Finite Group Databasebeta

← #104138

Subgroups — #104138

Order 512.

676 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 96 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#29757C211yesmaximal, characteristic
2#25811C211yesmaximal, characteristic
3#15230C211yesmaximal, characteristic
4#9909C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7576C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#498—21no—
10C2×#484—21no—
11#2857C2211yescharacteristic
12C2×#389C2211yescharacteristic
13C2×#382—21no—
14#2083—21no—
15#2068—22no—
16#2058—22no—
17#2043—21no—
18#1990—21no—
19#1872—21no—
20#1853—22no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23C4×C22⋊C8C2211yescharacteristic
24C2×#332—22no—
25#1446—22no—
26#1441—22no—
27C4⋊#95—21no—
28C4×#95C2211yescharacteristic
29#1294—22no—
30#1271—22no—
31C23×Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—44no—
37C2×C4⋊Q8D411yescharacteristic
38C2×C22⋊Q8—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#498—44no—
47#484—42no—
48#484—42no—
49C2×#107—22no—
50C2×#106—24no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D412yes—
53C2×#103—22no—
54C2×#103—22no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C4×C8—21no—
59#391—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#384D411yescharacteristic
67#382—44no—
68#380—21no—
69#380—21no—
70C2×#95—24no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—24no—
74C2×#95D412yes—
75#340—24no—
76#340—22no—
77#336—21no—
78#336—21no—
79#336D412yes—
80#332—44no—
81C22×Q8—21no—
82C22×Q8—44no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8—82no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—41no—
98C4⋊Q8—24no—
99C4⋊Q8—24no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C4⋊Q8—44no—
104C4⋊Q8—44no—
105C22⋊Q8—41no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
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—21no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4SD1612yes—
134C2×C22⋊C4—41no—
135C2×C42—24no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144#107—44no—
145#106—48no—
146C4⋊C8—42no—
147C4⋊C8—42no—
148#103—44no—
149#103—82no—
150C22⋊C8—24no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—42no—
154C4×C8—42no—
155#95—22no—
156#95—22no—
157#95—42no—
158#95—48no—
159#95—41no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95—84no—
164C24#12011yescharacteristic
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×Q8—82no—
173C2×Q8—88no—
174C2×Q8—88no—
175C2×Q8—82no—
176C2×Q8—81no—
177C2×Q8—81no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4#12011yescharacteristic
209C22×C4#12011yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12711yescharacteristic
214C22×C4C2×SD1611yescharacteristic
215C22×C4#13511yescharacteristic
216C22×C4#13512yes—
217C22×C4AGL(1,ℤ/8)11yescharacteristic
218C22×C4#13711yescharacteristic
219C22×C4#13712yes—
220C2×C8—24no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—81no—
227C2×C8—81no—
228C2×C8—81no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—44no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—81no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C42—22no—
254C42—22no—
255C42—41no—
256C42—41no—
257C42—44no—
258C42—44no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23#39211yescharacteristic
264C23#44811yescharacteristic
265C23#44911yescharacteristic
266C23#45212yes—
267C23#45612yes—
268Q8—84no—
269Q8—88no—
270Q8—84no—
271Q8—82no—
272Q8—82no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—84no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—84no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4#45011yescharacteristic
319C2×C4#45211yescharacteristic
320C2×C4#45412yes—
321C2×C4#45612yes—
322C2×C4B(4,𝔽2)12yes—
323C2×C4#46412yes—
324C2×C4#46511yescharacteristic
325C2×C4#46811yescharacteristic
326C2×C4#49511yescharacteristic
327C2×C4#49711yescharacteristic
328C8—44no—
329C8—82no—
330C8—161no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22#165212yes—
340C22#165312yes—
341C22#200011yescharacteristic
342C22#200311yescharacteristic
343C22#200511yescharacteristic
344C22#200511yescharacteristic
345C22#200811yescharacteristic
346C22#201511yescharacteristic
347C22#202111yescharacteristic
348C4—22no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—41no—
355C4—84no—
356C4—82no—
357C4—84no—
358C4—161no—
359C4—161no—
360C4—161no—
361C4#160312yes—
362C4#160412yes—
363C2—22no—
364C2—22no—
365C2#991711yescharacteristic
366C2#995511yescharacteristic
367C2#1006911yescharacteristic
368C2#1010111yescharacteristic
369C2#1024111yescharacteristic
370C2#1029611yescharacteristic
371C2#1033011yescharacteristic
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

Maximal subgroups

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

subgroupindex
#297572
#258112
#152302
#99092
#83402
#75762
#70282