Finite Group Databasebeta

← #103725

Subgroups — #103725

Order 512.

591 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 114 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#29219C211yesmaximal, characteristic, complemented
2#28438C211yesmaximal, characteristic
3#28252C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9865C211yesmaximal, characteristic, complemented
6#7569C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic, complemented
8C2×#492—21no—
9C2×#460—21no—
10C2×#461—22no—
11C2×#461—21no—
12#2868C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×#127C2211yescharacteristic
15#2099—22no—
16#2055—22no—
17#2013—22no—
18#1990—21no—
19#1988—22no—
20#1956—21no—
21#1842C2211yescharacteristic
22#1834—21no—
23#1803—21no—
24#1802—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1296—22no—
29#1279—22no—
30#1270—22no—
31C22×SD16—22no—
32C22×SD16—24no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8D412yes—
37C2×#127D411yescharacteristic
38C2×C4×Q8—21no—
39C2×C4×D4—21no—
40C2×C4×D4—21no—
41C2×C4×D4—21no—
42#492—42no—
43#492—42no—
44#461—44no—
45#461—42no—
46#461—42no—
47#461—44no—
48#460—42no—
49#460—42no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8D412yes—
55C2×C4⋊C8D412yes—
56C2×#103—22no—
57C2×#103—24no—
58C2×#102—22no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62#391—21no—
63#390—21no—
64#390—21no—
65#390—21no—
66#389—21no—
67#389—21no—
68#384—21no—
69#384—21no—
70#384—21no—
71#384—21no—
72#384D411yescharacteristic
73#384D411yescharacteristic
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C43C2311yescharacteristic
77#340—22no—
78#340—24no—
79#340—24no—
80#336—21no—
81#336—21no—
82#336D412yes—
83C22×Q8—21no—
84C22×Q8—22no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—21no—
88C23×C4—41no—
89C2×SD16—48no—
90C2×SD16—44no—
91C2×SD16—48no—
92C2×SD16—44no—
93C2×SD16—84no—
94C2×SD16—82no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—41no—
99C4⋊Q8—21no—
100C4⋊Q8—21no—
101C4⋊Q8—21no—
102C4⋊Q8—44no—
103C4⋊Q8—41no—
104C4⋊Q8—42no—
105C4⋊Q8—41no—
106C4⋊Q8—41no—
107C4⋊Q8—42no—
108C4⋊Q8—41no—
109C4⋊Q8SD1612yescomplemented
110#127—21no—
111#127—21no—
112#127—21no—
113#127D812yes—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—41no—
118C4×D4—41no—
119C4×D4—42no—
120C4×D4—41no—
121C4×D4—41no—
122C4×D4—42no—
123C4×D4—41no—
124C4×D4—82no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C42—41no—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42#4111yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—42no—
154C4⋊C8—41no—
155C4⋊C8—42no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8SD1614yescomplemented
161#103—48no—
162#103—44no—
163#102—44no—
164C4×C8—41no—
165C4×C8—41no—
166C4×C8—41no—
167C4×C8—41no—
168C4×C8—42no—
169#95—41no—
170#95—41no—
171C24—41no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—42no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—44no—
179C2×Q8—41no—
180C2×Q8—44no—
181C2×Q8—41no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—42no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—81no—
195C2×D4—81no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—82no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—82no—
204C22×C4#12011yescharacteristic
205C22×C4#12011yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12711yescharacteristic
211SD16—816no—
212SD16—88no—
213C2×C8—24no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—81no—
224C2×C8—81no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—81no—
238C4⋊C4—82no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—82no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C22⋊C4—82no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C42—21no—
254C42—22no—
255C42—21no—
256C42—22no—
257C42—41no—
258C42—81no—
259C42—81no—
260C42C2×D811yescharacteristic
261C42C2×SD1611yescharacteristic
262C42C2×SD1612yes—
263C42#13511yescharacteristic
264C42#13511yescharacteristic
265C42AGL(1,ℤ/8)11yescharacteristic
266C42AGL(1,ℤ/8)11yescharacteristic
267C42AGL(1,ℤ/8)11yescharacteristic
268C42AGL(1,ℤ/8)12yes—
269C42#13711yescharacteristic
270C23—42no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23#39211yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—81no—
281Q8—81no—
282Q8—84no—
283Q8—84no—
284D4—44no—
285D4—42no—
286D4—42no—
287D4—82no—
288D4—82no—
289D4—82no—
290D4—81no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—162no—
314C2×C4—162no—
315C2×C4#44711yescharacteristic
316C2×C4#44711yescharacteristic
317C2×C4#44811yescharacteristic
318C2×C4#44911yescharacteristic
319C2×C4#44911yescharacteristic
320C2×C4#44911yescharacteristic
321C2×C4#45012yes—
322C2×C4#45011yescharacteristic
323C2×C4#45211yescharacteristic
324C2×C4#45512yes—
325C2×C4#45511yescharacteristic
326C2×C4#45611yescharacteristic
327C2×C4#46012yes—
328C2×C4#46511yescharacteristic
329C2×C4#46611yescharacteristic
330C2×C4#46811yescharacteristic
331C2×C4#46911yescharacteristic
332C2×C4#47112yes—
333C2×C4#49211yescharacteristic
334C2×C4#49611yescharacteristic
335C8—44no—
336C8—82no—
337C8—82no—
338C8—82no—
339C22—82no—
340C22—82no—
341C22—82no—
342C22—161no—
343C22—161no—
344C22—161no—
345C22—161no—
346C22#199911yescharacteristic
347C22#200011yescharacteristic
348C22#200411yescharacteristic
349C22#200711yescharacteristic
350C22#200911yescharacteristic
351C22#201411yescharacteristic
352C22#201911yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—82no—
361C4—161no—
362C4—162no—
363C4—161no—
364C4#163912yes—
365C4#163911yescharacteristic
366C4#164212yes—
367C4#164611yescharacteristic
368C4#164711yescharacteristic
369C4#165411yescharacteristic
370C4#166612yes—
371C4#168612yes—
372C2—162no—
373C2#1006211yescharacteristic
374C2#1011411yescharacteristic
375C2#1018411yescharacteristic
376C2#1020611yescharacteristic
377C2#1023111yescharacteristic
378C2#1024411yescharacteristic
379C2#1026811yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29219C2semidirect
4. #9898C2semidirect
5. #9865C2semidirect
7. #7031C2semidirect
109. C4⋊Q8SD16semidirect
160. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#292192
#284382
#282522
#98982
#98652
#75692
#70312