Finite Group Databasebeta

← #103688

Subgroups — #103688

Order 512.

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

normal subgroups only

This group has 541 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#28716C211yesmaximal, characteristic
2#28687C211yesmaximal, characteristic, complemented
3#28280C211yesmaximal, characteristic
4#9692C211yesmaximal, characteristic, complemented
5#9674C211yesmaximal, characteristic
6#7571C211yesmaximal, characteristic, complemented
7#7031C211yesmaximal, characteristic, complemented
8C2×#498—21no—
9C2×#486—21no—
10C2×#476—21no—
11#2868C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13C4×#127C2211yescharacteristic
14#2084—22no—
15#2071—22no—
16#2065—22no—
17#1990—21no—
18#1990—21no—
19#1982—21no—
20#1948—22no—
21#1948—21no—
22#1842C2211yescharacteristic
23#1802—22no—
24#1802—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×#332—22no—
29C2×D4⋊C8—22no—
30#1272—22no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×#127D411yescharacteristic
36C2×C4×Q8—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40#498—42no—
41#498—42no—
42#486—42no—
43#486—42no—
44#476—42no—
45#476—42no—
46C2×#106—22no—
47C2×#106—24no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8D412yes—
54C2×#102—24no—
55C2×#102—22no—
56C2×#102—22no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60#391—21no—
61#390—21no—
62#390—21no—
63#390—21no—
64#389—21no—
65#389—21no—
66#384—21no—
67#384—21no—
68#384—21no—
69#384D411yescharacteristic
70#384D411yescharacteristic
71#384D412yes—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C43C2311yescharacteristic
75#340—22no—
76#340—24no—
77#340—24no—
78#336—21no—
79#336—21no—
80#336D412yes—
81#332—44no—
82D4⋊C8—44no—
83C22×Q8—21no—
84C22×Q8—41no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—21no—
88C23×C4—41no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—41no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—42no—
97C4⋊Q8—42no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—41no—
106C4⋊Q8D812yes—
107#127—21no—
108#127—21no—
109#127—21no—
110#127SD1612yes—
111C4×Q8—82no—
112C4×D4—44no—
113C4×D4—41no—
114C4×D4—42no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—42no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C42—41no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145#106—44no—
146#106—48no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—42no—
150C4⋊C8—42no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—41no—
156C4⋊C8SD1614yes—
157#102—44no—
158#102—48no—
159#102—82no—
160C4×C8—41no—
161C4×C8—41no—
162C4×C8—42no—
163C4×C8—41no—
164C4×C8—41no—
165#95—41no—
166#95—41no—
167C24—41no—
168C2×Q8—22no—
169C2×Q8—41no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—41no—
191C2×D4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—81no—
197C22×C4—82no—
198C22×C4—81no—
199C22×C4—82no—
200C22×C4#12011yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12711yescharacteristic
207C2×C8—24no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—81no—
218C2×C8—81no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—81no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C22⋊C4—82no—
243C22⋊C4—81no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C42—21no—
248C42—22no—
249C42—21no—
250C42—22no—
251C42—41no—
252C42—82no—
253C42C2×D811yescharacteristic
254C42C2×SD1611yescharacteristic
255C42C2×SD1612yes—
256C42#13511yescharacteristic
257C42#13511yescharacteristic
258C42AGL(1,ℤ/8)11yescharacteristic
259C42AGL(1,ℤ/8)11yescharacteristic
260C42AGL(1,ℤ/8)11yescharacteristic
261C42#13711yescharacteristic
262C42#13712yes—
263C23—42no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23#39211yescharacteristic
271Q8—82no—
272Q8—82no—
273Q8—161no—
274Q8—161no—
275Q8—161no—
276Q8—161no—
277D4—44no—
278D4—42no—
279D4—42no—
280D4—44no—
281D4—82no—
282D4—81no—
283D4—81no—
284D4—82no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—162no—
307C2×C4—162no—
308C2×C4#44711yescharacteristic
309C2×C4#44711yescharacteristic
310C2×C4#44811yescharacteristic
311C2×C4#44812yes—
312C2×C4#44911yescharacteristic
313C2×C4#44911yescharacteristic
314C2×C4#44911yescharacteristic
315C2×C4#45011yescharacteristic
316C2×C4#45211yescharacteristic
317C2×C4#45311yescharacteristic
318C2×C4#45412yes—
319C2×C4#45411yescharacteristic
320C2×C4#46112yes—
321C2×C4#46511yescharacteristic
322C2×C4#46611yescharacteristic
323C2×C4#46811yescharacteristic
324C2×C4#46911yescharacteristic
325C2×C4#47112yes—
326C2×C4#49211yescharacteristic
327C2×C4#49711yescharacteristic
328C8—44no—
329C8—82no—
330C8—82no—
331C8—82no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—161no—
336C22—161no—
337C22—161no—
338C22—161no—
339C22#199911yescharacteristic
340C22#200011yescharacteristic
341C22#200311yescharacteristic
342C22#200611yescharacteristic
343C22#200911yescharacteristic
344C22#201311yescharacteristic
345C22#201911yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—41no—
353C4—82no—
354C4—161no—
355C4—161no—
356C4—162no—
357C4#162111yescharacteristic
358C4#162211yescharacteristic
359C4#162612yes—
360C4#162712yes—
361C4#162711yescharacteristic
362C4#162811yescharacteristic
363C4#167112yes—
364C4#169112yes—
365C2—162no—
366C2#1005111yescharacteristic
367C2#1010811yescharacteristic
368C2#1018411yescharacteristic
369C2#1020111yescharacteristic
370C2#1022911yescharacteristic
371C2#1024711yescharacteristic
372C2#1027111yescharacteristic
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
2. #28687C2semidirect
4. #9692C2semidirect
6. #7571C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#287162
#286872
#282802
#96922
#96742
#75712
#70312