Finite Group Databasebeta

← #111900

Subgroups — #111900

Order 512.

1120 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1120 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#2545C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic, complemented
3C2×#1533C212yesmaximal
4C2×#1467C211yesmaximal, characteristic, complemented
5C2×#1285C212yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic
7C22×#124C411yescharacteristic
8C22×#124C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11C2×#432C2212yes—
12#2545—24no—
13#2545—24no—
14C2×#396—22no—
15C2×#391C411yescharacteristic
16C2×#390C411yescharacteristic
17C2×#388—22no—
18C2×C4×C4⋊C4C2211yescharacteristic
19#1985—24no—
20#1817—22no—
21#1533—24no—
22#1533—24no—
23C2×#329—24no—
24#1467—24no—
25C2×#324—22no—
26#1285—24no—
27C23×Q8C2×C411yescharacteristic
28C23×Q8C2×C411yescharacteristic
29C23×D4C2×C411yescharacteristic
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—44no—
33C2×#124—22no—
34C2×#124—22no—
35C2×#124—22no—
36C2×#124—22no—
37C2×#124—22no—
38C2×#124—22no—
39C2×#124—22no—
40C2×#124—44no—
41C2×#124D414yes—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4D412yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42D412yes—
59C22×C42C2311yescharacteristic
60#432—22no—
61#432—22no—
62#432—22no—
63#432—22no—
64#432—22no—
65#432—22no—
66#432—22no—
67#432D414yes—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×#97—22no—
73C2×#97D414yes—
74#396—216no—
75#391—22no—
76#391—22no—
77#390—22no—
78#390—22no—
79#388—216no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4D414yes—
82C2×#95—22no—
83C2×#95—21no—
84#336—28no—
85#329—216no—
86#324—28no—
87C25—21no—
88C22×Q8—24no—
89C22×Q8—21no—
90C22×Q8—21no—
91C22×Q8—21no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—42no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8C22⋊C412yes—
100C22×D4—21no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—42no—
104C22×D4—42no—
105C22×D4C22⋊C412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C412yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C22×C8—24no—
115C22×C8—42no—
116C4⋊Q8—48no—
117C4⋊Q8—48no—
118#124—24no—
119#124—24no—
120#124—24no—
121#124—24no—
122#124—48no—
123#124—48no—
124#124—42no—
125#124—42no—
126#124—42no—
127#124—42no—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—48no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42#4112yes—
177C2×C42#4112yes—
178C4⋊C8—24no—
179C4⋊C8—42no—
180C4⋊C8—42no—
181C4⋊C8—42no—
182C22⋊C8—24no—
183C22⋊C8—42no—
184C22⋊C8—42no—
185C22⋊C8—42no—
186#97—24no—
187#97—24no—
188#97—24no—
189#97—42no—
190#95—28no—
191#95—24no—
192C24—22no—
193C24—44no—
194C24—44no—
195C24—41no—
196C24—41no—
197C24—42no—
198C24—42no—
199C24C2×C22⋊C411yescharacteristic
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—48no—
204C2×Q8—48no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—84no—
215C2×Q8—84no—
216C2×D4—24no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—84no—
223C2×D4—84no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4#10014yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C4×D412yes—
255C22×C4C4×D412yes—
256C22×C4#12011yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12012yes—
259C22×C4#12012yes—
260C22×C4#12112yes—
261C22×C4#12112yes—
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C2×C8—48no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—84no—
268C4⋊C4—48no—
269C4⋊C4—416no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C22⋊C4—416no—
275C22⋊C4—416no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—44no—
285C42—44no—
286C42#12012yes—
287C42#12112yes—
288C42#12112yes—
289C42#12112yes—
290C42#12112yes—
291C42#12312yes—
292C42#12412yes—
293C42#12712yes—
294C23—22no—
295C23—22no—
296C23—44no—
297C23—44no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—82no—
302C23—88no—
303C23—88no—
304C23—82no—
305C23—82no—
306C23—82no—
307C23—82no—
308C23—82no—
309C23#37911yescharacteristic
310C23#38611yescharacteristic
311C23#38611yescharacteristic
312C23C2×#10012yes—
313C23#41312yes—
314Q8—48no—
315Q8—48no—
316Q8—84no—
317Q8—84no—
318D4—84no—
319D4—84no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—84no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4#38612yes—
359C2×C4#39012yes—
360C2×C4#39212yes—
361C2×C4#39212yes—
362C2×C4#39412yes—
363C2×C4#39412yes—
364C8—84no—
365C8—84no—
366C22—21no—
367C22—21no—
368C22—24no—
369C22—24no—
370C22—21no—
371C22—21no—
372C22—84no—
373C22—84no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22—164no—
378C22—164no—
379C22#143811yescharacteristic
380C22#178311yescharacteristic
381C22#178511yescharacteristic
382C22#188412yes—
383C22#190012yes—
384C22#190012yes—
385C22#191212yes—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—44no—
393C4—44no—
394C4—82no—
395C4—82no—
396C4—82no—
397C2—21no—
398C2—21no—
399C2—22no—
400C2—162no—
401C2#944814yes—
402C2#1096711yescharacteristic
403C2#1097911yescharacteristic
404C2#1099011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21822C2semidirect
4. C2×#1467C2semidirect
5. C2×#1285C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25452
#218222
C2×#15332
C2×#14672
C2×#12852