Finite Group Databasebeta

← #100548

Subgroups — #100548

Order 512.

436 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 98 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#33114C211yesmaximal, characteristic
2#20754C211yesmaximal, characteristic
3#8708C211yesmaximal, characteristic
4#7685C211yesmaximal, characteristic
5#7668C211yesmaximal, characteristic
6#7045C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C2×#485—21no—
9C2×#424—21no—
10#2857C2211yescharacteristic
11C4×#117C2211yescharacteristic
12#1981C411yescharacteristic
13#1981C411yescharacteristic
14#1948C411yescharacteristic
15#1947C411yescharacteristic
16#1928—21no—
17#1927—21no—
18#1873—21no—
19#1832—22no—
20#1831—22no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23C4×C22⋊C8C2211yescharacteristic
24#1729—21no—
25#1728C2211yescharacteristic
26C2×#336C2211yescharacteristic
27#1279—21no—
28#1279—21no—
29#1271—22no—
30#1271—22no—
31C22×M16—21no—
32C23×C8—21no—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C22⋊Q8—21no—
35C2×#117D411yescharacteristic
36C2×#117D411yescharacteristic
37C22×C42C2311yescharacteristic
38#485—42no—
39#485—42no—
40C2×#107—21no—
41C2×#106—21no—
42#424—42no—
43#424—42no—
44C2×C4⋊C8—24no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8C2×C411yescharacteristic
47C2×C4⋊C8C2×C411yescharacteristic
48C2×#103—21no—
49C2×#103—21no—
50C2×#103—21no—
51C2×#103—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C4×C8—21no—
56#391—21no—
57#389—21no—
58#389—21no—
59#384C2×C411yescharacteristic
60#380—21no—
61#380—21no—
62C4×C4⋊C4C2×C411yescharacteristic
63C4×C4⋊C4C2×C411yescharacteristic
64C4×C4⋊C4D411yescharacteristic
65C4×C4⋊C4D411yescharacteristic
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68#376—21no—
69C43—21no—
70#340—21no—
71#340—21no—
72#336—24no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336D412yes—
79#336D412yes—
80C22×Q8—21no—
81C23×C4C2×D411yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C22⋊Q8—41no—
105C22⋊Q8—41no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108#117—21no—
109#117—21no—
110#117—41no—
111#117—41no—
112#117D812yes—
113#117SD1612yes—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—41no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139#107—42no—
140#106—42no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—48no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146#103—42no—
147#103—42no—
148#103—42no—
149#103—42no—
150C22⋊C8—22no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—81no—
155C4×C8—41no—
156C4×C8—41no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—41no—
162#95—41no—
163C24#12311yescharacteristic
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4C4×D411yescharacteristic
191C22×C4#12011yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4C2×D811yescharacteristic
195C22×C4C2×SD1611yescharacteristic
196C22×C4AGL(1,ℤ/8)11yescharacteristic
197C22×C4#13711yescharacteristic
198M16—81no—
199M16—81no—
200M16—81no—
201M16—81no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—81no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C2×C8—81no—
222C2×C8—81no—
223C2×C8—81no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4#10212yes—
240C4⋊C4#10212yes—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C42—42no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—81no—
260C42—81no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23#38611yescharacteristic
268C23#48011yescharacteristic
269C23#48111yescharacteristic
270Q8—81no—
271Q8—81no—
272Q8—81no—
273Q8—81no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
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—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4#41011yescharacteristic
317C2×C4#41011yescharacteristic
318C2×C4C2×#10211yescharacteristic
319C2×C4#41711yescharacteristic
320C2×C4#42111yescharacteristic
321C2×C4#42111yescharacteristic
322C2×C4#44311yescharacteristic
323C2×C4#44411yescharacteristic
324C2×C4#44711yescharacteristic
325C2×C4#44811yescharacteristic
326C2×C4#47111yescharacteristic
327C2×C4#47311yescharacteristic
328C8—82no—
329C8—82no—
330C8—82no—
331C8—81no—
332C8—81no—
333C8—161no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#187411yescharacteristic
345C22#187611yescharacteristic
346C22#188111yescharacteristic
347C22#189711yescharacteristic
348C22#189911yescharacteristic
349C22#190211yescharacteristic
350C22#190811yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—41no—
358C4—42no—
359C4—41no—
360C4—42no—
361C4—81no—
362C4—81no—
363C4—161no—
364C4—161no—
365C4#151211yescharacteristic
366C4#151211yescharacteristic
367C4#151711yescharacteristic
368C4#151711yescharacteristic
369C2—41no—
370C2—41no—
371C2#899511yescharacteristic
372C2#915611yescharacteristic
373C2#917111yescharacteristic
374C2#921411yescharacteristic
375C2#924011yescharacteristic
376C2#931411yescharacteristic
377C2#934611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×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
#331142
#207542
#87082
#76852
#76682
#70452
#70292