Finite Group Databasebeta

← #117869

Subgroups — #117869

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21866C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3C2×#1847C211yesmaximal, characteristic
4C2×#1531C211yesmaximal, characteristic
5C2×#1302C211yesmaximal, characteristic
6#8040C211yesmaximal, characteristic
7#7914C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×#107—21no—
13C22×#106—21no—
14C22×#97C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#1993—21no—
18#1990—21no—
19#1960—21no—
20#1956—21no—
21#1872—22no—
22#1872—21no—
23#1872—21no—
24#1847—22no—
25#1847—22no—
26#1805—22no—
27#1803—22no—
28#1531—22no—
29#1531C414yescomplemented
30C2×#330—22no—
31#1302—22no—
32#1296—22no—
33#1280—22no—
34C23×C8—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533Q812yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
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—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69C2×#107—24no—
70C2×#107—44no—
71C2×#106—24no—
72C2×#106—44no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76C2×#103—24no—
77C2×#103—22no—
78C2×#103—22no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8D412yes—
82C2×#97—21no—
83C2×#97—21no—
84C2×#97—21no—
85C2×#97—21no—
86C2×#97—21no—
87C2×#97C2×C412yes—
88C2×#97D412yes—
89C2×#97D412yes—
90C2×C4×C8—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384D412yes—
95#382—21no—
96#382—21no—
97#382—21no—
98#382—21no—
99C2×#95—21no—
100C2×#95—21no—
101#340—24no—
102#340—22no—
103#340—22no—
104#330—28no—
105C22×Q8—21no—
106C22×Q8—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—41no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C22×C8—24no—
114C22×C8—24no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—41no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C4⋊C412yes—
128C4⋊Q8C4⋊C412yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C22⋊C4—41no—
176C2×C42—41no—
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×Q811yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189#107—48no—
190#106—48no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197#103—84no—
198#103—82no—
199#103—82no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—22no—
207#97—22no—
208#97—22no—
209#97—22no—
210#97—41no—
211#97—41no—
212#97—41no—
213#97C22⋊C414yes—
214C4×C8—41no—
215C4×C8—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220C24#12111yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—42no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4#11711yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12411yescharacteristic
264C22×C4#12611yescharacteristic
265C22×C4#12611yescharacteristic
266C22×C4#12711yescharacteristic
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—28no—
270C2×C8—28no—
271C2×C8—24no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—81no—
280C2×C8—81no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C42—42no—
307C42—42no—
308C42C2×C22⋊C411yescharacteristic
309C42C2×C4⋊C411yescharacteristic
310C42C4×D411yescharacteristic
311C42C4×D411yescharacteristic
312C42#12111yescharacteristic
313C42#12111yescharacteristic
314C42#12111yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42C22⋊Q811yescharacteristic
318C42#12311yescharacteristic
319C42#12311yescharacteristic
320C42#12311yescharacteristic
321C42#12311yescharacteristic
322C42#12411yescharacteristic
323C42#12411yescharacteristic
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23#38811yescharacteristic
329C23#39211yescharacteristic
330C23#39611yescharacteristic
331C23#47112yes—
332C23#47312yes—
333Q8—44no—
334Q8—44no—
335Q8—82no—
336Q8—82no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—164no—
375C2×C4—162no—
376C2×C4—162no—
377C2×C4#38011yescharacteristic
378C2×C4#38511yescharacteristic
379C2×C4#38511yescharacteristic
380C2×C4#38611yescharacteristic
381C2×C4#38811yescharacteristic
382C2×C4#39011yescharacteristic
383C2×C4#39311yescharacteristic
384C2×C4#39411yescharacteristic
385C2×C4#39411yescharacteristic
386C2×C4#39611yescharacteristic
387C2×C4#39711yescharacteristic
388C2×C4#39911yescharacteristic
389C2×C4#45312yes—
390C2×C4#45612yes—
391C8—28no—
392C8—82no—
393C8—82no—
394C8—82no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22#143911yescharacteristic
404C22#193012yes—
405C22#200211yescharacteristic
406C22#200511yescharacteristic
407C22#200811yescharacteristic
408C22#200911yescharacteristic
409C22#201911yescharacteristic
410C22#202011yescharacteristic
411C22#206412yes—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—82no—
423C4—82no—
424C4—161no—
425C4—162no—
426C4—161no—
427C4#185912yes—
428C4#203912yes—
429C2—22no—
430C2—22no—
431C2#1012911yescharacteristic
432C2#1013311yescharacteristic
433C2#1013911yescharacteristic
434C2#1014311yescharacteristic
435C2#1107811yescharacteristic
436C2#1111711yescharacteristic
437C2#1114511yescharacteristic
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. #1531C4semidirect

Maximal subgroups

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

subgroupindex
#218662
#218602
C2×#18472
C2×#15312
C2×#13022
#80402
#79142