Finite Group Databasebeta

← #99474

Subgroups — #99474

Order 512.

412 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 94 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#22297C211yesmaximal, characteristic
2#11326C211yesmaximal, characteristic
3#8942C211yesmaximal, characteristic
4#8833C211yesmaximal, characteristic
5#7566C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic
7#7026C211yesmaximal, characteristic
8C22×#108—21no—
9C2×#423C2211yescharacteristic
10C4×C22⋊Q8C2211yescharacteristic
11#1985—21no—
12#1939—21no—
13#1917—21no—
14#1916—21no—
15#1851—21no—
16#1797C2211yescharacteristic
17#1753—21no—
18C4×C22⋊C8C2211yescharacteristic
19C4×C22⋊C8C2211yescharacteristic
20C2×#341—21no—
21#1728C2211yescharacteristic
22C2×#336C2211yescharacteristic
23C2×#334—21no—
24C2×Q8⋊C8—21no—
25#1279—21no—
26#1277—21no—
27#1272C411yescharacteristic
28#1272C411yescharacteristic
29#1271C411yescharacteristic
30#1271C411yescharacteristic
31C22×M16—21no—
32C22×M16D411yescharacteristic
33C22×M16D411yescharacteristic
34C23×C8—21no—
35C2×C22⋊Q8Q811yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C22×C42C2311yescharacteristic
38C2×#108—42no—
39C2×#108—42no—
40C2×#108—42no—
41#423—42no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×#97—21no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55#391C2×C411yescharacteristic
56#389C2×C411yescharacteristic
57#387—21no—
58#385—21no—
59#380Q811yescharacteristic
60C4×C4⋊C4—21no—
61C4×C4⋊C4D411yescharacteristic
62C4×C22⋊C4—21no—
63#341—42no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336C2×C411yescharacteristic
74#336C2×C411yescharacteristic
75#336C2×C411yescharacteristic
76#336C2×C411yescharacteristic
77#334—42no—
78Q8⋊C8—42no—
79C22×Q8C4⋊C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×Q811yescharacteristic
82C23×C4#4111yescharacteristic
83C2×M16—21no—
84C2×M16—22no—
85C2×M16—21no—
86C2×M16—21no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16SD1612yes—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C22⋊Q8—22no—
114C22⋊Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4C4⋊C411yescharacteristic
121C2×C4⋊C4C4⋊C411yescharacteristic
122C2×C4⋊C4C4⋊C411yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42C22×C411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133#108—48no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—81no—
141C4⋊C8—81no—
142C22⋊C8—22no—
143C22⋊C8—42no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147#97—81no—
148C4×C8—41no—
149C4×C8—41no—
150C4×C8—41no—
151C4×C8—41no—
152C4×C8—81no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158C24C22⋊Q811yescharacteristic
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—41no—
162C2×Q8—41no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4C2×C4⋊C411yescharacteristic
180C22×C4C4×D411yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4#12311yescharacteristic
185C22×C4C2×SD1611yescharacteristic
186C22×C4#13511yescharacteristic
187C22×C4#13711yescharacteristic
188C22×C4#13711yescharacteristic
189M16—42no—
190M16—42no—
191M16—42no—
192M16—42no—
193M16—81no—
194M16—81no—
195M16—81no—
196M16—81no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—42no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—81no—
228C2×C8—81no—
229C2×C8—81no—
230C2×C8—81no—
231C4⋊C4—21no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—81no—
238C4⋊C4#10812yescomplemented
239C22⋊C4—42no—
240C22⋊C4—81no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42—41no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23#38511yescharacteristic
258C23#47511yescharacteristic
259C23#47911yescharacteristic
260Q8—42no—
261Q8—81no—
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—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4C2×#10811yescharacteristic
299C2×C4#43011yescharacteristic
300C2×C4C4×SD1611yescharacteristic
301C2×C4#44111yescharacteristic
302C2×C4#44311yescharacteristic
303C2×C4#44411yescharacteristic
304C2×C4#44811yescharacteristic
305C2×C4#45211yescharacteristic
306C2×C4#45311yescharacteristic
307C2×C4#45611yescharacteristic
308C2×C4B(4,𝔽2)11yescharacteristic
309C2×C4#45811yescharacteristic
310C8—42no—
311C8—42no—
312C8—42no—
313C8—42no—
314C8—81no—
315C8—82no—
316C8—82no—
317C8—81no—
318C8—81no—
319C8—81no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22#185511yescharacteristic
331C22#185811yescharacteristic
332C22#185911yescharacteristic
333C22#186111yescharacteristic
334C22#186511yescharacteristic
335C22#186611yescharacteristic
336C22#186711yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—42no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4#159611yescharacteristic
350C4#160311yescharacteristic
351C4#161111yescharacteristic
352C4#161811yescharacteristic
353C2—41no—
354C2—41no—
355C2#886011yescharacteristic
356C2#887111yescharacteristic
357C2#887511yescharacteristic
358C2#889811yescharacteristic
359C2#890911yescharacteristic
360C2#891611yescharacteristic
361C2#893111yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#385
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
238. C4⋊C4#108semidirect

Maximal subgroups

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

subgroupindex
#222972
#113262
#89422
#88332
#75662
#70292
#70262