Finite Group Databasebeta

← #103550

Subgroups — #103550

Order 512.

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

normal subgroups only

This group has 644 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#29757C211yesmaximal, characteristic
2#25813C211yesmaximal, characteristic
3C2×#2043C211yesmaximal, characteristic
4#9879C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7668C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#500—21no—
10C2×#484—21no—
11C22×#103—21no—
12C22×#103—21no—
13#2857C2211yescharacteristic
14C2×#389C2211yescharacteristic
15#2083—21no—
16#2068—22no—
17#2057—22no—
18#2043—22no—
19#2043—21no—
20#2043—22no—
21#1990—21no—
22#1872—21no—
23#1852—22no—
24#1797C2211yescharacteristic
25#1797C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#332—22no—
29#1280—22no—
30#1271—22no—
31C23×Q8—21no—
32C23×C8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—44no—
37C2×C4⋊Q8D412yes—
38C2×C4⋊Q8D411yescharacteristic
39C2×C22⋊Q8—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42C2311yescharacteristic
46#500—44no—
47#484—42no—
48#484—42no—
49C2×#107—24no—
50C2×#107—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D412yes—
53C2×#103—22no—
54C2×#103—24no—
55C2×#103—24no—
56C2×#103—22no—
57C2×#103—42no—
58C2×#103—44no—
59C2×#103—42no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8D412yes—
63C2×C4×C8—21no—
64#391—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#384D411yescharacteristic
72#380—21no—
73#380—21no—
74C2×#95—21no—
75#340—24no—
76#340—22no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336D412yes—
82#332—44no—
83C22×Q8—24no—
84C22×Q8—22no—
85C22×Q8—21no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8—41no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8—41no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—41no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—42no—
104C22×C8—42no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113C4⋊Q8—44no—
114C4⋊Q8—44no—
115C22⋊Q8—41no—
116C22⋊Q8—41no—
117C22⋊Q8—41no—
118C22⋊Q8—41no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
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—44no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4D812yes—
143C2×C22⋊C4—41no—
144C2×C42—42no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149#107—44no—
150#107—48no—
151C4⋊C8—42no—
152C4⋊C8—42no—
153#103—48no—
154#103—44no—
155#103—48no—
156#103—82no—
157C22⋊C8—22no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8—42no—
161C22⋊C8Q1614yes—
162C4×C8—42no—
163#95—41no—
164#95—41no—
165#95—41no—
166#95—41no—
167C24#12111yescharacteristic
168C2×Q8—24no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—42no—
178C2×Q8—41no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—82no—
184C2×Q8—82no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8—81no—
188C2×Q8—81no—
189C2×Q8—82no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—44no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—82no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4#12011yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4#12011yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12711yescharacteristic
219C22×C4C2×D811yescharacteristic
220C22×C4C2×Q1612yes—
221C22×C4#13511yescharacteristic
222C22×C4AGL(1,ℤ/8)11yescharacteristic
223C22×C4AGL(1,ℤ/8)11yescharacteristic
224C22×C4#13712yes—
225C2×C8—24no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—81no—
237C2×C8—81no—
238C2×C8—81no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—81no—
257C4⋊C4—82no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—82no—
261C22⋊C4—81no—
262C22⋊C4—81no—
263C42—22no—
264C42—22no—
265C42—41no—
266C42—41no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23#39211yescharacteristic
272C23#46212yes—
273C23#46611yescharacteristic
274C23#46911yescharacteristic
275C23#47312yes—
276Q8—44no—
277Q8—44no—
278Q8—82no—
279Q8—82no—
280Q8—84no—
281Q8—84no—
282Q8—82no—
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—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—82no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—81no—
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—162no—
322C2×C4#44711yescharacteristic
323C2×C4#44711yescharacteristic
324C2×C4#44911yescharacteristic
325C2×C4#44911yescharacteristic
326C2×C4#45112yes—
327C2×C4#45412yes—
328C2×C4B(4,𝔽2)12yes—
329C2×C4#47112yes—
330C2×C4#49411yescharacteristic
331C2×C4#49711yescharacteristic
332C8—44no—
333C8—82no—
334C8—82no—
335C8—161no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22#167412yes—
345C22#169412yes—
346C22#199911yescharacteristic
347C22#200311yescharacteristic
348C22#200911yescharacteristic
349C22#201611yescharacteristic
350C22#201911yescharacteristic
351C22#202011yescharacteristic
352C22#202111yescharacteristic
353C4—22no—
354C4—21no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—84no—
361C4—82no—
362C4—161no—
363C4—161no—
364C4—161no—
365C4#159712yes—
366C4#159812yes—
367C2—22no—
368C2—22no—
369C2#991511yescharacteristic
370C2#995711yescharacteristic
371C2#998911yescharacteristic
372C2#1013511yescharacteristic
373C2#1015711yescharacteristic
374C2#1018411yescharacteristic
375C2#1026511yescharacteristic
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

Maximal subgroups

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

subgroupindex
#297572
#258132
C2×#20432
#98792
#83402
#76682
#70292