Finite Group Databasebeta

← #110686

Subgroups — #110686

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1986C211yesmaximal, characteristic
2#20534C211yesmaximal, characteristic
3#17734C211yesmaximal, characteristic
4#17496C211yesmaximal, characteristic
5#8087C211yesmaximal, characteristic
6#7993C211yesmaximal, characteristic
7#7950C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1986—22no—
16#1986C414yes—
17#1771—22no—
18#1771—22no—
19#1749—21no—
20C4×C22⋊C8—21no—
21C2×#339—21no—
22C2×#339—21no—
23#1559—22no—
24#1558C412yescomplemented
25#1556C412yescomplemented
26#1542—22no—
27#1497C414yescomplemented
28#1491—22no—
29#1481—22no—
30#1302—22no—
31#1291—22no—
32#1285—22no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
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×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432C2×C412yes—
62#432C2×C412yes—
63#432C2×C412yes—
64#432D412yes—
65#432D412yes—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2×C412yes—
70#423—21no—
71#423—21no—
72C2×C4⋊C8D412yes—
73C2×C4⋊C8D412yes—
74#407—21no—
75#407—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16C2×C412yes—
84C2×#97—21no—
85C2×#97—21no—
86C2×#97—21no—
87C2×#97—22no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4D412yes—
94C2×#95—21no—
95#339—24no—
96#339—24no—
97#336—22no—
98#336—22no—
99#333—24no—
100#332—24no—
101Q8⋊C8—28no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C2×D411yescharacteristic
107C23×C4#4111yescharacteristic
108C23×C4#4111yescharacteristic
109C2×M16—24no—
110C2×M16—22no—
111C2×M16—42no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—41no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8C4212yes—
125C4⋊Q8C4212yes—
126#125—22no—
127#125—22no—
128#125—41no—
129#125—41no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—22no—
147#117—22no—
148#117—41no—
149#117—41no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—41no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8C22⋊C414yes—
186C4⋊C8C22⋊C414yes—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97—41no—
202#97—41no—
203#97—82no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—41no—
207C4×C8—41no—
208C4×C8—82no—
209#95—22no—
210#95—22no—
211C24#12311yescharacteristic
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—82no—
217C2×Q8—82no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4#11711yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12411yescharacteristic
247C22×C4#12411yescharacteristic
248C22×C4#12611yescharacteristic
249C22×C4#12611yescharacteristic
250M16—24no—
251M16—42no—
252M16—44no—
253M16—82no—
254M16—82no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—42no—
286C42—42no—
287C42C2×C4211yescharacteristic
288C42C2×C22⋊C411yescharacteristic
289C42C2×C22⋊C411yescharacteristic
290C42#11711yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42#12111yescharacteristic
298C42#12111yescharacteristic
299C42#12311yescharacteristic
300C42#12311yescharacteristic
301C42#12411yescharacteristic
302C42#12711yescharacteristic
303C23—21no—
304C23—21no—
305C23—22no—
306C23—41no—
307C23#38611yescharacteristic
308C23#38811yescharacteristic
309C23#38811yescharacteristic
310Q8—44no—
311Q8—82no—
312Q8—82no—
313Q8—82no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—84no—
341C2×C4—82no—
342C2×C4—84no—
343C2×C4—82no—
344C2×C4C4×C22⋊C411yescharacteristic
345C2×C4C4×C22⋊C411yescharacteristic
346C2×C4C4×C22⋊C411yescharacteristic
347C2×C4#38611yescharacteristic
348C2×C4#38611yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#38811yescharacteristic
352C2×C4#38811yescharacteristic
353C2×C4#39011yescharacteristic
354C2×C4#39011yescharacteristic
355C2×C4#39011yescharacteristic
356C2×C4#45512yes—
357C2×C4#45612yes—
358C8—44no—
359C8—44no—
360C8—44no—
361C8—82no—
362C8—82no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—42no—
371C22—42no—
372C22#143711yescharacteristic
373C22#188511yescharacteristic
374C22#188611yescharacteristic
375C22#193212yes—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—44no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4#188212yes—
391C4#188212yes—
392C2—21no—
393C2—21no—
394C2—22no—
395C2—41no—
396C2#1094611yescharacteristic
397C2#1095011yescharacteristic
398C2#1095711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
24. #1558C4semidirect
25. #1556C4semidirect
27. #1497C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19862
#205342
#177342
#174962
#80872
#79932
#79502