Finite Group Databasebeta

← #100990

Subgroups — #100990

Order 512.

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

normal subgroups only

This group has 720 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#21902C211yesmaximal, characteristic
2#15125C211yesmaximal, characteristic
3#8557C211yesmaximal, characteristic
4#8342C212yesmaximal, complemented
5#7039C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C2×#393C412yes—
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1923—24no—
11#1797C2212yes—
12#1797C2212yes—
13#1779—21no—
14C4×C22⋊C8—21no—
15C2×#333—24no—
16C2×#323—21no—
17#1446C412yes—
18#1442—21no—
19#1439—21no—
20C4⋊#95—21no—
21C4⋊#95C2211yescharacteristic
22#1280—21no—
23C23×Q8C2×C412yes—
24C24×C4D411yescharacteristic
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—44no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C22⋊C4—22no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4D411yescharacteristic
34C22×C42D411yescharacteristic
35C22×C42C2311yescharacteristic
36C2×C4⋊C8—22no—
37C2×C4⋊C8D414yes—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C4×C8—22no—
43#393—44no—
44#380—44no—
45C4×C22⋊C4—22no—
46C4×C22⋊C4—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—21no—
50C2×#95—22no—
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C412yes—
53C2×#95D411yescharacteristic
54#336—22no—
55#336—22no—
56#336—22no—
57#333—48no—
58#323—42no—
59C25C22⋊C411yescharacteristic
60C22×Q8—24no—
61C22×Q8—44no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—82no—
65C23×C4—22no—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C22×C8—24no—
80C22×C8—42no—
81C4⋊Q8—28no—
82C4⋊Q8—44no—
83C4⋊Q8—48no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—44no—
99C2×C22⋊C4—44no—
100C2×C22⋊C4—82no—
101C2×C22⋊C4SD1612yes—
102C2×C22⋊C4Q1612yes—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42C2×D412yes—
108C2×C42#4112yes—
109C4⋊C8—44no—
110C4⋊C8—44no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—82no—
116C22⋊C8—82no—
117C4×C8—82no—
118#95—22no—
119#95—22no—
120#95—42no—
121#95—44no—
122#95—44no—
123#95—42no—
124#95—82no—
125C24—21no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24—41no—
135C24#10312yes—
136C24C2×C22⋊C411yescharacteristic
137C2×Q8—28no—
138C2×Q8—44no—
139C2×Q8—84no—
140C2×Q8—84no—
141C2×Q8—88no—
142C2×Q8—88no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—48no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
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—81no—
184C22×C4—81no—
185C22×C4—84no—
186C22×C4#10012yes—
187C22×C4#10312yes—
188C22×C4C4×D412yes—
189C22×C4#12011yescharacteristic
190C22×C4#12112yes—
191C22×C4#12311yescharacteristic
192C22×C4C2×SD1611yescharacteristic
193C22×C4C2×Q1611yescharacteristic
194C22×C4#13512yes—
195C22×C4#13512yes—
196C22×C4#13712yes—
197C2×C8—44no—
198C2×C8—44no—
199C2×C8—44no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8—82no—
203C4⋊C4—44no—
204C4⋊C4—48no—
205C4⋊C4—44no—
206C4⋊C4—84no—
207C22⋊C4—24no—
208C22⋊C4—24no—
209C22⋊C4—44no—
210C22⋊C4—88no—
211C42—24no—
212C42—42no—
213C42—42no—
214C42—81no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23—81no—
240C23—81no—
241C23—81no—
242C23#38611yescharacteristic
243C23C2×#10011yescharacteristic
244C23#41311yescharacteristic
245C23C2×#10311yescharacteristic
246C23#41611yescharacteristic
247C23#41611yescharacteristic
248C23#41911yescharacteristic
249Q8—88no—
250Q8—88no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—41no—
280C2×C4—84no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—84no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4#45212yes—
297C2×C4#45512yes—
298C2×C4B(4,𝔽2)12yes—
299C2×C4#46512yes—
300C2×C4#46712yes—
301C2×C4#48512yes—
302C8—84no—
303C8—162no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—44no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#154211yescharacteristic
328C22#154311yescharacteristic
329C22#154811yescharacteristic
330C22#155411yescharacteristic
331C22#187311yescharacteristic
332C22#188511yescharacteristic
333C22#188911yescharacteristic
334C22#189211yescharacteristic
335C22#190011yescharacteristic
336C22#190911yescharacteristic
337C22#191411yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—84no—
345C4—81no—
346C4—162no—
347C4#160614yes—
348C2—21no—
349C2—21no—
350C2—21no—
351C2—21no—
352C2—82no—
353C2#901611yescharacteristic
354C2#912811yescharacteristic
355C2#919211yescharacteristic
356C2#939111yescharacteristic
357C2#945111yescharacteristic
358C2#946711yescharacteristic
359C2#950411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8342C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#219022
#151252
#85572
#83422
#70392