Finite Group Databasebeta

← #118536

Subgroups — #118536

Order 512.

650 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21312C211yesmaximal, characteristic
2C2×#1571C211yesmaximal, characteristic
3C2×#1542C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#10850C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7908C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1989—22no—
16#1988—21no—
17#1987—21no—
18#1736—22no—
19C2×#340—22no—
20#1571—22no—
21#1571—22no—
22C2×#335—22no—
23#1542—22no—
24#1542—22no—
25C2×#333—21no—
26C2×#332—21no—
27#1518—22no—
28#1518C414yescomplemented
29C2×#328—22no—
30#1286—22no—
31#1280—22no—
32C22×M16—21no—
33C23×C8—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533Q812yes—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D412yes—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431C2×C412yes—
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⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D412yes—
75C2×C4⋊C8D412yes—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80#382—21no—
81#382—21no—
82#382—21no—
83#382C2×C412yes—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91#340—24no—
92#340—24no—
93#340—44no—
94#335—28no—
95#333—24no—
96#332—24no—
97#328—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4—22no—
101C23×C4—41no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—42no—
108C2×M16—42no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—42no—
116C22×C8—42no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8C4⋊C412yes—
124C4⋊Q8C4⋊C412yes—
125#125—22no—
126#125—22no—
127#125—41no—
128#125—41no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145#117—22no—
146#117—22no—
147#117—41no—
148#117—41no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4Q1614yes—
165C2×C22⋊C4—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—82no—
170C2×C42C22×C411yescharacteristic
171C2×C42C2×D411yescharacteristic
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
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—41no—
197C22⋊C8C22⋊C414yes—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202#95—22no—
203#95—22no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208C24#12111yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—81no—
214C2×Q8—81no—
215C2×Q8—82no—
216C22×C4—24no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4C22⋊Q811yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12411yescharacteristic
246C22×C4#12411yescharacteristic
247C22×C4#12611yescharacteristic
248C22×C4#12711yescharacteristic
249C22×C4C2×Q1612yes—
250C22×C4#13512yes—
251C22×C4AGL(1,ℤ/8)12yes—
252C22×C4#13712yes—
253M16—82no—
254M16—81no—
255M16—81no—
256C2×C8—24no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—82no—
264C2×C8—82no—
265C2×C8—81no—
266C2×C8—81no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C42—42no—
289C42—42no—
290C42—82no—
291C42—82no—
292C42C2×C4⋊C411yescharacteristic
293C42#11711yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42#12011yescharacteristic
297C42#12011yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42C22⋊Q811yescharacteristic
303C42#12311yescharacteristic
304C42#12311yescharacteristic
305C42#12311yescharacteristic
306C42#12411yescharacteristic
307C42#12611yescharacteristic
308C23—21no—
309C23—21no—
310C23—22no—
311C23#39011yescharacteristic
312C23#39411yescharacteristic
313C23#39611yescharacteristic
314C23#46712yes—
315C23#46812yes—
316Q8—82no—
317Q8—82no—
318Q8—82no—
319Q8—82no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—84no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—84no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—162no—
348C2×C4—162no—
349C2×C4#38011yescharacteristic
350C2×C4#38511yescharacteristic
351C2×C4#38511yescharacteristic
352C2×C4#38611yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#39211yescharacteristic
356C2×C4#39311yescharacteristic
357C2×C4#39411yescharacteristic
358C2×C4#39611yescharacteristic
359C2×C4#39711yescharacteristic
360C2×C4#39911yescharacteristic
361C2×C4C4×Q1612yes—
362C2×C4#44012yes—
363C2×C4#44912yes—
364C2×C4#45112yes—
365C2×C4#46212yes—
366C2×C4#46312yes—
367C2×C4#47712yes—
368C2×C4#47912yes—
369C2×C4#48412yes—
370C2×C4#48512yes—
371C2×C4#49411yescharacteristic
372C2×C4#49511yescharacteristic
373C2×C4#49611yescharacteristic
374C2×C4#49711yescharacteristic
375C8—44no—
376C8—82no—
377C8—82no—
378C8—82no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22#143911yescharacteristic
388C22#196911yescharacteristic
389C22#197311yescharacteristic
390C22#203612yes—
391C22#204811yescharacteristic
392C22#204911yescharacteristic
393C22#205312yes—
394C22#206812yes—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—82no—
404C4—82no—
405C4—84no—
406C4—162no—
407C4#186312yes—
408C4#192812yes—
409C4#201211yescharacteristic
410C4#201611yescharacteristic
411C4#209511yescharacteristic
412C4#209711yescharacteristic
413C2—21no—
414C2—22no—
415C2—21no—
416C2#1046912yes—
417C2#1047112yes—
418C2#1109211yescharacteristic
419C2#1112011yescharacteristic
420C2#1112411yescharacteristic
subgroupG/H
center, socleC23#394
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
28. #1518C4semidirect

Maximal subgroups

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

subgroupindex
#213122
C2×#15712
C2×#15422
C2×#15182
#108502
C2×#12862
#79082