Finite Group Databasebeta

← #142156

Subgroups — #142156

Order 512.

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

normal subgroups only

This group has 782 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#15125C211yesmaximal, characteristic
2C4×#342C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#7250C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic, complemented
6C2×#393—21no—
7C2×#380—21no—
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1446—21no—
15#1442C412yescomplemented
16#1439C2212yes—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic, complemented
19#1280—21no—
20C23×Q8—21no—
21C24×C4D411yescharacteristic
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—44no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C412yes—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—22no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4C2311yescharacteristic
32C22×C42—41no—
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37#393—44no—
38#380—44no—
39C4×C22⋊C4—24no—
40C4×C22⋊C4—41no—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—42no—
43C2×#95—22no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—41no—
48C2×#95C2×C412yes—
49C2×#95C2×C411yescharacteristic
50C2×#95D411yescharacteristic
51#342—24no—
52#342—22no—
53#342Q814yescomplemented
54#336—41no—
55#323—24no—
56C25C2×D411yescharacteristic
57C22×Q8—42no—
58C22×Q8—44no—
59C22×Q8—42no—
60C22×Q8—42no—
61C22×Q8—82no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4SD1612yes—
75C23×C4Q1612yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—42no—
79C4⋊Q8—48no—
80C4⋊Q8—44no—
81C4⋊Q8—44no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—44no—
104C2×C22⋊C4—82no—
105C2×C22⋊C4C2×Q812yes—
106C2×C22⋊C4#4112yes—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42—44no—
114C2×C42—82no—
115C2×C42—82no—
116C4⋊C8—82no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#95—41no—
120#95—41no—
121#95—42no—
122#95—44no—
123#95—42no—
124#95—44no—
125#95—44no—
126#95—82no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24—41no—
135C24C2×C22⋊C411yescharacteristic
136C24C22⋊Q812yes—
137C24C2×SD1611yescharacteristic
138C24C2×Q1611yescharacteristic
139C24AGL(1,ℤ/8)12yes—
140C2×Q8—48no—
141C2×Q8—84no—
142C2×Q8—88no—
143C2×Q8—88no—
144C2×Q8—84no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
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—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—84no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—82no—
197C22×C4—82no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4#9911yescharacteristic
201C22×C4#9911yescharacteristic
202C22×C4#10312yes—
203C22×C4#10312yes—
204C22×C4#10412yes—
205C22×C4#10412yes—
206C22×C4C4×Q812yes—
207C22×C4#12411yescharacteristic
208C22×C4C4⋊Q811yescharacteristic
209C2×C8—82no—
210C2×C8—82no—
211C2×C8—82no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—84no—
217C22⋊C4—28no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—88no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42—81no—
229C42—88no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—41no—
248C23—41no—
249C23—42no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23#39111yescharacteristic
261C23C2×#9911yescharacteristic
262C23#41011yescharacteristic
263C23C2×#10311yescharacteristic
264C23#41811yescharacteristic
265C23C2×#10411yescharacteristic
266C23C2×#10411yescharacteristic
267C23#47612yes—
268C23#47712yes—
269Q8—816no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
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—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4#32812yes—
327C2×C4#32812yes—
328C2×C4AΣL(1,𝔽16)12yes—
329C2×C4#35412yes—
330C8—162no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—82no—
356C22—82no—
357C22—81no—
358C22—81no—
359C22C2×#32811yescharacteristic
360C22#150311yescharacteristic
361C22#150511yescharacteristic
362C22#150711yescharacteristic
363C22#198711yescharacteristic
364C22#199411yescharacteristic
365C22#199611yescharacteristic
366C22C2×AΣL(1,𝔽16)11yescharacteristic
367C22#212311yescharacteristic
368C22C2×#35411yescharacteristic
369C22#212811yescharacteristic
370C4—22no—
371C4—42no—
372C4—42no—
373C4—41no—
374C4—41no—
375C4—42no—
376C4—88no—
377C4—81no—
378C4—84no—
379C4—162no—
380C4#135114yes—
381C2—21no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2#990811yescharacteristic
388C2C2×#135111yescharacteristic
389C2#1171811yescharacteristic
390C2#1177211yescharacteristic
391C2#1177711yescharacteristic
392C2#1329611yescharacteristic
393C2#1329911yescharacteristic
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
5. C22×C4⋊Q8C4semidirect
15. #1442C4semidirect
18. C4⋊#95C4semidirect
53. #342Q8semidirect

Maximal subgroups

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

subgroupindex
#151252
C4×#3422
#85572
#72502