Finite Group Databasebeta

← #117844

Subgroups — #117844

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21838C211yesmaximal, characteristic
2#20764C211yesmaximal, characteristic
3C2×#1528C211yesmaximal, characteristic
4#14283C211yesmaximal, characteristic
5C2×#1300C211yesmaximal, characteristic
6#8040C211yesmaximal, characteristic
7#8022C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1992—21no—
16#1992—21no—
17#1989—22no—
18#1836—22no—
19#1834—21no—
20#1834—21no—
21#1824—21no—
22#1823—21no—
23#1736—22no—
24#1528—22no—
25#1528C414yescomplemented
26C2×#330—22no—
27#1441—22no—
28#1441—22no—
29C4⋊#95—21no—
30C4⋊#95—21no—
31#1300—22no—
32#1296—22no—
33#1294—22no—
34C22×M16—21no—
35C2×#125—21no—
36C2×#125—21no—
37C2×#125—21no—
38C2×#125—21no—
39C2×#125—21no—
40C2×#125—21no—
41C2×#125—21no—
42C2×#125—21no—
43C2×#125—21no—
44C2×#125—21no—
45C2×#125C2×C412yes—
46C2×#125Q812yes—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D412yes—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16C2×C412yes—
72C4×M16D412yes—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97D412yes—
76C2×C4×C8—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384D411yescharacteristic
82#384D411yescharacteristic
83#382—21no—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88#376—21no—
89#376—21no—
90#376—21no—
91#376—21no—
92C2×#95—21no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95D412yes—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#330—28no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—42no—
113C2×M16—42no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—41no—
117#125—22no—
118#125—22no—
119#125—22no—
120#125—22no—
121#125—22no—
122#125—22no—
123#125—22no—
124#125—22no—
125#125—41no—
126#125—41no—
127#125—41no—
128#125—41no—
129#125—41no—
130#125—41no—
131#125—41no—
132#125—41no—
133#117—44no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
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—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4C4⋊C412yes—
172C2×C4⋊C4C4⋊C412yes—
173C2×C22⋊C4—41no—
174C2×C42—24no—
175C2×C42—42no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×Q811yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—41no—
195C22⋊C8—22no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#97—22no—
200#97—22no—
201#97—41no—
202#97—41no—
203#97—41no—
204#97C22⋊C414yes—
205C4×C8—22no—
206C4×C8—22no—
207#95—22no—
208#95—22no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—84no—
214#95—84no—
215C24#12011yescharacteristic
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12411yescharacteristic
262C22×C4#12711yescharacteristic
263M16—44no—
264M16—44no—
265C2×C8—24no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—81no—
273C2×C8—81no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—82no—
303C42—44no—
304C42—44no—
305C42—82no—
306C42C2×C22⋊C411yescharacteristic
307C42#11711yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42#12111yescharacteristic
311C42#12111yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12411yescharacteristic
319C42#12411yescharacteristic
320C42#12611yescharacteristic
321C42#12611yescharacteristic
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23#38511yescharacteristic
327C23#39211yescharacteristic
328C23#39411yescharacteristic
329C23#45412yes—
330C23#45512yes—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—88no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4#38011yescharacteristic
381C2×C4#38511yescharacteristic
382C2×C4#38611yescharacteristic
383C2×C4#38811yescharacteristic
384C2×C4#38811yescharacteristic
385C2×C4#39011yescharacteristic
386C2×C4#39311yescharacteristic
387C2×C4#39411yescharacteristic
388C2×C4#39611yescharacteristic
389C2×C4#39611yescharacteristic
390C2×C4#39711yescharacteristic
391C2×C4#39911yescharacteristic
392C2×C4#45412yes—
393C2×C4#45612yes—
394C8—44no—
395C8—82no—
396C8—82no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—21no—
404C22—21no—
405C22#143911yescharacteristic
406C22#186012yes—
407C22#200311yescharacteristic
408C22#200311yescharacteristic
409C22#200411yescharacteristic
410C22#200511yescharacteristic
411C22#200811yescharacteristic
412C22#200911yescharacteristic
413C22#204012yes—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—84no—
425C4—82no—
426C4—82no—
427C4—161no—
428C4—161no—
429C4—162no—
430C4#185712yes—
431C4#203812yes—
432C2—22no—
433C2—22no—
434C2#1009311yescharacteristic
435C2#1009511yescharacteristic
436C2#1009711yescharacteristic
437C2#1009911yescharacteristic
438C2#1107911yescharacteristic
439C2#1111711yescharacteristic
440C2#1113611yescharacteristic
subgroupG/H
center, socleC23#392
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
25. #1528C4semidirect

Maximal subgroups

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

subgroupindex
#218382
#207642
C2×#15282
#142832
C2×#13002
#80402
#80222