Finite Group Databasebeta

← #121870

Subgroups — #121870

Order 512.

441 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#21360C211yesmaximal, characteristic
2#8068C211yesmaximal, characteristic
3#8007C211yesmaximal, characteristic
4#7973C211yesmaximal, characteristic
5#7932C211yesmaximal, characteristic
6#7870C211yesmaximal, characteristic
7#7857C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1949—21no—
16#1949—21no—
17#1854—21no—
18#1854—21no—
19C4×#107C411yescharacteristic
20C4×#107C411yescharacteristic
21C4×#106C411yescharacteristic
22C4×#106C411yescharacteristic
23#1300—22no—
24#1293—21no—
25#1293—21no—
26#1288—22no—
27#1283—21no—
28#1283—21no—
29#1276—22no—
30#1274—22no—
31C22×M16—21no—
32C2×#117—21no—
33C22×C42C2311yescharacteristic
34#432—21no—
35#432—21no—
36#432—21no—
37#432Q812yes—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431D412yes—
49#431Q812yes—
50C2×#107—21no—
51C2×#107—21no—
52C2×#106—21no—
53C2×#106—21no—
54#423—21no—
55#423—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8D411yescharacteristic
58C2×C4⋊C8D411yescharacteristic
59#407—21no—
60#407—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8D411yescharacteristic
63C2×C22⋊C8D411yescharacteristic
64C4×M16—21no—
65C4×M16—21no—
66C4×M16—21no—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16D412yes—
70C2×#97—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8C2×C411yescharacteristic
74C2×C4×C8C2×C411yescharacteristic
75#383—21no—
76#382—21no—
77#381—21no—
78#380—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4C2×C411yescharacteristic
81C4×C4⋊C4C2×C411yescharacteristic
82C4×C4⋊C4C2×C411yescharacteristic
83C4×C4⋊C4C2×C411yescharacteristic
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87#376—21no—
88#376—21no—
89C43—21no—
90#340—21no—
91#340—21no—
92#340—21no—
93#340—21no—
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C2×M16—24no—
98C2×M16—22no—
99C2×M16—42no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C22×C8C22⋊C411yescharacteristic
108C22×C8C22⋊C411yescharacteristic
109#117—42no—
110#117—42no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42C22×C411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×Q811yescharacteristic
128C2×C42C2×Q811yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134C2×C42#4111yescharacteristic
135#107—42no—
136#107—42no—
137#106—42no—
138#106—42no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—22no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C22⋊C8—21no—
150C22⋊C8—21no—
151C22⋊C8—21no—
152C22⋊C8—21no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159#97—22no—
160#97—22no—
161#97—41no—
162#97—41no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—41no—
171C4×C8—41no—
172C4×C8—41no—
173#95—21no—
174#95—21no—
175#95—21no—
176#95—21no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—41no—
181C24#12411yescharacteristic
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—81no—
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4C4×D411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4#12311yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12511yescharacteristic
201C22×C4#12511yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12511yescharacteristic
204C22×C4#12611yescharacteristic
205C22×C4#12611yescharacteristic
206C22×C4#12611yescharacteristic
207C22×C4#12611yescharacteristic
208C22×C4#12711yescharacteristic
209M16—24no—
210M16—42no—
211M16—44no—
212M16—81no—
213M16—81no—
214M16—81no—
215M16—81no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—44no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—82no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C22⋊C4—44no—
239C22⋊C4—82no—
240C42—44no—
241C42—42no—
242C42—42no—
243C42—42no—
244C42—42no—
245C42—82no—
246C42#11711yescharacteristic
247C42#11711yescharacteristic
248C42C4×Q811yescharacteristic
249C42C4×Q811yescharacteristic
250C42C22⋊Q811yescharacteristic
251C42C22⋊Q811yescharacteristic
252C42C22⋊Q811yescharacteristic
253C42C22⋊Q811yescharacteristic
254C42#12311yescharacteristic
255C42#12311yescharacteristic
256C42#12311yescharacteristic
257C42#12311yescharacteristic
258C42#12311yescharacteristic
259C42#12311yescharacteristic
260C42#12411yescharacteristic
261C42C4⋊Q811yescharacteristic
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23#39011yescharacteristic
267C23#40011yescharacteristic
268C23#40011yescharacteristic
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4#38111yescharacteristic
303C2×C4#38711yescharacteristic
304C2×C4#38711yescharacteristic
305C2×C4#38711yescharacteristic
306C2×C4#38711yescharacteristic
307C2×C4#39111yescharacteristic
308C2×C4#39711yescharacteristic
309C2×C4#39711yescharacteristic
310C2×C4#39811yescharacteristic
311C2×C4#39811yescharacteristic
312C2×C4#39911yescharacteristic
313C2×C4#39911yescharacteristic
314C2×C4#44312yes—
315C2×C4#44412yes—
316C8—44no—
317C8—41no—
318C8—41no—
319C8—41no—
320C8—41no—
321C8—82no—
322C8—82no—
323C8—82no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22#144511yescharacteristic
335C22#197411yescharacteristic
336C22#197511yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—82no—
347C4—82no—
348C4—161no—
349C4—161no—
350C4#182711yescharacteristic
351C4#182711yescharacteristic
352C4#192212yes—
353C4#192212yes—
354C2—21no—
355C2—21no—
356C2—41no—
357C2—41no—
358C2#1134811yescharacteristic
359C2#1136011yescharacteristic
360C2#1137611yescharacteristic
subgroupG/H
centerC2×C4#387
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1445
prime radicalC24#124
LCSG ▷ C2×C42 ▷ 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
#213602
#80682
#80072
#79732
#79322
#78702
#78572