Finite Group Databasebeta

← #117426

Subgroups — #117426

Order 512.

656 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21986C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic
5C2×#1302C211yesmaximal, characteristic
6#8034C211yesmaximal, characteristic
7#7914C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×#97C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1993—22no—
16#1993—22no—
17#1990—21no—
18#1990—21no—
19#1988—22no—
20#1949—22no—
21#1853—21no—
22#1852—21no—
23#1847—21no—
24#1836—21no—
25#1823—22no—
26C2×#340—21no—
27C2×#340—21no—
28#1558—22no—
29#1558C414yescomplemented
30C2×#333—22no—
31#1302—22no—
32#1295—22no—
33#1280—22no—
34C23×Q8—21no—
35C23×C8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8D412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432C2×C412yes—
61#432D412yes—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66C2×#107—22no—
67C2×#106—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8D412yes—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8D412yes—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97—21no—
81C2×#97D412yes—
82C2×C4×C8—21no—
83#384D411yescharacteristic
84#384D411yescharacteristic
85#382—21no—
86#382—21no—
87#382—21no—
88#382—21no—
89#382—21no—
90#380—21no—
91#380—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C2×#95—21no—
95C2×#95—21no—
96#340—24no—
97#340—24no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#333—28no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×Q8—41no—
109C22×Q8—82no—
110C22×Q8—82no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—41no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C22×C8—24no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—42no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8C4⋊C414yes—
141#117—82no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C22⋊C4—41no—
167C2×C42—41no—
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×Q811yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180#107—82no—
181#106—82no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187C4⋊C8C22⋊C414yes—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—41no—
200#97—41no—
201#97—41no—
202C4×C8—41no—
203C4×C8—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210C24#12111yescharacteristic
211C2×Q8—24no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—84no—
216C2×Q8—84no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8—84no—
220C2×Q8—84no—
221C2×Q8—82no—
222C2×Q8—82no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4#11711yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12611yescharacteristic
260C22×C4#12611yescharacteristic
261C22×C4#12711yescharacteristic
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—81no—
275C2×C8—81no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C22⋊C4—82no—
295C42—82no—
296C42C2×C22⋊C411yescharacteristic
297C42C2×C4⋊C411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42C22⋊Q811yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42#12311yescharacteristic
307C42#12311yescharacteristic
308C42#12311yescharacteristic
309C42#12311yescharacteristic
310C42#12411yescharacteristic
311C42#12411yescharacteristic
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23#38811yescharacteristic
317C23#39211yescharacteristic
318C23#39611yescharacteristic
319C23#47112yes—
320C23#47212yes—
321Q8—84no—
322Q8—84no—
323Q8—84no—
324Q8—84no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—162no—
365C2×C4—162no—
366C2×C4#38011yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4#38511yescharacteristic
369C2×C4#38611yescharacteristic
370C2×C4#38811yescharacteristic
371C2×C4#39011yescharacteristic
372C2×C4#39311yescharacteristic
373C2×C4#39411yescharacteristic
374C2×C4#39411yescharacteristic
375C2×C4#39611yescharacteristic
376C2×C4#39711yescharacteristic
377C2×C4#39911yescharacteristic
378C2×C4#45512yes—
379C2×C4#45612yes—
380C8—44no—
381C8—44no—
382C8—82no—
383C8—82no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22#143911yescharacteristic
393C22#193112yes—
394C22#200411yescharacteristic
395C22#200511yescharacteristic
396C22#200911yescharacteristic
397C22#200911yescharacteristic
398C22#201811yescharacteristic
399C22#201911yescharacteristic
400C22#206312yes—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—84no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4#188212yes—
417C4#202812yes—
418C2—22no—
419C2—22no—
420C2#1014011yescharacteristic
421C2#1014111yescharacteristic
422C2#1014211yescharacteristic
423C2#1014311yescharacteristic
424C2#1106411yescharacteristic
425C2#1113011yescharacteristic
426C2#1114411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
29. #1558C4semidirect

Maximal subgroups

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

subgroupindex
#219862
#218332
C2×#19932
C2×#15582
C2×#13022
#80342
#79142