Finite Group Databasebeta

← #125344

Subgroups — #125344

Order 512.

788 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 98 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#15178C211yesmaximal, characteristic
2C4×#323C211yesmaximal, characteristic
3#7911C211yesmaximal, characteristic
4#7123C214yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×C22⋊Q8—21no—
7C2×#407—21no—
8C2×#391—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#323—21no—
12#1444—21no—
13#1444C2212yes—
14#1439—21no—
15#1439C412yes—
16C4⋊#95C411yescharacteristic
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C23×Q8—21no—
21C24×C4D411yescharacteristic
22C22×M16—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—41no—
25C2×C4⋊Q8—44no—
26C2×C22⋊Q8—44no—
27C2×C22⋊Q8—44no—
28C2×C22⋊Q8—42no—
29C2×C22⋊Q8—42no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4C2×C412yes—
34C22×C22⋊C4—22no—
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4D411yescharacteristic
37C22×C42C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39#407—24no—
40#407—42no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—41no—
44C2×C4×C8—41no—
45#391—44no—
46C4×C22⋊C4—41no—
47C4×C22⋊C4—41no—
48C2×#95—21no—
49C2×#95—22no—
50C2×#95—22no—
51C2×#95C2×C412yes—
52C2×#95D411yescharacteristic
53C2×#95C2311yescharacteristic
54#336—41no—
55#323—24no—
56C25C22⋊C411yescharacteristic
57C22×Q8—24no—
58C22×Q8—42no—
59C22×Q8—42no—
60C22×Q8—42no—
61C22×Q8—41no—
62C22×Q8—41no—
63C22×Q8—42no—
64C22×Q8—42no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4D812yes—
76C23×C4SD1612yes—
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C2×M16—44no—
81C2×M16—82no—
82C2×M16—82no—
83C22×C8—42no—
84C22×C8—42no—
85C4⋊Q8—44no—
86C4⋊Q8—48no—
87C4⋊Q8—44no—
88C22⋊Q8—416no—
89C22⋊Q8—416no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—44no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—44no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—42no—
115C2×C42—42no—
116C22⋊C8—24no—
117C22⋊C8—24no—
118C22⋊C8—44no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C4×C8—82no—
122#95—24no—
123#95—41no—
124#95—41no—
125#95—44no—
126#95—44no—
127#95—42no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—41no—
135C24—41no—
136C24—41no—
137C24—41no—
138C24#10212yes—
139C24C2×C22⋊C411yescharacteristic
140C2×Q8—48no—
141C2×Q8—44no—
142C2×Q8—48no—
143C2×Q8—44no—
144C2×Q8—82no—
145C2×Q8—84no—
146C2×Q8—84no—
147C2×Q8—82no—
148C2×Q8—82no—
149C2×Q8—82no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
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—42no—
194C22×C4—84no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4#9911yescharacteristic
198C22×C4#9911yescharacteristic
199C22×C4#10112yes—
200C22×C4#10212yes—
201C22×C4#10412yes—
202C22×C4#10412yes—
203C22×C4#13112yes—
204C22×C4C2×D811yescharacteristic
205C22×C4C2×SD1611yescharacteristic
206C22×C4#13712yes—
207M16—88no—
208C2×C8—44no—
209C2×C8—44no—
210C2×C8—82no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8—82no—
214C4⋊C4—48no—
215C4⋊C4—44no—
216C4⋊C4—48no—
217C4⋊C4—48no—
218C4⋊C4—44no—
219C4⋊C4—48no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—416no—
223C42—42no—
224C42—42no—
225C42—42no—
226C42—81no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—81no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23#40811yescharacteristic
255C23C2×#9911yescharacteristic
256C23C2×#10111yescharacteristic
257C23C2×#10211yescharacteristic
258C23#41911yescharacteristic
259C23C2×#10411yescharacteristic
260C23C2×#10411yescharacteristic
261Q8—88no—
262Q8—88no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
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—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4#32712yes—
313C2×C4#32712yes—
314C2×C4AΣL(1,𝔽16)12yes—
315C2×C4#35312yes—
316C8—84no—
317C8—84no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—44no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#147211yescharacteristic
342C22#148511yescharacteristic
343C22#149211yescharacteristic
344C22C2×#32711yescharacteristic
345C22#150811yescharacteristic
346C22#152611yescharacteristic
347C22#154611yescharacteristic
348C22C2×AΣL(1,𝔽16)11yescharacteristic
349C22#212311yescharacteristic
350C22C2×#35311yescharacteristic
351C22#213511yescharacteristic
352C4—22no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—84no—
360C4—84no—
361C4#134314yes—
362C2—21no—
363C2—21no—
364C2—21no—
365C2—21no—
366C2—82no—
367C2#768511yescharacteristic
368C2#1157011yescharacteristic
369C2#1158011yescharacteristic
370C2C2×#134311yescharacteristic
371C2#1170011yescharacteristic
372C2#1178811yescharacteristic
373C2#1179111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7123C2semidirect

Maximal subgroups

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

subgroupindex
#151782
C4×#3232
#79112
#71232