Finite Group Databasebeta

← #100606

Subgroups — #100606

Order 512.

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

normal subgroups only

This group has 409 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#29980C211yesmaximal, characteristic
2#22798C211yesmaximal, characteristic
3#9896C211yesmaximal, characteristic, complemented
4#9703C211yesmaximal, characteristic, complemented
5#7574C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic, complemented
7#7022C211yesmaximal, characteristic, complemented
8C2×#433—21no—
9#2868C2211yescharacteristic
10#2293C2211yescharacteristic
11#2099C412yes—
12#2093C412yes—
13#2057—21no—
14#2023—21no—
15#1955—21no—
16#1947—21no—
17#1924—21no—
18#1923—21no—
19#1842C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1802—21no—
23#1798—21no—
24#1771—21no—
25#1767C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C2×#332—21no—
28C2×#332—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#433—42no—
39#433—42no—
40C2×#107—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×#102—22no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—21no—
56C2×#97—21no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59#390C2×C411yescharacteristic
60#390D411yescharacteristic
61#389C2×C411yescharacteristic
62#389D411yescharacteristic
63#388—21no—
64#388—21no—
65#384—21no—
66#384—21no—
67#384—21no—
68#384C2×C412yes—
69C4×C4⋊C4D411yescharacteristic
70C4×C22⋊C4—21no—
71C43C2311yescharacteristic
72#340—22no—
73#340—22no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336C2×C412yes—
82#332—42no—
83#332—42no—
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—82no—
99C4⋊Q8—82no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104#117—41no—
105#117—41no—
106#117—41no—
107#117—41no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125C2×C42C22×C411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132#107—82no—
133C4⋊C8—22no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—41no—
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⋊C8D812yes—
149#102—82no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152C22⋊C8—81no—
153C22⋊C8—81no—
154#97—42no—
155#97—41no—
156#97—41no—
157C4×C8—42no—
158C4×C8—81no—
159#95—21no—
160#95—21no—
161#95—21no—
162C24—21no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8—81no—
166C2×Q8—81no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—41no—
170C2×D4—41no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4#12011yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12311yescharacteristic
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—41no—
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—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—81no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42C2×D811yescharacteristic
245C42#13511yescharacteristic
246C42#13511yescharacteristic
247C42#13511yescharacteristic
248C42AGL(1,ℤ/8)11yescharacteristic
249C42AGL(1,ℤ/8)11yescharacteristic
250C42#13711yescharacteristic
251C42#13711yescharacteristic
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#38611yescharacteristic
260Q8—162no—
261Q8—162no—
262D4—42no—
263D4—81no—
264C2×C4—21no—
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—41no—
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—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4#41611yescharacteristic
294C2×C4#41911yescharacteristic
295C2×C4C4×D811yescharacteristic
296C2×C4#44211yescharacteristic
297C2×C4#44711yescharacteristic
298C2×C4#44911yescharacteristic
299C2×C4#45211yescharacteristic
300C2×C4#45211yescharacteristic
301C2×C4#45511yescharacteristic
302C2×C4#45611yescharacteristic
303C2×C4#45911yescharacteristic
304C2×C4#46311yescharacteristic
305C2×C4#47111yescharacteristic
306C2×C4#47311yescharacteristic
307C2×C4#48511yescharacteristic
308C2×C4#48511yescharacteristic
309C8—42no—
310C8—82no—
311C8—82no—
312C8—81no—
313C8—81no—
314C8—81no—
315C8—161no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22#187311yescharacteristic
324C22#187911yescharacteristic
325C22#188211yescharacteristic
326C22#190311yescharacteristic
327C22#190811yescharacteristic
328C22#191011yescharacteristic
329C22#191511yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—162no—
345C4#164111yescharacteristic
346C4#164611yescharacteristic
347C4#164711yescharacteristic
348C4#165211yescharacteristic
349C2—161no—
350C2#898411yescharacteristic
351C2#917011yescharacteristic
352C2#919411yescharacteristic
353C2#927111yescharacteristic
354C2#929311yescharacteristic
355C2#933811yescharacteristic
356C2#937611yescharacteristic
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. #9896C2semidirect
4. #9703C2semidirect
6. #7040C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#299802
#227982
#98962
#97032
#75742
#70402
#70222