Finite Group Databasebeta

← #110228

Subgroups — #110228

Order 512.

581 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21302C211yesmaximal, characteristic
2#17739C211yesmaximal, characteristic
3#17664C211yesmaximal, characteristic
4#17334C211yesmaximal, characteristic
5#8002C211yesmaximal, characteristic
6#7988C211yesmaximal, characteristic
7#7920C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1985—22no—
16#1984—21no—
17#1984—21no—
18#1729C414yes—
19#1561—22no—
20#1556C412yes—
21#1556C412yes—
22#1544—22no—
23#1537—22no—
24#1534C414yes—
25#1526—22no—
26#1455—22no—
27#1453—22no—
28#1452—21no—
29#1450—21no—
30#1292—22no—
31#1290—22no—
32#1281—22no—
33C22×M16C2×C412yes—
34C23×C8C2×C412yes—
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×#125C2×C412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117C2×C412yes—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432C2×C412yes—
56#432C2×C412yes—
57#432C2×C412yes—
58#432C2×C412yes—
59#432D412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431D412yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423D412yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78#407—21no—
79#407—21no—
80#407—21no—
81#407—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16D412yes—
90C2×#97D412yes—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93#336—24no—
94#336—24no—
95#332—24no—
96#332—24no—
97#330—28no—
98#322—22no—
99#322—24no—
100C82—22no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4C22×C411yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—24no—
111C2×M16—42no—
112C2×M16—41no—
113C2×M16—41no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—44no—
118C22×C8C22⋊C412yes—
119C22×C8C22⋊C412yes—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128#125—22no—
129#125—22no—
130#125—22no—
131#125—22no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148#117—22no—
149#117—22no—
150#117—22no—
151#117—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C4212yes—
157C2×C4⋊C4C4212yes—
158C2×C22⋊C4—22no—
159C2×C42—22no—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
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⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8C22⋊C414yes—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192#97—22no—
193#97—22no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97C22⋊C414yes—
198C4×C8—21no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—41no—
205C4×C8—41no—
206C24#11711yescharacteristic
207C2×Q8—44no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—82no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—81no—
219C22×C4C2×C4211yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12611yescharacteristic
236C22×C4#12611yescharacteristic
237M16—44no—
238M16—44no—
239M16—82no—
240M16—82no—
241M16—82no—
242M16—82no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—44no—
254C2×C8—82no—
255C2×C8—82no—
256C2×C8#10114yes—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—42no—
272C42—42no—
273C42C2×C22⋊C411yescharacteristic
274C42C2×C22⋊C411yescharacteristic
275C42#11711yescharacteristic
276C42#11711yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42#12011yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12111yescharacteristic
286C42#12311yescharacteristic
287C42#12411yescharacteristic
288C42#12711yescharacteristic
289C23—22no—
290C23—41no—
291C23—41no—
292C23C4×C22⋊C411yescharacteristic
293C23#38811yescharacteristic
294C23#38811yescharacteristic
295Q8—82no—
296Q8—82no—
297Q8—82no—
298Q8—82no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4C4×C22⋊C411yescharacteristic
323C2×C4C4×C22⋊C411yescharacteristic
324C2×C4#38611yescharacteristic
325C2×C4#38611yescharacteristic
326C2×C4#38611yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#39011yescharacteristic
332C2×C4#39011yescharacteristic
333C2×C4#39011yescharacteristic
334C2×C4C2×#10112yes—
335C2×C4#41312yes—
336C8—24no—
337C8—44no—
338C8—44no—
339C8—42no—
340C8—82no—
341C8—82no—
342C8—82no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22#143711yescharacteristic
353C22C4×#10112yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—82no—
365C4#188112yes—
366C4#190112yes—
367C4#196611yescharacteristic
368C4#196711yescharacteristic
369C2—22no—
370C2—41no—
371C2—41no—
372C2#1095011yescharacteristic
373C2#1095611yescharacteristic
374C2#1096311yescharacteristic
subgroupG/H
centerC2×C4#386
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
#213022
#177392
#176642
#173342
#80022
#79882
#79202