Finite Group Databasebeta

← #110709

Subgroups — #110709

Order 512.

684 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21324C211yesmaximal, characteristic
2#20538C211yesmaximal, characteristic
3C2×#1548C211yesmaximal, characteristic
4C2×#1546C211yesmaximal, characteristic
5C2×#1299C211yesmaximal, characteristic
6C2×#1299C211yesmaximal, characteristic
7#8041C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#1993C414yes—
16#1990—22no—
17#1770—21no—
18#1770—21no—
19C8×C4⋊C4—21no—
20C8×C4⋊C4—21no—
21#1750—22no—
22#1736—22no—
23#1548—22no—
24#1548C414yescomplemented
25#1546—22no—
26#1546C414yescomplemented
27C2×#333—21no—
28C2×#333—21no—
29C2×#332—21no—
30C2×#332—21no—
31#1299—22no—
32#1299—22no—
33#1296—22no—
34C23×Q8—21no—
35C22×M16—21no—
36C22×M16—21no—
37C22×M16—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⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C412yes—
50C2×#117D412yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C42C2311yescharacteristic
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431C2×C412yes—
59#431C2×C412yes—
60#431D412yes—
61#431D412yes—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69#423C2×C412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8D411yescharacteristic
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C2×#97—22no—
85C2×#97C2×C412yes—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90#384D412yes—
91#382C2×C412yes—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96#340—28no—
97#336—22no—
98#336—22no—
99#333—24no—
100#333—24no—
101#332—24no—
102#332—24no—
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—41no—
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—82no—
126C2×M16—82no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—41no—
133C22×C8—41no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149C4⋊Q8C4214yes—
150#117—42no—
151#117—42no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4C22⋊C414yes—
167C2×C22⋊C4—22no—
168C2×C42—22no—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8C22⋊C412yes—
194C4⋊C8C22⋊C412yes—
195C4⋊C8C22⋊C412yes—
196C4⋊C8C22⋊C412yes—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201#97—22no—
202#97—22no—
203#97—82no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—41no—
209C4×C8—41no—
210C4×C8—82no—
211#95—22no—
212#95—22no—
213C24#12311yescharacteristic
214C2×Q8—24no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—84no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—84no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—84no—
225C2×Q8—82no—
226C2×Q8—82no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—84no—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12411yescharacteristic
255C22×C4#12411yescharacteristic
256C22×C4#12611yescharacteristic
257C22×C4#12611yescharacteristic
258M16—42no—
259M16—42no—
260M16—42no—
261M16—42no—
262M16—82no—
263M16—82no—
264M16—82no—
265M16—82no—
266C2×C8—24no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—82no—
284C2×C8—82no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C22⋊C4—82no—
299C42—42no—
300C42—42no—
301C42C2×C4211yescharacteristic
302C42C2×C22⋊C411yescharacteristic
303C42C2×C22⋊C411yescharacteristic
304C42#11711yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42#12111yescharacteristic
312C42#12111yescharacteristic
313C42#12311yescharacteristic
314C42#12311yescharacteristic
315C42#12411yescharacteristic
316C42#12711yescharacteristic
317C23—21no—
318C23—21no—
319C23—22no—
320C23—22no—
321C23#38611yescharacteristic
322C23#38811yescharacteristic
323C23#38811yescharacteristic
324Q8—84no—
325Q8—84no—
326Q8—84no—
327Q8—82no—
328Q8—82no—
329C2×C4—21no—
330C2×C4—21no—
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—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4C4×C22⋊C411yescharacteristic
361C2×C4C4×C22⋊C411yescharacteristic
362C2×C4C4×C22⋊C411yescharacteristic
363C2×C4#38611yescharacteristic
364C2×C4#38611yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#38811yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#39011yescharacteristic
370C2×C4#39011yescharacteristic
371C2×C4#39011yescharacteristic
372C2×C4#45511yescharacteristic
373C2×C4#45511yescharacteristic
374C2×C4#45611yescharacteristic
375C2×C4#45611yescharacteristic
376C2×C4#47112yes—
377C2×C4#47112yes—
378C2×C4#47212yes—
379C2×C4#47312yes—
380C8—44no—
381C8—42no—
382C8—42no—
383C8—42no—
384C8—42no—
385C8—82no—
386C8—82no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—42no—
394C22—42no—
395C22#143711yescharacteristic
396C22#188511yescharacteristic
397C22#188611yescharacteristic
398C22#190712yes—
399C22#190812yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—84no—
407C4—84no—
408C4#188211yescharacteristic
409C4#188211yescharacteristic
410C4#188211yescharacteristic
411C4#188211yescharacteristic
412C4#193012yes—
413C4#193112yes—
414C4#197711yescharacteristic
415C4#197711yescharacteristic
416C4#197811yescharacteristic
417C4#197911yescharacteristic
418C2—41no—
419C2—41no—
420C2#943312yes—
421C2#943412yes—
422C2#1094211yescharacteristic
423C2#1094311yescharacteristic
424C2#1095711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ 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
24. #1548C4semidirect
26. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#213242
#205382
C2×#15482
C2×#15462
C2×#12992
#80412