Finite Group Databasebeta

← #100618

Subgroups — #100618

Order 512.

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

normal subgroups only

This group has 411 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#29868C211yesmaximal, characteristic
2#22798C211yesmaximal, characteristic
3#9905C211yesmaximal, characteristic, complemented
4#9898C211yesmaximal, characteristic, complemented
5#7574C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic, complemented
7#7022C211yesmaximal, characteristic, complemented
8C2×#476—21no—
9C2×#433—21no—
10#2868C2211yescharacteristic
11#2293C2211yescharacteristic
12#2084—21no—
13#2084C412yes—
14#2071—21no—
15#2071C412yes—
16#1987—21no—
17#1955—21no—
18#1924—21no—
19#1923—21no—
20#1842C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1798—21no—
24#1771—21no—
25#1767C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27#1296—21no—
28#1296—21no—
29#1296—21no—
30#1296—21no—
31#1271—21no—
32#1270—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4×D4D411yescharacteristic
37C2×#117—21no—
38#476—42no—
39#476—42no—
40#433—42no—
41#433—42no—
42C2×#106—22no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×C4⋊C8D411yescharacteristic
52C2×#102—22no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×#97—21no—
58C2×#97—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61#390C2×C411yescharacteristic
62#390D411yescharacteristic
63#389C2×C411yescharacteristic
64#389D411yescharacteristic
65#388—21no—
66#388—21no—
67#384—21no—
68#384—21no—
69#384—21no—
70#384C2×C412yes—
71C4×C4⋊C4D411yescharacteristic
72C4×C22⋊C4—21no—
73C43C2311yescharacteristic
74#340—22no—
75#340—22no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336C2×C412yes—
84C22×Q8—41no—
85C22×D4C22⋊C411yescharacteristic
86C23×C4—21no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—41no—
93C22×C8—41no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—81no—
99C4⋊Q8—81no—
100C4⋊Q8—82no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105#117—41no—
106#117—41no—
107#117—41no—
108#117—41no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C42—21no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133#106—44no—
134C4⋊C8—22no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—21no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8SD1612yes—
150#102—44no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153C22⋊C8—81no—
154C22⋊C8—81no—
155#97—42no—
156#97—41no—
157#97—41no—
158C4×C8—42no—
159C4×C8—81no—
160#95—21no—
161#95—21no—
162#95—21no—
163C24—21no—
164C2×Q8—82no—
165C2×Q8—82no—
166C2×Q8—81no—
167C2×Q8—81no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—41no—
171C2×D4—41no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12311yescharacteristic
189C2×C8—22no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—81no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—81no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42C2×SD1611yescharacteristic
246C42#13511yescharacteristic
247C42#13511yescharacteristic
248C42#13511yescharacteristic
249C42AGL(1,ℤ/8)11yescharacteristic
250C42AGL(1,ℤ/8)11yescharacteristic
251C42AGL(1,ℤ/8)11yescharacteristic
252C42#13711yescharacteristic
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#38611yescharacteristic
261Q8—162no—
262Q8—162no—
263D4—42no—
264D4—81no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4#41611yescharacteristic
295C2×C4#41811yescharacteristic
296C2×C4C4×SD1611yescharacteristic
297C2×C4#44011yescharacteristic
298C2×C4#44911yescharacteristic
299C2×C4#44911yescharacteristic
300C2×C4#45011yescharacteristic
301C2×C4#45211yescharacteristic
302C2×C4#45511yescharacteristic
303C2×C4#45611yescharacteristic
304C2×C4#46111yescharacteristic
305C2×C4#46311yescharacteristic
306C2×C4#47111yescharacteristic
307C2×C4#47211yescharacteristic
308C2×C4#48211yescharacteristic
309C2×C4#48211yescharacteristic
310C8—42no—
311C8—82no—
312C8—82no—
313C8—81no—
314C8—81no—
315C8—81no—
316C8—161no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#187511yescharacteristic
325C22#187711yescharacteristic
326C22#188211yescharacteristic
327C22#190311yescharacteristic
328C22#190711yescharacteristic
329C22#190911yescharacteristic
330C22#191511yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—162no—
346C4#164211yescharacteristic
347C4#164611yescharacteristic
348C4#165011yescharacteristic
349C4#165411yescharacteristic
350C2—161no—
351C2#898811yescharacteristic
352C2#917411yescharacteristic
353C2#919611yescharacteristic
354C2#926911yescharacteristic
355C2#928711yescharacteristic
356C2#933811yescharacteristic
357C2#937511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9905C2semidirect
4. #9898C2semidirect
6. #7040C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#298682
#227982
#99052
#98982
#75742
#70402
#70222