Finite Group Databasebeta

← #100366

Subgroups — #100366

Order 512.

720 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 112 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#32957C211yesmaximal, characteristic
2#13817C211yesmaximal, characteristic
3#9908C211yesmaximal, characteristic
4#9679C211yesmaximal, characteristic
5#8311C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#477C412yescomplemented
10#2857C2211yescharacteristic
11C2×#391—21no—
12C2×C4×C4⋊C4—21no—
13#1863—22no—
14#1852—21no—
15#1803C412yescomplemented
16#1797C2211yescharacteristic
17#1797C2211yescharacteristic
18#1787—21no—
19C4×C22⋊C8C2211yescharacteristic
20C2×#328—22no—
21C2×Q8⋊C8—22no—
22#1436—22no—
23C4⋊#95C2211yescharacteristic
24#1434—21no—
25C4×#95—21no—
26C4×#95C2211yescharacteristic
27#1294—21no—
28#1294—21no—
29C23×Q8D411yescharacteristic
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×C22⋊Q8—21no—
32C2×C4×Q8—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8—44no—
35C2×C4×Q8C2×C412yes—
36C2×C4×Q8D412yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C42—21no—
41C22×C42—21no—
42C22×C42—22no—
43C22×C42C2311yescharacteristic
44#477—44no—
45C2×#107—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8C2×C412yes—
48C2×#103—22no—
49C2×#103—22no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C4×C8—21no—
54#391—28no—
55#391—21no—
56#389—21no—
57#389—21no—
58#384C2×C411yescharacteristic
59#380—21no—
60#380—21no—
61C4×C4⋊C4—44no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95D411yescharacteristic
68C2×#95D411yescharacteristic
69C2×#95D412yes—
70#340—22no—
71#336—21no—
72#336—21no—
73#336—21no—
74#328—44no—
75Q8⋊C8—44no—
76C22×Q8—24no—
77C22×Q8—21no—
78C22×Q8—21no—
79C22×Q8—21no—
80C22×Q8—21no—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—42no—
84C22×Q8C22⋊C412yes—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—22no—
94C22×C8—41no—
95C22×C8—41no—
96C4⋊Q8—42no—
97C22⋊Q8—42no—
98C22⋊Q8—42no—
99C4×Q8—24no—
100C4×Q8—24no—
101C4×Q8—42no—
102C4×Q8—42no—
103C4×Q8—44no—
104C4×Q8—44no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4C22⋊C412yes—
121C2×C22⋊C4—41no—
122C2×C42—24no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—42no—
126C2×C42—44no—
127C2×C42—42no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—44no—
131C2×C42—44no—
132C2×C42—82no—
133C2×C42C22×C411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137#107—44no—
138C4⋊C8—42no—
139C4⋊C8—42no—
140#103—44no—
141#103—44no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C4×C8—42no—
149#95—28no—
150#95—28no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—42no—
155#95—42no—
156#95—44no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95Q1612yes—
162C24#12011yescharacteristic
163C2×Q8—24no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—81no—
179C2×Q8#10414yes—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4C2×Q1611yescharacteristic
228C22×C4#13511yescharacteristic
229C22×C4#13511yescharacteristic
230C22×C4#13511yescharacteristic
231C22×C4#13711yescharacteristic
232C22×C4#13711yescharacteristic
233C22×C4#13711yescharacteristic
234C22×C4#13711yescharacteristic
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—81no—
240C2×C8—81no—
241C2×C8—81no—
242C2×C8—81no—
243C2×C8—81no—
244C4⋊C4—24no—
245C4⋊C4—22no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—81no—
254C4⋊C4—81no—
255C4⋊C4—82no—
256C4⋊C4#9914yes—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C42—22no—
260C42—22no—
261C42—41no—
262C42—41no—
263C42—44no—
264C42—44no—
265C42—44no—
266C42—44no—
267C42—88no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23#38611yescharacteristic
273C23#45111yescharacteristic
274C23#45211yescharacteristic
275C23#45211yescharacteristic
276C23#45211yescharacteristic
277C23#45311yescharacteristic
278C23#45411yescharacteristic
279Q8—24no—
280Q8—24no—
281Q8—44no—
282Q8—44no—
283Q8—42no—
284Q8—42no—
285Q8—162no—
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—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—24no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4C2×#9912yes—
346C2×C4#41611yescharacteristic
347C2×C4#41911yescharacteristic
348C2×C4C2×#10412yes—
349C2×C4C4×Q1611yescharacteristic
350C2×C4#44111yescharacteristic
351C2×C4#46211yescharacteristic
352C2×C4#46411yescharacteristic
353C2×C4#48511yescharacteristic
354C2×C4#48511yescharacteristic
355C8—82no—
356C8—82no—
357C8—161no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22#163011yescharacteristic
371C22#163211yescharacteristic
372C22#163611yescharacteristic
373C22#163811yescharacteristic
374C22#187311yescharacteristic
375C22#187811yescharacteristic
376C22#188311yescharacteristic
377C22#189611yescharacteristic
378C22#189811yescharacteristic
379C22#190511yescharacteristic
380C22#190611yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—44no—
388C4—41no—
389C4—41no—
390C4—44no—
391C4—44no—
392C4—84no—
393C4—82no—
394C4—161no—
395C4—161no—
396C4#150112yes—
397C4#150812yes—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#897411yescharacteristic
403C2#914611yescharacteristic
404C2#917811yescharacteristic
405C2#923211yescharacteristic
406C2#925811yescharacteristic
407C2#932611yescharacteristic
408C2#936511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ 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
9. C2×#477C4semidirect
15. #1803C4semidirect

Maximal subgroups

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

subgroupindex
#329572
#138172
#99082
#96792
#83112
#70392
#70282