Finite Group Databasebeta

← #140629

Subgroups — #140629

Order 512.

536 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#13909C211yesmaximal, characteristic
2C2×#1325C212yesmaximal
3C2×#1324C212yesmaximal
4#8558C211yesmaximal, characteristic
5C4×#324C211yesmaximal, characteristic
6C2×#384C2211yescharacteristic
7C2×#382C411yescharacteristic
8C2×#382C2211yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797—21no—
12C4×C22⋊C8—21no—
13C2×#324C2212yes—
14C2×#324C2212yes—
15#1441—21no—
16#1441—21no—
17#1436—21no—
18#1436—21no—
19C4⋊#95C411yescharacteristic
20C4⋊#95C411yescharacteristic
21#1325—24no—
22#1324—24no—
23#1280—21no—
24#1280—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2311yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C2×C4⋊C8—41no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C4×C8—41no—
41#384—22no—
42#384—22no—
43#382—22no—
44#382—22no—
45#382—41no—
46#382—41no—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—41no—
49C4×C4⋊C4—41no—
50C4×C4⋊C4C814yescomplemented
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95D411yescharacteristic
61C2×#95D411yescharacteristic
62#336—41no—
63#336—41no—
64#324—22no—
65#324—22no—
66#324—22no—
67#324D414yes—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C22×C8—42no—
80C22×C8—42no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4C2×D412yes—
103C2×C4⋊C4#4112yes—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—81no—
118C2×C42—81no—
119C2×C42C2×C812yes—
120C2×C42C2×C812yes—
121C2×C42C2×C812yes—
122C2×C42M1612yes—
123C4⋊C8—82no—
124C22⋊C8—24no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C4×C8—82no—
129#95—24no—
130#95—41no—
131#95—41no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—41no—
136#95—41no—
137#95—42no—
138#95—41no—
139#95—41no—
140#95—81no—
141#95—81no—
142C24C2×C22⋊C411yescharacteristic
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
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—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4C22⋊C812yes—
199C22×C4C22⋊C812yes—
200C22×C4#9911yescharacteristic
201C22×C4#9911yescharacteristic
202C22×C4#10112yes—
203C22×C4#10112yes—
204C22×C4C4×D412yes—
205C22×C4#12112yes—
206C22×C4#12411yescharacteristic
207C22×C4#12711yescharacteristic
208C22×C4C22×C811yescharacteristic
209C22×C4C2×M1611yescharacteristic
210C22×C4#13112yes—
211C2×C8—44no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—48no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—84no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—22no—
230C42—24no—
231C42—24no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42—42no—
236C42—81no—
237C42—81no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23#39011yescharacteristic
243C23C2×C22⋊C811yescharacteristic
244C23#40811yescharacteristic
245C23C2×#9911yescharacteristic
246C23#41011yescharacteristic
247C23C2×#10111yescharacteristic
248C23C2×#10111yescharacteristic
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4#32412yes—
319C2×C4#32412yes—
320C2×C4C8×D412yes—
321C2×C4#43612yes—
322C8—84no—
323C8—162no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22C2×#32411yescharacteristic
337C22#146111yescharacteristic
338C22#147011yescharacteristic
339C22#147211yescharacteristic
340C22#195411yescharacteristic
341C22#196411yescharacteristic
342C22#196611yescharacteristic
343C22#212611yescharacteristic
344C22#212711yescharacteristic
345C22#213211yescharacteristic
346C22#213311yescharacteristic
347C4—22no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—44no—
352C4—42no—
353C4—42no—
354C4—41no—
355C4—41no—
356C4—42no—
357C4—82no—
358C4—81no—
359C4—161no—
360C4—161no—
361C2—21no—
362C2—21no—
363C2—21no—
364C2—21no—
365C2#965011yescharacteristic
366C2#1144711yescharacteristic
367C2#1144811yescharacteristic
368C2#1157111yescharacteristic
369C2#1157211yescharacteristic
370C2#1322511yescharacteristic
371C2#1322811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
50. C4×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
#139092
C2×#13252
C2×#13242
#85582
C4×#3242