Finite Group Databasebeta

← #110474

Subgroups — #110474

Order 512.

593 conjugacy classes of subgroups.

The subgroup lattice has 362 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×#1452C211yesmaximal, 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—22no—
20#1493—22no—
21C2×Q8⋊C8C414yes—
22#1472C414yes—
23#1470—22no—
24#1461—22no—
25#1452—22no—
26#1452—22no—
27C2×#322—22no—
28#1301—22no—
29#1283—22no—
30#1278—22no—
31C22×M16—21no—
32C23×C8—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64#431D412yes—
65#431D412yes—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423C2×C412yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8C2×C412yes—
75C2×C4⋊C8D412yes—
76#407—21no—
77#407C2×C412yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8C2×C412yes—
80C4×M16—21no—
81C4×M16—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8C2×C412yes—
88C2×C4×C8D412yes—
89C2×C4×C8D412yes—
90#333—24no—
91#332—24no—
92Q8⋊C8—28no—
93#324—28no—
94#322—28no—
95C22×Q8—21no—
96C22×Q8C4212yes—
97C23×C4C22×C411yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—82no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—44no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120#125—41no—
121#125—41no—
122#125—41no—
123#125—41no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140#117—41no—
141#117—41no—
142#117—41no—
143#117—41no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4C4212yes—
150C2×C22⋊C4—22no—
151C2×C42—22no—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8C22⋊C414yes—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8C22⋊C414yes—
184#97—41no—
185#97—41no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—41no—
194C4×C8—41no—
195C4×C8C22⋊C414yes—
196C24#11711yescharacteristic
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—44no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—44no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4C2×C4211yescharacteristic
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4#12411yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12611yescharacteristic
227M16—44no—
228M16—44no—
229M16—82no—
230M16—82no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—82no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—42no—
259C42—42no—
260C42C2×C22⋊C411yescharacteristic
261C42C2×C22⋊C411yescharacteristic
262C42#11711yescharacteristic
263C42#11711yescharacteristic
264C42C4×D411yescharacteristic
265C42C4×D411yescharacteristic
266C42C4×D411yescharacteristic
267C42C4×D411yescharacteristic
268C42#12011yescharacteristic
269C42#12111yescharacteristic
270C42#12111yescharacteristic
271C42#12111yescharacteristic
272C42#12111yescharacteristic
273C42#12311yescharacteristic
274C42#12411yescharacteristic
275C42#12711yescharacteristic
276C23—22no—
277C23—22no—
278C23—41no—
279C23C4×C22⋊C411yescharacteristic
280C23#38811yescharacteristic
281C23#38811yescharacteristic
282Q8—44no—
283Q8—82no—
284Q8—82no—
285Q8—82no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—82no—
308C2×C4C4×C22⋊C411yescharacteristic
309C2×C4C4×C22⋊C411yescharacteristic
310C2×C4#38611yescharacteristic
311C2×C4#38611yescharacteristic
312C2×C4#38611yescharacteristic
313C2×C4#38811yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#38811yescharacteristic
316C2×C4#38811yescharacteristic
317C2×C4#39011yescharacteristic
318C2×C4#39011yescharacteristic
319C2×C4#39011yescharacteristic
320C2×C4#44414yes—
321C2×C4#47112yes—
322C2×C4#47312yes—
323C8—44no—
324C8—44no—
325C8—44no—
326C8—82no—
327C8—82no—
328C8—82no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—44no—
337C22#143711yescharacteristic
338C22#176512yes—
339C22#193012yes—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—82no—
348C4—82no—
349C4#175712yes—
350C4#190812yes—
351C4#193812yes—
352C4#197512yes—
353C4#197711yescharacteristic
354C4#197911yescharacteristic
355C2—22no—
356C2—22no—
357C2—41no—
358C2#1093011yescharacteristic
359C2#1094211yescharacteristic
360C2#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×#14522
#80812
#79342
#78852