Finite Group Databasebeta

← #101495

Subgroups — #101495

Order 512.

394 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 82 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#29262C211yesmaximal, characteristic
2#22393C211yesmaximal, characteristic
3#9868C211yesmaximal, characteristic, complemented
4#8581C211yesmaximal, characteristic
5#7624C211yesmaximal, characteristic, complemented
6#7041C211yesmaximal, characteristic, complemented
7#7022C211yesmaximal, characteristic, complemented
8C4×C4⋊Q8C2211yescharacteristic
9C4×#124C2211yescharacteristic
10#2096C411yescharacteristic
11#2093C411yescharacteristic
12#2024C411yescharacteristic
13#2023C411yescharacteristic
14#1985—21no—
15#1983—21no—
16#1949—21no—
17#1924—21no—
18#1923—21no—
19#1870—21no—
20#1864—21no—
21#1863—21no—
22#1851—21no—
23#1839C2211yescharacteristic
24#1831C2211yescharacteristic
25#1767C2211yescharacteristic
26#1767C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28#1753—21no—
29C2×#329—21no—
30#1295—21no—
31#1279—21no—
32#1279—21no—
33#1272—21no—
34#1272—21no—
35C22×M16—21no—
36C2×C4⋊Q8C2×C411yescharacteristic
37C2×#124D411yescharacteristic
38C2×C4×Q8—21no—
39C2×C4×Q8D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×#107—21no—
42C2×#106—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×#102—21no—
51C2×#102—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—21no—
56C2×#97—21no—
57C2×#97—21no—
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60#391—21no—
61#391C2×C411yescharacteristic
62#390C2×C411yescharacteristic
63#389—21no—
64#389—21no—
65#388—21no—
66#388—21no—
67#384C2×C411yescharacteristic
68#382D411yescharacteristic
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C43C2311yescharacteristic
74#340—21no—
75#340—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#329—42no—
89C22×Q8—21no—
90C22×Q8C22⋊C411yescharacteristic
91C22×D4C22⋊C411yescharacteristic
92C23×C4—21no—
93C2×M16—42no—
94C2×M16—81no—
95C2×M16—81no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—41no—
103C22×C8—41no—
104C4⋊Q8—42no—
105#124—21no—
106#124—21no—
107#124—21no—
108#124—21no—
109C4×Q8—81no—
110C4×Q8—82no—
111C4×D4—81no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C42—21no—
127C2×C42—41no—
128C2×C42C22×C411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42#4111yescharacteristic
134C2×C42#4111yescharacteristic
135#107—81no—
136#106—81no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—42no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—81no—
145#103—42no—
146#103—81no—
147#102—42no—
148#102—81no—
149C22⋊C8—42no—
150C22⋊C8—81no—
151C22⋊C8—81no—
152#97—41no—
153#97—41no—
154#97—41no—
155#97—41no—
156#97—81no—
157C4×C8—41no—
158C4×C8—41no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—41no—
164#95—41no—
165C24—21no—
166C2×Q8—22no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—81no—
173C2×Q8—81no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C4×D411yescharacteristic
191C22×C4C4×D411yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12311yescharacteristic
196M16—84no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—82no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42—41no—
251C42—41no—
252C42—82no—
253C42AGL(1,ℤ/8)11yescharacteristic
254C42#13711yescharacteristic
255C42#13711yescharacteristic
256C42#13711yescharacteristic
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#38611yescharacteristic
265Q8—42no—
266Q8—81no—
267Q8—82no—
268Q8—82no—
269D4—42no—
270D4—81no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4#41711yescharacteristic
307C2×C4#41911yescharacteristic
308C2×C4#44311yescharacteristic
309C2×C4#44411yescharacteristic
310C2×C4#45511yescharacteristic
311C2×C4#45611yescharacteristic
312C2×C4#46811yescharacteristic
313C2×C4#46811yescharacteristic
314C2×C4#47011yescharacteristic
315C2×C4#47011yescharacteristic
316C8—42no—
317C8—81no—
318C8—81no—
319C8—81no—
320C8—81no—
321C8—81no—
322C8—82no—
323C8—82no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#187611yescharacteristic
332C22#188211yescharacteristic
333C22#189111yescharacteristic
334C22#189411yescharacteristic
335C22#190311yescharacteristic
336C22#190611yescharacteristic
337C22#191111yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—82no—
352C4—81no—
353C4—81no—
354C4—161no—
355C2—161no—
356C2#905111yescharacteristic
357C2#920911yescharacteristic
358C2#928311yescharacteristic
359C2#931911yescharacteristic
360C2#937911yescharacteristic
361C2#948711yescharacteristic
362C2#953211yescharacteristic
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. #9868C2semidirect
5. #7624C2semidirect
6. #7041C2semidirect
7. #7022C2semidirect

Maximal subgroups

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

subgroupindex
#292622
#223932
#98682
#85812
#76242
#70412
#70222