Finite Group Databasebeta

← #101406

Subgroups — #101406

Order 512.

470 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 104 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#27648C211yesmaximal, characteristic
2#22295C211yesmaximal, characteristic
3#11357C211yesmaximal, characteristic
4#8724C211yesmaximal, characteristic
5#8572C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7020C211yesmaximal, characteristic
8C2×#484—21no—
9C4×C22⋊Q8C2211yescharacteristic
10C2×#384C2211yescharacteristic
11#2083C412yescomplemented
12#2072C412yescomplemented
13#1985—21no—
14#1947—21no—
15#1940—21no—
16#1917—21no—
17#1917—21no—
18#1850—21no—
19#1824—21no—
20#1803—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24#1752—21no—
25C2×#340—21no—
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#328—21no—
29C4⋊#95C2211yescharacteristic
30#1296—21no—
31#1296—21no—
32#1294—21no—
33C23×C8D411yescharacteristic
34C23×C8D411yescharacteristic
35C2×C22⋊Q8D411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C42C2311yescharacteristic
42#484—42no—
43#484—42no—
44C2×#107—22no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×#103—22no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—21no—
56C2×#97—21no—
57#391C2×C411yescharacteristic
58#389C2×C411yescharacteristic
59#387—21no—
60#385—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384C2×C412yes—
65#380D411yescharacteristic
66C4×C4⋊C4—21no—
67C4×C4⋊C4D411yescharacteristic
68C4×C22⋊C4—21no—
69C2×#95—21no—
70C2×#95—21no—
71#340—22no—
72#340—22no—
73#340—42no—
74#340—42no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336C2×C412yes—
84#328—42no—
85C22×Q8C22⋊C411yescharacteristic
86C23×C4—41no—
87C23×C4—41no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4#4111yescharacteristic
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8Q1612yes—
105C22⋊Q8—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C4×Q8—41no—
110C4×Q8—41no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C22⋊C411yescharacteristic
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42C22×C411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42#4111yescharacteristic
148#107—44no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—81no—
156C4⋊C8—81no—
157#103—44no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164#97—81no—
165#97—81no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—41no—
172#95—41no—
173#95—81no—
174C24#12111yescharacteristic
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C4×D411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4#12011yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12311yescharacteristic
213C22×C4C2×Q1611yescharacteristic
214C22×C4#13511yescharacteristic
215C22×C4AGL(1,ℤ/8)11yescharacteristic
216C22×C4AGL(1,ℤ/8)11yescharacteristic
217C22×C4AGL(1,ℤ/8)11yescharacteristic
218C22×C4#13711yescharacteristic
219C22×C4#13711yescharacteristic
220C22×C4#13711yescharacteristic
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—81no—
239C2×C8—81no—
240C2×C8—81no—
241C2×C8—81no—
242C2×C8—81no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—82no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—82no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—41no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—41no—
278C42—41no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23#38611yescharacteristic
284C23#46211yescharacteristic
285C23#46411yescharacteristic
286C23#46811yescharacteristic
287C23#46911yescharacteristic
288C23#47111yescharacteristic
289C23#47311yescharacteristic
290Q8—42no—
291Q8—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—161no—
337C2×C4—161no—
338C2×C4—161no—
339C2×C4—161no—
340C2×C4—161no—
341C2×C4—161no—
342C2×C4—161no—
343C2×C4—161no—
344C2×C4#41711yescharacteristic
345C2×C4#41811yescharacteristic
346C2×C4C4×Q1611yescharacteristic
347C2×C4#44111yescharacteristic
348C2×C4#45111yescharacteristic
349C2×C4#45211yescharacteristic
350C2×C4#45311yescharacteristic
351C2×C4#45411yescharacteristic
352C2×C4#45811yescharacteristic
353C2×C4#45811yescharacteristic
354C2×C4#46811yescharacteristic
355C2×C4#46911yescharacteristic
356C8—42no—
357C8—82no—
358C8—81no—
359C8—81no—
360C8—81no—
361C8—161no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22#169211yescharacteristic
375C22#169411yescharacteristic
376C22#170011yescharacteristic
377C22#170211yescharacteristic
378C22#187811yescharacteristic
379C22#188311yescharacteristic
380C22#189011yescharacteristic
381C22#189511yescharacteristic
382C22#190311yescharacteristic
383C22#190811yescharacteristic
384C22#190911yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—162no—
398C4—162no—
399C4#161411yescharacteristic
400C4#161511yescharacteristic
401C4#161711yescharacteristic
402C4#161811yescharacteristic
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2#904611yescharacteristic
408C2#922611yescharacteristic
409C2#928811yescharacteristic
410C2#932211yescharacteristic
411C2#937711yescharacteristic
412C2#947411yescharacteristic
413C2#952111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
11. #2083C4semidirect
12. #2072C4semidirect

Maximal subgroups

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

subgroupindex
#276482
#222952
#113572
#87242
#85722
#70392
#70202