Finite Group Databasebeta

← #100367

Subgroups — #100367

Order 512.

836 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 128 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
2Q8×#95C211yesmaximal, characteristic
3#9908C211yesmaximal, characteristic
4#9679C211yesmaximal, characteristic
5#8311C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C22×C4×Q8—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#477C412yescomplemented
11#2857C2211yescharacteristic
12#1863—22no—
13#1852—21no—
14#1803C412yescomplemented
15#1797C2211yescharacteristic
16#1797C2211yescharacteristic
17#1787—21no—
18C4×C22⋊C8C2211yescharacteristic
19C2×#328—22no—
20C2×Q8⋊C8—22no—
21C4⋊#95—21no—
22C4⋊#95—22no—
23C4⋊#95—21no—
24C4⋊#95C2211yescharacteristic
25C4×#95—21no—
26C4×#95C2211yescharacteristic
27#1294—21no—
28#1294—21no—
29C23×Q8D411yescharacteristic
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×C22⋊Q8—21no—
32C2×C4×Q8—28no—
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—44no—
36C2×C4×Q8—44no—
37C2×C4×Q8—42no—
38C2×C4×Q8—42no—
39C2×C4×Q8C2×C412yes—
40C2×C4×Q8D412yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D411yescharacteristic
45C22×C42—22no—
46C22×C42—21no—
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#477—44no—
50C2×#107—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8C2×C412yes—
53C2×#103—22no—
54C2×#103—22no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C4×C8—21no—
59#391—21no—
60#389—21no—
61#389—21no—
62#384C2×C411yescharacteristic
63#380—21no—
64#380—21no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—22no—
69C2×#95D411yescharacteristic
70C2×#95D411yescharacteristic
71C2×#95D412yes—
72#340—22no—
73#336—21no—
74#336—21no—
75#336—21no—
76#328—44no—
77Q8⋊C8—44no—
78C22×Q8—24no—
79C22×Q8—24no—
80C22×Q8—21no—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8C22⋊C412yes—
85C22×Q8SD1612yes—
86C22×Q8Q1612yes—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C22×C8—22no—
96C22×C8—41no—
97C22×C8—41no—
98C4⋊Q8—42no—
99C22⋊Q8—42no—
100C22⋊Q8—42no—
101C4×Q8—24no—
102C4×Q8—24no—
103C4×Q8—42no—
104C4×Q8—42no—
105C4×Q8—44no—
106C4×Q8—44no—
107C4×Q8—416no—
108C4×Q8—416no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4C22⋊C412yes—
128C2×C22⋊C4—41no—
129C2×C42—28no—
130C2×C42—24no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—42no—
134C2×C42—44no—
135C2×C42—44no—
136C2×C42—44no—
137C2×C42—42no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42C22×C411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144#107—44no—
145C4⋊C8—42no—
146C4⋊C8—42no—
147#103—44no—
148#103—44no—
149C22⋊C8—22no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C4×C8—42no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—42no—
160#95—42no—
161#95—44no—
162#95—41no—
163#95—41no—
164#95—41no—
165#95—41no—
166#95—82no—
167#95Q1612yes—
168C24#12011yescharacteristic
169C2×Q8—28no—
170C2×Q8—24no—
171C2×Q8—28no—
172C2×Q8—24no—
173C2×Q8—28no—
174C2×Q8—28no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—42no—
181C2×Q8—81no—
182C2×Q8—81no—
183C2×Q8#9914yes—
184C2×Q8#10314yes—
185C2×Q8#10414yes—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4C2×SD1611yescharacteristic
234C22×C4C2×Q1611yescharacteristic
235C22×C4C2×Q1611yescharacteristic
236C22×C4#13511yescharacteristic
237C22×C4AGL(1,ℤ/8)11yescharacteristic
238C22×C4#13711yescharacteristic
239C22×C4#13711yescharacteristic
240C22×C4#13711yescharacteristic
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—81no—
246C2×C8—81no—
247C2×C8—81no—
248C2×C8—81no—
249C2×C8—81no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—416no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—82no—
264C22⋊C4—81no—
265C22⋊C4—81no—
266C42—22no—
267C42—22no—
268C42—41no—
269C42—41no—
270C42—44no—
271C42—44no—
272C42—416no—
273C42—44no—
274C42—44no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23#38611yescharacteristic
280C23#45011yescharacteristic
281C23#45111yescharacteristic
282C23#45111yescharacteristic
283C23#45211yescharacteristic
284C23#45311yescharacteristic
285C23#45411yescharacteristic
286Q8—28no—
287Q8—28no—
288Q8—24no—
289Q8—24no—
290Q8—24no—
291Q8—162no—
292Q8#32818yescomplemented
293C2×C4—28no—
294C2×C4—28no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4C2×#9912yes—
353C2×C4C2×#10311yescharacteristic
354C2×C4#41711yescharacteristic
355C2×C4C2×#10412yes—
356C2×C4C4×Q1611yescharacteristic
357C2×C4#44111yescharacteristic
358C2×C4#46211yescharacteristic
359C2×C4#46411yescharacteristic
360C2×C4#48311yescharacteristic
361C2×C4#48411yescharacteristic
362C8—82no—
363C8—82no—
364C8—161no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22#163111yescharacteristic
378C22#163311yescharacteristic
379C22#163611yescharacteristic
380C22#163711yescharacteristic
381C22#187211yescharacteristic
382C22#187811yescharacteristic
383C22#188311yescharacteristic
384C22#189611yescharacteristic
385C22#189811yescharacteristic
386C22#190511yescharacteristic
387C22#190611yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—28no—
393C4—21no—
394C4—21no—
395C4—44no—
396C4—41no—
397C4—41no—
398C4—82no—
399C4—84no—
400C4—82no—
401C4—161no—
402C4—161no—
403C4C2×#32812yes—
404C4#150412yes—
405C2—21no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2#897511yescharacteristic
410C2#914311yescharacteristic
411C2#917611yescharacteristic
412C2#923211yescharacteristic
413C2#925811yescharacteristic
414C2#932611yescharacteristic
415C2#936811yescharacteristic
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
10. C2×#477C4semidirect
14. #1803C4semidirect
292. Q8#328semidirect

Maximal subgroups

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

subgroupindex
#329572
Q8×#952
#99082
#96792
#83112
#70392
#70282