Finite Group Databasebeta

← #140609

Subgroups — #140609

Order 512.

734 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 120 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#13904C211yesmaximal, characteristic
2C2×#1317C214yesmaximal
3#8557C211yesmaximal, characteristic
4C4×#323C211yesmaximal, characteristic
5C23×C4⋊C4C411yescharacteristic
6C2×#381C412yes—
7C2×#381C2212yes—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C4×C22⋊C8—21no—
12C2×#323C2212yes—
13C2×#323C2212yes—
14#1445—21no—
15#1445—21no—
16C4⋊#95—21no—
17C4⋊#95—21no—
18#1317—28no—
19#1280—21no—
20#1280—21no—
21C24×C4C2×C412yes—
22C24×C4C2×C411yescharacteristic
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4—41no—
29C22×C4⋊C4—42no—
30C22×C4⋊C4C814yescomplemented
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4C2×C412yes—
33C22×C22⋊C4C2311yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C4×C8—41no—
40#381—24no—
41#381—24no—
42#381—44no—
43#380—42no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—21no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95D411yescharacteristic
52C2×#95D411yescharacteristic
53#336—41no—
54#336—41no—
55#323—22no—
56#323—24no—
57#323D414yes—
58C25C22×C411yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—24no—
65C23×C4—24no—
66C23×C4—22no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C2×C814yes—
78C23×C4C2×C812yes—
79C23×C4M1612yes—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C22×C8—42no—
83C22×C8—42no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4C2×D412yes—
107C2×C22⋊C4#4112yes—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—42no—
113C2×C42—42no—
114C4⋊C8—82no—
115C22⋊C8—24no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C4×C8—82no—
120#95—41no—
121#95—41no—
122#95—42no—
123#95—44no—
124#95—44no—
125#95—44no—
126#95—42no—
127#95—42no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—41no—
133C24—41no—
134C24—41no—
135C24—41no—
136C24C2×C22⋊C411yescharacteristic
137C24C4×D412yes—
138C24C22×C811yescharacteristic
139C24C2×M1611yescharacteristic
140C24#13112yes—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—28no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—48no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
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—44no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—84no—
196C22×C4—84no—
197C22×C4—84no—
198C22×C4—84no—
199C22×C4C22⋊C812yes—
200C22×C4C22⋊C812yes—
201C22×C4#9911yescharacteristic
202C22×C4#9911yescharacteristic
203C22×C4#10012yes—
204C22×C4#10012yes—
205C22×C4#12112yes—
206C22×C4#12411yescharacteristic
207C22×C4#12711yescharacteristic
208C2×C8—44no—
209C2×C8—82no—
210C2×C8—82no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8—82no—
214C4⋊C4—48no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—84no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—44no—
221C22⋊C4—84no—
222C42—42no—
223C42—42no—
224C42—81no—
225C42—81no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#39011yescharacteristic
257C23C2×C22⋊C811yescharacteristic
258C23#40811yescharacteristic
259C23C2×#9911yescharacteristic
260C23#41011yescharacteristic
261C23C2×#10011yescharacteristic
262C23C2×#10011yescharacteristic
263C23C8×D412yes—
264C23#43612yes—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—88no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—88no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4#32312yes—
317C2×C4#32312yes—
318C2×C4#35212yes—
319C2×C4#35212yes—
320C8—84no—
321C8—162no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—22no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—44no—
339C22—42no—
340C22—42no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—42no—
350C22C2×#32311yescharacteristic
351C22#146011yescharacteristic
352C22#146911yescharacteristic
353C22#147111yescharacteristic
354C22#195411yescharacteristic
355C22#196411yescharacteristic
356C22#196511yescharacteristic
357C22C2×#35211yescharacteristic
358C22C2×#35211yescharacteristic
359C22#212311yescharacteristic
360C22#212311yescharacteristic
361C4—22no—
362C4—42no—
363C4—42no—
364C4—41no—
365C4—41no—
366C4—42no—
367C4—81no—
368C4—84no—
369C4—162no—
370C4#131714yescomplemented
371C2—21no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—42no—
376C2—42no—
377C2#964911yescharacteristic
378C2C2×#131711yescharacteristic
379C2#1142811yescharacteristic
380C2#1155811yescharacteristic
381C2#1156211yescharacteristic
382C2#1321711yescharacteristic
383C2#1322011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C22×C8
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. C22×C4⋊C4C8semidirect
370. C4#1317semidirect

Maximal subgroups

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

subgroupindex
#139042
C2×#13172
#85572
C4×#3232