Finite Group Databasebeta

← #98968

Subgroups — #98968

Order 512.

473 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 110 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#21487C211yesmaximal, characteristic
2#10864C211yesmaximal, characteristic, complemented
3#8596C211yesmaximal, characteristic, complemented
4#7616C212yesmaximal
5#7045C212yesmaximal, complemented
6C2×#436C411yescharacteristic
7C2×#435—21no—
8C22×#108—21no—
9C22×C4×C8C2211yescharacteristic
10C4×#121C2211yescharacteristic
11#1954C411yescharacteristic
12#1939C411yescharacteristic
13#1923—21no—
14#1869—21no—
15#1797C2212yes—
16#1797C2211yescharacteristic
17#1770C411yescharacteristic
18#1753—21no—
19C2×#341—21no—
20#1728C2212yes—
21C2×#335—21no—
22#1277—21no—
23#1271—22no—
24#1270—22no—
25C22×M16—21no—
26C22×M16—22no—
27C23×C8D411yescharacteristic
28C23×C8D411yescharacteristic
29C2×#121Q811yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4C2×C411yescharacteristic
32C22×C42C2311yescharacteristic
33#436—42no—
34#435—42no—
35#435—42no—
36C2×#108—24no—
37C2×#108—42no—
38C2×#108—42no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—22no—
48C2×#97—21no—
49C2×C4×C8—42no—
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8C2×C411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#390D411yescharacteristic
55#389D411yescharacteristic
56#388—21no—
57#386—21no—
58#380Q811yescharacteristic
59C4×C4⋊C4C2×C411yescharacteristic
60C4×C22⋊C4—21no—
61#341—42no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—22no—
66#336—22no—
67#336—22no—
68#335—42no—
69C22×D4—21no—
70C22×D4C4⋊C411yescharacteristic
71C23×C4—21no—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×Q811yescharacteristic
74C23×C4#4111yescharacteristic
75C2×M16—24no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C2×M16—42no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8D812yes—
99C22×C8SD1612yes—
100#121—42no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—41no—
104C4×D4—41no—
105C4×D4—41no—
106C4×D4—41no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C4⋊C411yescharacteristic
109C2×C4⋊C4C4⋊C411yescharacteristic
110C2×C4⋊C4C4⋊C411yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117C2×C42—41no—
118C2×C42—41no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42#4111yescharacteristic
123#108—48no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—42no—
127C4⋊C8—42no—
128C4⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—42no—
134C22⋊C8—82no—
135#97—42no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—41no—
139C4×C8—41no—
140#95—21no—
141#95—21no—
142#95—21no—
143C24—21no—
144C24C22⋊Q811yescharacteristic
145C2×D4—22no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—41no—
151C2×D4—41no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C2×C4⋊C411yescharacteristic
174C22×C4#11711yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4C2×D811yescharacteristic
179C22×C4C2×SD1611yescharacteristic
180C22×C4AGL(1,ℤ/8)11yescharacteristic
181C22×C4#13711yescharacteristic
182M16—44no—
183M16—44no—
184M16—82no—
185M16—82no—
186M16—84no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—24no—
190C2×C8—24no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—44no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8—82no—
210C2×C8—82no—
211C2×C8#9912yes—
212C2×C8#10212yes—
213C2×C8#10212yes—
214C2×C8#10412yes—
215C4⋊C4—22no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—41no—
233C42—41no—
234C42—41no—
235C42—41no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#38011yescharacteristic
250C23#47411yescharacteristic
251C23#47611yescharacteristic
252D4—82no—
253D4—82no—
254D4—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
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—44no—
276C2×C4—42no—
277C2×C4—42no—
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×C4C2×#9911yescharacteristic
292C2×C4#41011yescharacteristic
293C2×C4C2×#10211yescharacteristic
294C2×C4#41711yescharacteristic
295C2×C4C2×#10411yescharacteristic
296C2×C4#42111yescharacteristic
297C2×C4#43012yes—
298C2×C4#45911yescharacteristic
299C2×C4#46111yescharacteristic
300C2×C4#47211yescharacteristic
301C2×C4#47311yescharacteristic
302C8—22no—
303C8—41no—
304C8—41no—
305C8—84no—
306C8—81no—
307C8—81no—
308C8—162no—
309C8#32714yes—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#179911yescharacteristic
328C22#180211yescharacteristic
329C22#180611yescharacteristic
330C22#181111yescharacteristic
331C22#181311yescharacteristic
332C22#181411yescharacteristic
333C22#181611yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4C2×#32711yescharacteristic
345C4#150911yescharacteristic
346C4#151311yescharacteristic
347C4#151711yescharacteristic
348C2—41no—
349C2—41no—
350C2—161no—
351C2#857111yescharacteristic
352C2#861311yescharacteristic
353C2#862811yescharacteristic
354C2#866211yescharacteristic
355C2#867311yescharacteristic
356C2#868411yescharacteristic
357C2#869311yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10864C2semidirect
3. #8596C2semidirect
5. #7045C2semidirect

Maximal subgroups

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

subgroupindex
#214872
#108642
#85962
#76162
#70452