Finite Group Databasebeta

← #141108

Subgroups — #141108

Order 512.

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

normal subgroups only

This group has 742 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#15135C211yesmaximal, characteristic
2C2×#1352C214yesmaximal
3C4×#342C211yesmaximal, characteristic
4#8557C211yesmaximal, characteristic
5C2×#399C2212yes—
6C2×#389—21no—
7C2×#384C411yescharacteristic, complemented
8C2×#380—21no—
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1797—21no—
12C2×#342—21no—
13C2×#342C2212yes—
14C2×#323C2212yes—
15#1445C412yescomplemented
16#1444—21no—
17#1439—21no—
18C4⋊#95C411yescharacteristic, complemented
19#1352—28no—
20#1280—21no—
21C24×C4D411yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C412yes—
27C22×C22⋊C4—22no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4D411yescharacteristic
30C22×C22⋊C4C2311yescharacteristic
31C22×C42—41no—
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—41no—
35C2×C22⋊C8—22no—
36#399—24no—
37#399—24no—
38#389—44no—
39#384—41no—
40#384—41no—
41#380—44no—
42C4×C22⋊C4—24no—
43C4×C22⋊C4—41no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—42no—
46C2×#95—21no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—41no—
50C2×#95C2×C412yes—
51C2×#95C2×C411yescharacteristic
52C2×#95D411yescharacteristic
53#342—24no—
54#342—22no—
55#342D414yes—
56#336—41no—
57#323—24no—
58C25C2×D411yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4SD1612yes—
72C23×C4Q1612yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C22×C8—42no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—82no—
91C2×C4⋊C4—82no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—44no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—82no—
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—
114C2×C42—44no—
115C2×C42—82no—
116C2×C42—82no—
117C4⋊C8—82no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120#95—24no—
121#95—41no—
122#95—41no—
123#95—44no—
124#95—44no—
125#95—44no—
126#95—42no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24—41no—
135C24C2×C22⋊C411yescharacteristic
136C24#12112yes—
137C24C2×SD1611yescharacteristic
138C24C2×Q1611yescharacteristic
139C24#13712yes—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—42no—
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—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—84no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—82no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4#9911yescharacteristic
196C22×C4#9911yescharacteristic
197C22×C4#10312yes—
198C22×C4#10312yes—
199C22×C4#10412yes—
200C22×C4#10412yes—
201C22×C4C4×D412yes—
202C22×C4#12411yescharacteristic
203C22×C4#12711yescharacteristic
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C4⋊C4—48no—
208C4⋊C4—84no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C4⋊C4—84no—
212C4⋊C4—84no—
213C4⋊C4—82no—
214C22⋊C4—28no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—88no—
222C42—42no—
223C42—42no—
224C42—42no—
225C42—81no—
226C42—88no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23#39011yescharacteristic
258C23C2×#9911yescharacteristic
259C23#41011yescharacteristic
260C23C2×#10311yescharacteristic
261C23#41911yescharacteristic
262C23C2×#10411yescharacteristic
263C23C2×#10411yescharacteristic
264C23#46112yes—
265C23#46212yes—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
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—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4#32812yes—
323C2×C4#32812yes—
324C2×C4#35212yes—
325C2×C4#35312yes—
326C8—162no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—82no—
352C22—81no—
353C22—81no—
354C22—82no—
355C22C2×#32811yescharacteristic
356C22#150311yescharacteristic
357C22#150611yescharacteristic
358C22#150811yescharacteristic
359C22#196211yescharacteristic
360C22#196411yescharacteristic
361C22#198111yescharacteristic
362C22C2×#35211yescharacteristic
363C22#212311yescharacteristic
364C22C2×#35311yescharacteristic
365C22#212811yescharacteristic
366C4—22no—
367C4—42no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—84no—
373C4—88no—
374C4—81no—
375C4—162no—
376C4#135214yescomplemented
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—42no—
383C2#971111yescharacteristic
384C2C2×#135211yescharacteristic
385C2#1171811yescharacteristic
386C2#1178611yescharacteristic
387C2#1178811yescharacteristic
388C2#1322011yescharacteristic
389C2#1322311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ 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
7. C2×#384C4semidirect
15. #1445C4semidirect
18. C4⋊#95C4semidirect
376. C4#1352semidirect

Maximal subgroups

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

subgroupindex
#151352
C2×#13522
C4×#3422
#85572