Finite Group Databasebeta

← #110559

Subgroups — #110559

Order 512.

595 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20545C211yesmaximal, characteristic
2C2×#1550C211yesmaximal, characteristic
3#17590C211yesmaximal, characteristic
4#17385C211yesmaximal, characteristic
5#8082C211yesmaximal, characteristic
6#7960C211yesmaximal, characteristic
7#7934C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14#2283C2211yescharacteristic
15#1770—21no—
16C8×C4⋊C4—21no—
17#1768—22no—
18#1752—21no—
19C8×C22⋊C4—21no—
20#1550—22no—
21#1550—21no—
22#1550—21no—
23C2×#333C412yescomplemented
24C2×#332C412yescomplemented
25#1518—22no—
26#1511C414yes—
27#1506—22no—
28#1472—21no—
29#1472—21no—
30#1470—22no—
31#1461C414yescomplemented
32#1301—22no—
33#1286—22no—
34#1283—22no—
35C22×M16—21no—
36C23×C8—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C412yes—
52C2×#117—21no—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431C2×C412yes—
65#431D412yes—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8C2×C412yes—
78C2×C4⋊C8C2×C412yes—
79C2×C4⋊C8D412yes—
80C2×C4⋊C8D411yescharacteristic
81C2×C4⋊C8D411yescharacteristic
82#407—21no—
83#407C2×C412yes—
84C2×C22⋊C8D411yescharacteristic
85C2×C22⋊C8D411yescharacteristic
86C4×M16—21no—
87C4×M16—21no—
88C2×#97—22no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91#383—21no—
92#382D412yes—
93#381C2×C412yes—
94#380—21no—
95C4×C4⋊C4—21no—
96C4×C22⋊C4—21no—
97#376D412yes—
98#336—22no—
99#336—22no—
100#333—22no—
101#333—22no—
102#332—22no—
103#332—22no—
104#328—28no—
105#324—24no—
106#324—24no—
107C22×Q8—21no—
108C22×Q8C4212yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—82no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—82no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133#125—41no—
134#125—41no—
135#125—41no—
136#125—41no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153#117—41no—
154#117—41no—
155#117—41no—
156#117—41no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4C4212yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—22no—
167C2×C42—22no—
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
179C2×C42#4111yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193C4⋊C8—41no—
194C4⋊C8C22⋊C412yes—
195C4⋊C8C22⋊C412yes—
196C22⋊C8—21no—
197C22⋊C8—21no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8C22⋊C412yes—
203C22⋊C8C22⋊C412yes—
204#97—41no—
205#97—41no—
206#97—82no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—41no—
211C4×C8—82no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95C22⋊C414yes—
216C24C4×D411yescharacteristic
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—81no—
223C2×Q8—81no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4C2×C4211yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4#11711yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12411yescharacteristic
249C22×C4#12611yescharacteristic
250M16—44no—
251M16—81no—
252M16—81no—
253M16—82no—
254M16—82no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—82no—
269C2×C8—82no—
270C2×C8—82no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42C2×C22⋊C411yescharacteristic
293C42#11711yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12311yescharacteristic
304C42#12411yescharacteristic
305C42#12411yescharacteristic
306C42#12611yescharacteristic
307C42#12711yescharacteristic
308C23—22no—
309C23—22no—
310C23—41no—
311C23C4×C22⋊C411yescharacteristic
312C23#38611yescharacteristic
313C23#38811yescharacteristic
314Q8—82no—
315Q8—82no—
316Q8—82no—
317Q8—81no—
318Q8—81no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—82no—
348C2×C4C4×C22⋊C411yescharacteristic
349C2×C4C4×C22⋊C411yescharacteristic
350C2×C4#38611yescharacteristic
351C2×C4#38611yescharacteristic
352C2×C4#38811yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#39011yescharacteristic
358C2×C4#39011yescharacteristic
359C2×C4#39011yescharacteristic
360C2×C4#47112yes—
361C2×C4#47312yes—
362C8—42no—
363C8—42no—
364C8—42no—
365C8—42no—
366C8—82no—
367C8—82no—
368C8—82no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—44no—
377C22#143711yescharacteristic
378C22#175712yes—
379C22#190812yes—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—84no—
390C4—82no—
391C4—82no—
392C4#190411yescharacteristic
393C4#190511yescharacteristic
394C4#193012yes—
395C4#197711yescharacteristic
396C4#197911yescharacteristic
397C2—22no—
398C2—22no—
399C2—41no—
400C2#1094211yescharacteristic
401C2#1095411yescharacteristic
402C2#1096111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
23. C2×#333C4semidirect
24. C2×#332C4semidirect
31. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#205452
C2×#15502
#175902
#173852
#80822
#79602
#79342