Finite Group Databasebeta

← #100004

Subgroups — #100004

Order 512.

420 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 84 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#23344C211yesmaximal, characteristic
2#22050C211yesmaximal, characteristic
3#9878C211yesmaximal, characteristic
4#7595C211yesmaximal, characteristic
5#7583C211yesmaximal, characteristic
6#7033C211yesmaximal, characteristic
7#7021C211yesmaximal, characteristic
8C2×#451C411yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10C2×#391C2211yescharacteristic
11#1985—21no—
12#1971—21no—
13#1940—21no—
14#1935—21no—
15#1927—21no—
16#1924—21no—
17#1924—21no—
18#1917—21no—
19#1917—21no—
20#1878—21no—
21#1872C411yescharacteristic
22#1853C411yescharacteristic
23#1817C2211yescharacteristic
24#1817C2211yescharacteristic
25#1797C2211yescharacteristic
26#1787C411yescharacteristic
27#1749C2211yescharacteristic
28C4×C22⋊C8C2211yescharacteristic
29#1280—21no—
30#1280—21no—
31#1272—21no—
32#1272—21no—
33#1272—21no—
34#1271—21no—
35C23×Q8C2×C411yescharacteristic
36C22×Q16—21no—
37C2×C22⋊Q8C2×C411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C42C2311yescharacteristic
41#451—22no—
42#451—22no—
43C2×#106—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×#103—21no—
48C2×#103—21no—
49C2×#103—21no—
50C2×#103—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C411yescharacteristic
55C2×C22⋊C8C2×C411yescharacteristic
56C2×C22⋊C8D411yescharacteristic
57C2×C22⋊C8D411yescharacteristic
58C2×#97—21no—
59C2×#97—21no—
60C2×#97—21no—
61C2×C4×C8—21no—
62#391—21no—
63#391—21no—
64#391—21no—
65#391—21no—
66#391D411yescharacteristic
67#389D411yescharacteristic
68#387—21no—
69#385—21no—
70#380C2×C411yescharacteristic
71C4×C4⋊C4—21no—
72C4×C4⋊C4D411yescharacteristic
73C4×C22⋊C4—21no—
74C2×#95—21no—
75C2×#95—21no—
76#340—21no—
77#340—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336D411yescharacteristic
89#336D411yescharacteristic
90C22×Q8—21no—
91C22×Q8—21no—
92C22×Q8—21no—
93C22×Q8—21no—
94C22×Q8—21no—
95C22×Q8—81no—
96C22×Q8—81no—
97C22×Q8C22⋊C411yescharacteristic
98C22×Q8M1612yes—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C2×Q16—24no—
105C2×Q16—81no—
106C2×Q16—81no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—41no—
114C22×C8—41no—
115C22⋊Q8—21no—
116C22⋊Q8—21no—
117C22⋊Q8—21no—
118C22⋊Q8—21no—
119C4×Q8—81no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—81no—
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—81no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138#106—81no—
139C4⋊C8—81no—
140C4⋊C8—81no—
141C4⋊C8—81no—
142#103—24no—
143#103—81no—
144#103—81no—
145#103—81no—
146C22⋊C8—21no—
147C22⋊C8—22no—
148C22⋊C8—21no—
149C22⋊C8—21no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—81no—
154C22⋊C8—81no—
155C22⋊C8M1612yes—
156#97—81no—
157#97—81no—
158#97—81no—
159C4×C8—81no—
160#95—21no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—41no—
166#95—41no—
167#95—41no—
168#95—41no—
169C24C4×D411yescharacteristic
170C2×Q8—22no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—81no—
183C2×Q8—81no—
184C2×Q8—82no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12311yescharacteristic
216C22×C4C2×M1611yescharacteristic
217C22×C4C2×M1611yescharacteristic
218C22×C4#13111yescharacteristic
219C22×C4#13111yescharacteristic
220Q16—84no—
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—42no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—81no—
241C2×C8—81no—
242C2×C8—81no—
243C2×C8—81no—
244C2×C8—81no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—81no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—81no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—41no—
263C42—41no—
264C42—41no—
265C42—41no—
266C42—41no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23#38611yescharacteristic
274C23#43511yescharacteristic
275C23#43611yescharacteristic
276Q8—42no—
277Q8—42no—
278Q8—42no—
279Q8—81no—
280Q8—82no—
281Q8—81no—
282Q8—81no—
283Q8—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—162no—
327C2×C4#40711yescharacteristic
328C2×C4#40811yescharacteristic
329C2×C4#43511yescharacteristic
330C2×C4#43611yescharacteristic
331C2×C4#45511yescharacteristic
332C2×C4#45611yescharacteristic
333C8—42no—
334C8—81no—
335C8—81no—
336C8—81no—
337C8—81no—
338C8—81no—
339C8—82no—
340C8—161no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22#187011yescharacteristic
352C22#188111yescharacteristic
353C22#188211yescharacteristic
354C22#190211yescharacteristic
355C22#190511yescharacteristic
356C22#191311yescharacteristic
357C22#191411yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—82no—
371C4—81no—
372C4—161no—
373C2—41no—
374C2—41no—
375C2#895511yescharacteristic
376C2#909211yescharacteristic
377C2#910911yescharacteristic
378C2#933111yescharacteristic
379C2#933411yescharacteristic
380C2#933711yescharacteristic
381C2#934211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#233442
#220502
#98782
#75952
#75832
#70332
#70212