Finite Group Databasebeta

← #118517

Subgroups — #118517

Order 512.

550 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21371C211yesmaximal, characteristic
2#17648C211yesmaximal, characteristic
3C2×#1515C211yesmaximal, characteristic
4#17496C211yesmaximal, characteristic
5#8120C211yesmaximal, characteristic
6#7975C211yesmaximal, characteristic
7C2×#1288C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×#383C2211yescharacteristic
15#1950—22no—
16#1855—22no—
17#1778—22no—
18#1531—22no—
19#1528—22no—
20#1526—22no—
21#1515—22no—
22#1515C414yes—
23C2×#328—22no—
24#1497—22no—
25#1491—22no—
26#1481—22no—
27#1307—22no—
28#1288—22no—
29#1288—22no—
30C22×M16—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533C2×C412yes—
38#533D412yes—
39C2×C4⋊Q8Q811yescharacteristic
40C2×C4⋊Q8Q811yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×#117—21no—
47C22×C42C2311yescharacteristic
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#432D412yes—
59#432D412yes—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64C2×#108—24no—
65#423—21no—
66#423—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69#407—21no—
70#407—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8D412yes—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16D412yes—
77C4×M16D412yes—
78C4×M16D412yes—
79C2×#97—21no—
80C2×#97—21no—
81C2×#97—21no—
82C2×C4×C8—21no—
83#383—21no—
84#383—21no—
85#383—21no—
86#383C2×C412yes—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90#341—24no—
91#330—28no—
92#328—28no—
93Q8⋊C8—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C23×C4—41no—
97C23×C4C2×D411yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—44no—
104C2×M16—41no—
105C22×C8—22no—
106C22×C8—41no—
107C22×C8—41no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8C4⋊C412yes—
115C4⋊Q8C4⋊C412yes—
116#125—22no—
117#125—22no—
118#125—41no—
119#125—41no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136#117—22no—
137#117—22no—
138#117—41no—
139#117—41no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C42—41no—
148C2×C42C22×C411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×Q811yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160#108—48no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8C22⋊C414yes—
177#97—22no—
178#97—22no—
179#97—22no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—41no—
184#97—41no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—41no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—41no—
193#95—41no—
194#95—41no—
195C24#12411yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—81no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12411yescharacteristic
225C22×C4#12411yescharacteristic
226C22×C4#12611yescharacteristic
227C22×C4#12611yescharacteristic
228C22×C4#12711yescharacteristic
229M16—24no—
230M16—42no—
231M16—44no—
232M16—44no—
233M16—82no—
234C2×C8—24no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—82no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C42—42no—
257C42—42no—
258C42C2×C4⋊C411yescharacteristic
259C42#11711yescharacteristic
260C42C4×D411yescharacteristic
261C42C4×D411yescharacteristic
262C42#12011yescharacteristic
263C42#12011yescharacteristic
264C42#12111yescharacteristic
265C42#12111yescharacteristic
266C42#12111yescharacteristic
267C42#12111yescharacteristic
268C42#12111yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42C22⋊Q811yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42#12311yescharacteristic
273C42#12311yescharacteristic
274C23—22no—
275C23—22no—
276C23—41no—
277C23#39011yescharacteristic
278C23#39311yescharacteristic
279C23#39711yescharacteristic
280Q8—44no—
281Q8—82no—
282Q8—82no—
283Q8—82no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—164no—
310C2×C4#38011yescharacteristic
311C2×C4#38511yescharacteristic
312C2×C4#38511yescharacteristic
313C2×C4#38611yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#38811yescharacteristic
316C2×C4#39211yescharacteristic
317C2×C4#39411yescharacteristic
318C2×C4#39411yescharacteristic
319C2×C4#39611yescharacteristic
320C2×C4#39611yescharacteristic
321C2×C4#39911yescharacteristic
322C8—44no—
323C8—44no—
324C8—82no—
325C8—82no—
326C8—82no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—44no—
335C22#143911yescharacteristic
336C22#203112yes—
337C22#208212yes—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—82no—
351C4—82no—
352C4—162no—
353C2—22no—
354C2—22no—
355C2—41no—
356C2#1109111yescharacteristic
357C2#1111811yescharacteristic
358C2#1113411yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ 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
#213712
#176482
C2×#15152
#174962
#81202
#79752
C2×#12882