Finite Group Databasebeta

← #104091

Subgroups — #104091

Order 512.

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

normal subgroups only

This group has 742 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#29953C211yesmaximal, characteristic
2#25722C211yesmaximal, characteristic
3#15191C211yesmaximal, characteristic
4#9908C211yesmaximal, characteristic
5#8342C211yesmaximal, characteristic
6#7575C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#494—21no—
10C2×#451—22no—
11#2857C2211yescharacteristic
12C2×#391—21no—
13C2×#391C2211yescharacteristic
14#2095—21no—
15#2068—22no—
16#2024—21no—
17#2016—22no—
18#1956—21no—
19#1852—21no—
20#1797C2211yescharacteristic
21#1797C2211yescharacteristic
22#1787—21no—
23C4×C22⋊C8C2211yescharacteristic
24C2×#333—22no—
25#1446—22no—
26#1442—22no—
27C4⋊#95—21no—
28C4×#95C2211yescharacteristic
29#1294—22no—
30#1272—22no—
31C23×Q8D411yescharacteristic
32C23×Q8D412yes—
33C22×Q16—24no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—44no—
39C2×C4⋊Q8D411yescharacteristic
40C2×C22⋊Q8—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#494—44no—
47#451—48no—
48C2×#107—22no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×#103—24no—
52C2×#103—22no—
53C2×#103—22no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C4×C8—21no—
58#391—21no—
59#391—21no—
60#391—21no—
61#391—21no—
62#391—21no—
63#391—44no—
64#389—21no—
65#389—21no—
66#384D411yescharacteristic
67#380—21no—
68#380—21no—
69C2×#95—24no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—24no—
74C2×#95D412yes—
75#340—22no—
76#336—21no—
77#336—21no—
78#336D412yes—
79#333—44no—
80C22×Q8—21no—
81C22×Q8—22no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8—22no—
85C22×Q8—21no—
86C22×Q8—21no—
87C22×Q8—21no—
88C22×Q8—44no—
89C22×Q8—42no—
90C22×Q8—42no—
91C22×Q8—44no—
92C22×Q8—44no—
93C22×Q8Q1612yescomplemented
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×Q16—48no—
102C2×Q16—48no—
103C2×Q16—84no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—41no—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—44no—
113C4⋊Q8—44no—
114C22⋊Q8—44no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141#107—82no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144#103—48no—
145#103—82no—
146#103—82no—
147C22⋊C8—24no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—42no—
151C4×C8—42no—
152#95—22no—
153#95—42no—
154#95—48no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—84no—
162#95Q1614yescomplemented
163C24#12011yescharacteristic
164C2×Q8—24no—
165C2×Q8—24no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—44no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—44no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—82no—
182C2×Q8—82no—
183C2×Q8—81no—
184C2×Q8—81no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8—816no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4#12011yescharacteristic
219C22×C4#12011yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12711yescharacteristic
224C22×C4C2×Q1611yescharacteristic
225C22×C4C2×Q1612yes—
226C22×C4#13511yescharacteristic
227C22×C4#13711yescharacteristic
228C22×C4#13711yescharacteristic
229C22×C4#13712yes—
230Q16—816no—
231C2×C8—24no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—81no—
238C2×C8—81no—
239C2×C8—81no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—82no—
252C22⋊C4—82no—
253C42—22no—
254C42—22no—
255C42—41no—
256C42—41no—
257C42—82no—
258C42—82no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23#39211yescharacteristic
264C23#45112yes—
265C23#45111yescharacteristic
266C23#45211yescharacteristic
267C23#45312yes—
268Q8—48no—
269Q8—44no—
270Q8—82no—
271Q8—84no—
272Q8—82no—
273Q8—84no—
274Q8—84no—
275Q8—84no—
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—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4#45111yescharacteristic
323C2×C4#45211yescharacteristic
324C2×C4#45512yes—
325C2×C4#45512yes—
326C2×C4B(4,𝔽2)12yes—
327C2×C4#46212yes—
328C2×C4#46711yescharacteristic
329C2×C4#47011yescharacteristic
330C2×C4#49411yescharacteristic
331C2×C4#49711yescharacteristic
332C8—44no—
333C8—82no—
334C8—161no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22#163212yes—
344C22#163312yes—
345C22#200111yescharacteristic
346C22#200211yescharacteristic
347C22#200411yescharacteristic
348C22#200411yescharacteristic
349C22#200911yescharacteristic
350C22#201611yescharacteristic
351C22#202111yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—41no—
359C4—84no—
360C4—84no—
361C4—82no—
362C4—161no—
363C4—161no—
364C4—161no—
365C4#160212yes—
366C4#160612yes—
367C2—22no—
368C2—22no—
369C2#992411yescharacteristic
370C2#995711yescharacteristic
371C2#1005711yescharacteristic
372C2#1009011yescharacteristic
373C2#1025211yescharacteristic
374C2#1029711yescharacteristic
375C2#1032911yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#120
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
93. C22×Q8Q16semidirect
162. #95Q16semidirect

Maximal subgroups

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

subgroupindex
#299532
#257222
#151912
#99082
#83422
#75752
#70282