Finite Group Databasebeta

← #100016

Subgroups — #100016

Order 512.

447 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 108 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#22639C211yesmaximal, characteristic
2#21034C211yesmaximal, characteristic
3#9879C211yesmaximal, characteristic
4#8708C211yesmaximal, characteristic
5#7573C211yesmaximal, characteristic
6#7034C211yesmaximal, characteristic
7#7026C211yesmaximal, characteristic
8C2×#428C411yescharacteristic
9C2×#389C2211yescharacteristic
10C4×#117C2211yescharacteristic
11#1939—21no—
12#1923—22no—
13#1921—21no—
14#1920—21no—
15#1919—21no—
16#1918—21no—
17#1839—21no—
18#1797C2211yescharacteristic
19#1797C2211yescharacteristic
20C8×C4⋊C4—21no—
21C4×C4⋊C8—21no—
22C4×C22⋊C8C2211yescharacteristic
23C4×C22⋊C8C2211yescharacteristic
24#1736C411yescharacteristic
25#1736C411yescharacteristic
26#1736C411yescharacteristic
27#1728C2211yescharacteristic
28#1280—22no—
29#1279—21no—
30#1279—21no—
31#1271—21no—
32#1271—21no—
33C22×M16C2×C411yescharacteristic
34C22×M16C2×C411yescharacteristic
35C2×#117C2×C411yescharacteristic
36C2×#117C2×C411yescharacteristic
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C42C2311yescharacteristic
41#428—22no—
42#428C814yescomplemented
43C2×#107—21no—
44C2×#106—21no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8D412yes—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—22no—
55#389—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59C4×C4⋊C4D411yescharacteristic
60C4×C4⋊C4D411yescharacteristic
61C4×C4⋊C4D411yescharacteristic
62C4×C4⋊C4D411yescharacteristic
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65#376—21no—
66C2×#95—21no—
67C43—21no—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#336—22no—
75#336—22no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336D411yescharacteristic
82#336D411yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4#4111yescharacteristic
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—21no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16C2×C812yes—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—41no—
102C22×C8—41no—
103#117—21no—
104#117—21no—
105#117—22no—
106#117—41no—
107#117C2×C812yes—
108#117M1612yes—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—81no—
121C2×C4⋊C4—81no—
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C4⋊C4C22⋊C411yescharacteristic
126C2×C4⋊C4C2×C812yes—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141#107—24no—
142#106—24no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—44no—
148C22⋊C8—22no—
149C22⋊C8—42no—
150C22⋊C8—41no—
151C22⋊C8—42no—
152C22⋊C8—81no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8—82no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—41no—
163#95—41no—
164C24C4×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C4×D411yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4C22×C811yescharacteristic
197C22×C4C2×M1611yescharacteristic
198C22×C4#13111yescharacteristic
199C22×C4#13111yescharacteristic
200M16—42no—
201M16—42no—
202M16—81no—
203M16—81no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—41no—
220C2×C8—81no—
221C2×C8—81no—
222C2×C8—81no—
223C2×C8—81no—
224C2×C8—81no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—81no—
237C4⋊C4—82no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4C22⋊C812yes—
241C4⋊C4C22⋊C812yes—
242C4⋊C4#10112yes—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C42—42no—
255C42—42no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—81no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23#38611yescharacteristic
268C23C8×D411yescharacteristic
269C23#43511yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
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—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—162no—
315C2×C4C2×C22⋊C811yescharacteristic
316C2×C4#40811yescharacteristic
317C2×C4#41011yescharacteristic
318C2×C4#41011yescharacteristic
319C2×C4C2×#10111yescharacteristic
320C2×C4#41311yescharacteristic
321C2×C4C8×D411yescharacteristic
322C2×C4#43511yescharacteristic
323C2×C4#45511yescharacteristic
324C2×C4#45611yescharacteristic
325C2×C4#47111yescharacteristic
326C2×C4#47211yescharacteristic
327C8—42no—
328C8—42no—
329C8—82no—
330C8—81no—
331C8—82no—
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#186911yescharacteristic
345C22#188111yescharacteristic
346C22#188211yescharacteristic
347C22#189911yescharacteristic
348C22#190111yescharacteristic
349C22#190611yescharacteristic
350C22#190711yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—41no—
358C4—42no—
359C4—42no—
360C4—41no—
361C4—42no—
362C4—82no—
363C4—81no—
364C4—161no—
365C4#145811yescharacteristic
366C4#146111yescharacteristic
367C4#147011yescharacteristic
368C4#147311yescharacteristic
369C2—41no—
370C2—41no—
371C2#895311yescharacteristic
372C2#908211yescharacteristic
373C2#909511yescharacteristic
374C2#934811yescharacteristic
375C2#935211yescharacteristic
376C2#935511yescharacteristic
377C2#936211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
42. #428C8semidirect

Maximal subgroups

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

subgroupindex
#226392
#210342
#98792
#87082
#75732
#70342
#70262