Finite Group Databasebeta

← #137707

Subgroups — #137707

Order 512.

522 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2163C211yesmaximal, characteristic, complemented
2C2×#1923C211yesmaximal, characteristic
3#12844C211yesmaximal, characteristic, complemented
4C2×#1314C212yesmaximal
5#7183C212yesmaximal, complemented
6C22×C4⋊C8C411yescharacteristic
7C22×C22⋊C8C411yescharacteristic
8C22×C22⋊C8C2211yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C2×C4×C22⋊C4C411yescharacteristic
11#2163—24no—
12C2×C4⋊C16—21no—
13C2×C22⋊C16C2212yes—
14C2×C22⋊C16C2212yes—
15C2×#346—21no—
16#1923—21no—
17#1923—21no—
18#1923—21no—
19#1923—21no—
20C2×#336C411yescharacteristic
21C2×#336C2211yescharacteristic
22#1374—21no—
23#1374—21no—
24#1314—24no—
25C24×C4C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8C2311yescharacteristic
30C22×C22⋊C4C811yescharacteristic
31C22×C22⋊C4C811yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C16—22no—
35C22×C16—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—42no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—42no—
46C2×C4×C8—22no—
47C2×C4×C8—22no—
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55C4×C22⋊C4—21no—
56C2×#95C811yescharacteristic
57C2×#95C811yescharacteristic
58C4⋊C16—24no—
59C22⋊C16—22no—
60C22⋊C16—22no—
61C22⋊C16—22no—
62C22⋊C16D414yes—
63#346—24no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C25C2×C811yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4C2×C811yescharacteristic
78C23×C4C2×C811yescharacteristic
79C23×C4C2×C811yescharacteristic
80C23×C4C2×C811yescharacteristic
81C23×C4C2×C811yescharacteristic
82C23×C4C22×C411yescharacteristic
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D412yes—
101C22×C8#4112yes—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—42no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C22⋊C411yescharacteristic
116C2×C42C22⋊C411yescharacteristic
117C2×C42M1611yescharacteristic
118C2×C42M1611yescharacteristic
119C2×C16—24no—
120C2×C16—42no—
121C2×C16—42no—
122C2×C16—42no—
123C2×C16—42no—
124C2×C16—42no—
125C4⋊C8—42no—
126C4⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—41no—
130C22⋊C8—42no—
131C22⋊C8—41no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—21no—
137C4×C8—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142C24—21no—
143C24—21no—
144C24—22no—
145C24—22no—
146C24—41no—
147C24—41no—
148C24—41no—
149C24—41no—
150C24M3212yes—
151C24C22×C811yescharacteristic
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4C22⋊C811yescharacteristic
189C22×C4C22⋊C811yescharacteristic
190C22×C4C22⋊C811yescharacteristic
191C22×C4C22⋊C811yescharacteristic
192C22×C4M3212yes—
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C4×D412yes—
195C22×C4C2×M1611yescharacteristic
196C22×C4#13112yes—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—44no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8#10012yes—
218C2×C8#12011yescharacteristic
219C2×C8#12112yes—
220C2×C8#12311yescharacteristic
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—22no—
230C42—22no—
231C42—42no—
232C42—41no—
233C42—41no—
234C16—44no—
235C16—82no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—24no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23C2×C22⋊C811yescharacteristic
258C23C8×D412yes—
259C23C2×M3211yescharacteristic
260C23C2×M3211yescharacteristic
261C23#50412yes—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—84no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4#34912yes—
301C2×C4#38611yescharacteristic
302C2×C4#40811yescharacteristic
303C2×C4#41011yescharacteristic
304C2×C4#41011yescharacteristic
305C2×C4C2×#10011yescharacteristic
306C2×C4#41311yescharacteristic
307C2×C4#43512yes—
308C8—21no—
309C8—22no—
310C8—21no—
311C8—22no—
312C8—41no—
313C8—81no—
314C8—81no—
315C22—22no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—84no—
327C22—84no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22#145811yescharacteristic
333C22#146011yescharacteristic
334C22#186911yescharacteristic
335C22#211211yescharacteristic
336C22#211311yescharacteristic
337C22C2×#34911yescharacteristic
338C22#211511yescharacteristic
339C22#216812yes—
340C22#216912yes—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—42no—
347C4—81no—
348C4—81no—
349C4#146911yescharacteristic
350C4#147311yescharacteristic
351C4#189911yescharacteristic
352C4#190011yescharacteristic
353C2—22no—
354C2—22no—
355C2—82no—
356C2#908011yescharacteristic
357C2#1146011yescharacteristic
358C2#1146111yescharacteristic
359C2#1148811yescharacteristic
360C2#1148911yescharacteristic
361C2#1297711yescharacteristic
362C2#1298511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC25C2×C8
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2163C2semidirect
3. #12844C2semidirect
5. #7183C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21632
C2×#19232
#128442
C2×#13142
#71832