4#209711yescharacteristic
428C4#209711yescharacteristic
429C2—21no—
430C2—21no—
431C2—22no—
432C2#917712yes—
433C2#917812yes—
434C2#1105211yescharacteristic
435C2#1105711yescharacteristic
436C2#1106011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1988C4semidirect
17. #1987C4semidirect

Maximal subgroups

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

subgroupindex
#213122
C2×#15182
C2×#15032
#139552
C2×#13032
C2×#12862
#78922