Finite Group Databasebeta

← #110472

Subgroups — #110472

Order 512.

613 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17718C211yesmaximal, characteristic
2C2×#1493C211yesmaximal, characteristic
3#17385C211yesmaximal, characteristic
4C2×#1450C211yesmaximal, characteristic
5#8081C211yesmaximal, characteristic
6#7934C211yesmaximal, characteristic
7#7885C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#1554—22no—
16#1548C412yes—
17#1546C412yes—
18#1542—22no—
19#1493—21no—
20#1493—21no—
21#1493—22no—
22C2×Q8⋊C8C414yes—
23#1472C414yes—
24#1470—22no—
25#1461—22no—
26#1450—22no—
27#1450—22no—
28C2×#322—21no—
29C2×C82—21no—
30#1301—22no—
31#1283—22no—
32#1278—22no—
33C22×M16—21no—
34C23×C8—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#431D412yes—
66#431D412yes—
67#431D412yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72#423—21no—
73#423C2×C412yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2×C412yes—
77C2×C4⋊C8D412yes—
78#407—21no—
79#407C2×C412yes—
80C2×C22⋊C8—21no—
81C2×C22⋊C8C2×C412yes—
82C4×M16—21no—
83C4×M16—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2×C412yes—
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D412yes—
93#333—24no—
94#332—24no—
95Q8⋊C8—24no—
96Q8⋊C8—24no—
97#324—28no—
98#322—24no—
99C82—24no—
100C22×Q8—21no—
101C22×Q8C4212yes—
102C23×C4C22×C411yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—82no—
110C22×C8—28no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—41no—
127#125—41no—
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—41no—
147#117—41no—
148#117—41no—
149#117—41no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C4212yes—
156C2×C22⋊C4—22no—
157C2×C42—22no—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8C22⋊C414yes—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8C22⋊C414yes—
190#97—41no—
191#97—41no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—21no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—21no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—41no—
201C4×C8—41no—
202C4×C8C22⋊C412yes—
203C4×C8C22⋊C412yes—
204C24#11711yescharacteristic
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—44no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4C2×C4211yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12611yescharacteristic
234C22×C4#12611yescharacteristic
235M16—44no—
236M16—44no—
237M16—82no—
238M16—82no—
239C2×C8—28no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—28no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—44no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—82no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C42—42no—
269C42—42no—
270C42C2×C22⋊C411yescharacteristic
271C42C2×C22⋊C411yescharacteristic
272C42#11711yescharacteristic
273C42#11711yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42#12011yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42#12311yescharacteristic
284C42#12411yescharacteristic
285C42#12711yescharacteristic
286C23—22no—
287C23—22no—
288C23—41no—
289C23C4×C22⋊C411yescharacteristic
290C23#38811yescharacteristic
291C23#38811yescharacteristic
292Q8—44no—
293Q8—82no—
294Q8—82no—
295Q8—82no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
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—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—82no—
318C2×C4C4×C22⋊C411yescharacteristic
319C2×C4C4×C22⋊C411yescharacteristic
320C2×C4#38611yescharacteristic
321C2×C4#38611yescharacteristic
322C2×C4#38611yescharacteristic
323C2×C4#38811yescharacteristic
324C2×C4#38811yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#38811yescharacteristic
327C2×C4#39011yescharacteristic
328C2×C4#39011yescharacteristic
329C2×C4#39011yescharacteristic
330C2×C4#44314yes—
331C2×C4#47112yes—
332C2×C4#47312yes—
333C8—24no—
334C8—24no—
335C8—44no—
336C8—44no—
337C8—82no—
338C8—82no—
339C8—82no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22#143711yescharacteristic
349C22#176412yes—
350C22#193012yes—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—82no—
359C4—82no—
360C4#175712yes—
361C4#190812yes—
362C4#193712yes—
363C4#197412yes—
364C4#197711yescharacteristic
365C4#197911yescharacteristic
366C2—22no—
367C2—22no—
368C2—41no—
369C2#1092911yescharacteristic
370C2#1094211yescharacteristic
371C2#1096111yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#177182
C2×#14932
#173852
C2×#14502
#80812
#79342
#78852