Finite Group Databasebeta

← #121451

Subgroups — #121451

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21365C211yesmaximal, characteristic
2#20765C211yesmaximal, characteristic
3#14298C211yesmaximal, characteristic
4#8030C211yesmaximal, characteristic
5#8018C211yesmaximal, characteristic
6C2×#1288C211yesmaximal, characteristic
7#7904C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×C4×M16C2211yescharacteristic
11#2285C2211yescharacteristic
12C2×#383C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1948—21no—
16#1947—21no—
17#1853—21no—
18#1852—21no—
19#1836—21no—
20#1836—21no—
21#1835—22no—
22#1776—22no—
23#1736—21no—
24#1736—21no—
25#1444—21no—
26#1444—21no—
27#1442—21no—
28#1442—21no—
29#1436—22no—
30#1295—22no—
31#1294—22no—
32#1288C414yescomplemented
33#1279—22no—
34C22×M16—21no—
35C2×#117—21no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C42—21no—
42C22×C42C2311yescharacteristic
43#432—21no—
44#432—21no—
45#432—21no—
46#432—21no—
47#432—21no—
48#432C2×C412yes—
49#432Q812yes—
50#432Q812yes—
51C2×#107—22no—
52C2×#106—22no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C4×M16—21no—
58C4×M16—21no—
59C4×M16C2×C412yes—
60C4×M16Q812yes—
61C2×#97—21no—
62C2×#97—21no—
63C2×#97D412yes—
64C2×C4×C8D412yes—
65#384D411yescharacteristic
66#384D411yescharacteristic
67#383—21no—
68#383—21no—
69#383C2×C412yes—
70#383Q812yes—
71#382—21no—
72#382—21no—
73#382—21no—
74#382—21no—
75#382—21no—
76#380—21no—
77#380—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D412yes—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—42no—
110C2×M16—42no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114#117—44no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4—82no—
136C2×C22⋊C4—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—82no—
141C2×C42C22×C411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×Q811yescharacteristic
146C2×C42C2×Q811yescharacteristic
147C2×C42C2×Q811yescharacteristic
148C2×C42C2×Q811yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4111yescharacteristic
153#107—44no—
154#106—44no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—41no—
163#97—22no—
164#97—22no—
165#97—22no—
166#97—22no—
167#97—41no—
168#97C4⋊C414yescomplemented
169C4×C8—22no—
170C4×C8C4⋊C414yescomplemented
171#95—22no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—44no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—82no—
186#95—82no—
187#95C4⋊C414yes—
188C24#12011yescharacteristic
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C2×C4⋊C411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12711yescharacteristic
238C22×C4C4⋊Q811yescharacteristic
239C22×C4C4⋊Q811yescharacteristic
240C22×C4C4⋊Q811yescharacteristic
241M16—44no—
242M16—44no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—81no—
251C2×C8—81no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C4⋊C4—88no—
271C22⋊C4—82no—
272C42—82no—
273C42—82no—
274C42—82no—
275C42C2×C4⋊C411yescharacteristic
276C42C2×C4⋊C411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×Q811yescharacteristic
279C42C22⋊Q811yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12511yescharacteristic
286C42#12511yescharacteristic
287C42#12511yescharacteristic
288C42C4⋊Q811yescharacteristic
289C42C4⋊Q811yescharacteristic
290C42C4⋊Q811yescharacteristic
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#38511yescharacteristic
296C23#39211yescharacteristic
297C23#39511yescharacteristic
298C23#45412yes—
299C23#45512yes—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—84no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4#38411yescharacteristic
357C2×C4#38511yescharacteristic
358C2×C4#38511yescharacteristic
359C2×C4#38911yescharacteristic
360C2×C4#38911yescharacteristic
361C2×C4#38911yescharacteristic
362C2×C4#39511yescharacteristic
363C2×C4#39511yescharacteristic
364C2×C4#39811yescharacteristic
365C2×C4#39811yescharacteristic
366C2×C4#39811yescharacteristic
367C2×C4#39911yescharacteristic
368C8—44no—
369C8—44no—
370C8—82no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22#144411yescharacteristic
380C22#186012yes—
381C22#200311yescharacteristic
382C22#200411yescharacteristic
383C22#200811yescharacteristic
384C22#200911yescharacteristic
385C22#201011yescharacteristic
386C22#201011yescharacteristic
387C22#206112yes—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—84no—
401C4—82no—
402C4—82no—
403C4—161no—
404C4—161no—
405C4—161no—
406C4—161no—
407C2—22no—
408C2—22no—
409C2#1001911yescharacteristic
410C2#1002111yescharacteristic
411C2#1002211yescharacteristic
412C2#1002411yescharacteristic
413C2#1133311yescharacteristic
414C2#1133511yescharacteristic
415C2#1133911yescharacteristic
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
32. #1288C4semidirect
168. #97C4⋊C4semidirect
170. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#213652
#207652
#142982
#80302
#80182
C2×#12882
#79042