Finite Group Databasebeta

← #103610

Subgroups — #103610

Order 512.

654 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 100 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#29715C211yesmaximal, characteristic
2#25811C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic
4#9878C211yesmaximal, characteristic
5#8342C211yesmaximal, characteristic
6#7668C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#498—21no—
10C2×#451—21no—
11C22×#103—21no—
12#2857C2211yescharacteristic
13C2×#391C2211yescharacteristic
14#2072—21no—
15#2058—22no—
16#2056—22no—
17#2024—22no—
18#2024—22no—
19#2016—21no—
20#1990—21no—
21#1872—21no—
22#1853—22no—
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#333—22no—
29#1280—22no—
30#1271—22no—
31C23×Q8—21no—
32C23×Q8D411yescharacteristic
33C22×Q16—22no—
34C23×C8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—44no—
39C2×C4⋊Q8D412yes—
40C2×C4⋊Q8D411yescharacteristic
41C2×C22⋊Q8—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D411yescharacteristic
45C22×C42C2311yescharacteristic
46#498—44no—
47#451—42no—
48#451—42no—
49C2×#106—24no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8D412yes—
52C2×#103—22no—
53C2×#103—24no—
54C2×#103—22no—
55C2×#103—44no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8D412yes—
59C2×C4×C8—21no—
60#391—21no—
61#391—21no—
62#391—21no—
63#391—21no—
64#391—21no—
65#389—21no—
66#389—21no—
67#384D411yescharacteristic
68#380—21no—
69#380—21no—
70C2×#95—21no—
71C2×#95—21no—
72#340—22no—
73#340—24no—
74#340—24no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336D412yes—
80#333—44no—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8—21no—
85C22×Q8—22no—
86C22×Q8—21no—
87C22×Q8—21no—
88C22×Q8—42no—
89C22×Q8—42no—
90C22×Q8—44no—
91C22×Q8—41no—
92C22×Q8—82no—
93C22×Q8—82no—
94C22×Q8SD1612yes—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×Q16—44no—
102C2×Q16—82no—
103C2×Q16—82no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C4⋊Q8—24no—
113C4⋊Q8—24no—
114C4⋊Q8—42no—
115C4⋊Q8—42no—
116C4⋊Q8—42no—
117C4⋊Q8—44no—
118C4⋊Q8—44no—
119C22⋊Q8—41no—
120C22⋊Q8—41no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C42—42no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145#106—48no—
146C4⋊C8—42no—
147C4⋊C8—42no—
148#103—48no—
149#103—44no—
150#103—82no—
151C22⋊C8—22no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—42no—
155C22⋊C8Q1614yescomplemented
156C4×C8—42no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163C24#12111yescharacteristic
164C2×Q8—24no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—82no—
182C2×Q8—82no—
183C2×Q8—82no—
184C2×Q8—82no—
185C2×Q8—82no—
186C2×Q8—816no—
187C2×Q8—82no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4#12011yescharacteristic
211C22×C4#12011yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12711yescharacteristic
216C22×C4C2×SD1611yescharacteristic
217C22×C4C2×Q1612yes—
218C22×C4#13511yescharacteristic
219C22×C4AGL(1,ℤ/8)11yescharacteristic
220C22×C4#13711yescharacteristic
221C22×C4#13712yes—
222Q16—88no—
223C2×C8—24no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—81no—
235C2×C8—81no—
236C2×C8—81no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C42—22no—
256C42—22no—
257C42—41no—
258C42—41no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23#39211yescharacteristic
264C23#46212yes—
265C23#46511yescharacteristic
266C23#46811yescharacteristic
267C23#47112yes—
268Q8—44no—
269Q8—44no—
270Q8—82no—
271Q8—88no—
272Q8—84no—
273Q8—84no—
274Q8—82no—
275Q8—82no—
276Q8—82no—
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—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—88no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—162no—
314C2×C4#44811yescharacteristic
315C2×C4#44911yescharacteristic
316C2×C4#45011yescharacteristic
317C2×C4#45112yes—
318C2×C4#45211yescharacteristic
319C2×C4#45512yes—
320C2×C4B(4,𝔽2)12yes—
321C2×C4#47112yes—
322C2×C4#49411yescharacteristic
323C2×C4#49711yescharacteristic
324C8—44no—
325C8—82no—
326C8—82no—
327C8—161no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22#167212yes—
337C22#169212yes—
338C22#200011yescharacteristic
339C22#200411yescharacteristic
340C22#200911yescharacteristic
341C22#201611yescharacteristic
342C22#201911yescharacteristic
343C22#201911yescharacteristic
344C22#202111yescharacteristic
345C4—22no—
346C4—21no—
347C4—21no—
348C4—22no—
349C4—21no—
350C4—21no—
351C4—41no—
352C4—84no—
353C4—82no—
354C4—161no—
355C4—161no—
356C4—161no—
357C4#160112yes—
358C4#160612yes—
359C2—22no—
360C2—22no—
361C2#991611yescharacteristic
362C2#995711yescharacteristic
363C2#999011yescharacteristic
364C2#1014111yescharacteristic
365C2#1016311yescharacteristic
366C2#1018911yescharacteristic
367C2#1027311yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C22×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
155. C22⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#297152
#258112
C2×#20242
#98782
#83422
#76682
#70292