Finite Group Databasebeta

← #141109

Subgroups — #141109

Order 512.

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

normal subgroups only

This group has 798 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
2C2×#1351C214yesmaximal
3C4×#342C211yesmaximal, characteristic
4#8557C211yesmaximal, characteristic
5C22×C4⋊Q8C411yescharacteristic, complemented
6C2×#393C2212yes—
7C2×#380—21no—
8C2×C4×C22⋊C4—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1446C412yescomplemented
15#1442—21no—
16#1439—21no—
17C4⋊#95—21no—
18C4⋊#95C411yescharacteristic, complemented
19#1351—28no—
20#1280—21no—
21C23×Q8C2×C412yes—
22C24×C4D411yescharacteristic
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—41no—
25C2×C4⋊Q8—44no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C22⋊C4—22no—
30C22×C22⋊C4—22no—
31C22×C22⋊C4D411yescharacteristic
32C22×C22⋊C4C2311yescharacteristic
33C22×C42—41no—
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C4⋊C8—41no—
37C2×C22⋊C8—22no—
38#393—24no—
39#393—24no—
40#380—44no—
41C4×C22⋊C4—24no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C4×C22⋊C4—42no—
45C2×#95—21no—
46C2×#95—22no—
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
59C22×Q8—24no—
60C22×Q8—44no—
61C22×Q8—42no—
62C22×Q8—42no—
63C22×Q8—82no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4SD1612yes—
77C23×C4Q1612yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—42no—
81C4⋊Q8—48no—
82C4⋊Q8—44no—
83C4⋊Q8—44no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—44no—
106C2×C22⋊C4—82no—
107C2×C22⋊C4C2×D412yes—
108C2×C22⋊C4#4112yes—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—44no—
116C2×C42—82no—
117C2×C42—82no—
118C4⋊C8—82no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121#95—24no—
122#95—41no—
123#95—41no—
124#95—44no—
125#95—44no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—82no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—41no—
135C24—41no—
136C24—41no—
137C24—41no—
138C24C2×C22⋊C411yescharacteristic
139C24#12112yes—
140C24C2×SD1611yescharacteristic
141C24C2×Q1611yescharacteristic
142C24#13712yes—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—84no—
146C2×Q8—88no—
147C2×Q8—88no—
148C2×Q8—84no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
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—44no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—44no—
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—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—84no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4#9911yescharacteristic
205C22×C4#9911yescharacteristic
206C22×C4#10312yes—
207C22×C4#10312yes—
208C22×C4#10412yes—
209C22×C4#10412yes—
210C22×C4C4×D412yes—
211C22×C4#12411yescharacteristic
212C22×C4#12711yescharacteristic
213C2×C8—82no—
214C2×C8—82no—
215C2×C8—82no—
216C4⋊C4—44no—
217C4⋊C4—48no—
218C4⋊C4—44no—
219C4⋊C4—84no—
220C22⋊C4—28no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—88no—
228C42—42no—
229C42—42no—
230C42—42no—
231C42—81no—
232C42—88no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23#39011yescharacteristic
264C23C2×#9911yescharacteristic
265C23#41011yescharacteristic
266C23C2×#10311yescharacteristic
267C23#41911yescharacteristic
268C23C2×#10411yescharacteristic
269C23C2×#10411yescharacteristic
270C23#46112yes—
271C23#46212yes—
272Q8—88no—
273Q8—88no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4#32812yes—
331C2×C4#32812yes—
332C2×C4AΣL(1,𝔽16)12yes—
333C2×C4#35412yes—
334C8—162no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—82no—
360C22—81no—
361C22—81no—
362C22—82no—
363C22C2×#32811yescharacteristic
364C22#150311yescharacteristic
365C22#150611yescharacteristic
366C22#150811yescharacteristic
367C22#196211yescharacteristic
368C22#196411yescharacteristic
369C22#198111yescharacteristic
370C22C2×AΣL(1,𝔽16)11yescharacteristic
371C22#212211yescharacteristic
372C22C2×#35411yescharacteristic
373C22#212911yescharacteristic
374C4—22no—
375C4—42no—
376C4—42no—
377C4—41no—
378C4—41no—
379C4—42no—
380C4—88no—
381C4—81no—
382C4—84no—
383C4—162no—
384C4#135114yescomplemented
385C2—21no—
386C2—21no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2#971111yescharacteristic
392C2C2×#135111yescharacteristic
393C2#1171711yescharacteristic
394C2#1178511yescharacteristic
395C2#1178711yescharacteristic
396C2#1322111yescharacteristic
397C2#1322411yescharacteristic
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
14. #1446C4semidirect
18. C4⋊#95C4semidirect
384. C4#1351semidirect

Maximal subgroups

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

subgroupindex
#151252
C2×#13512
C4×#3422
#85572