Finite Group Databasebeta

← #168672

Subgroups — #168672

Order 512.

714 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C2×#2810C211yesmaximal, characteristic
2#29969C211yesmaximal, characteristic
3#29968C211yesmaximal, characteristic
4#29329C211yesmaximal, characteristic
5#7832C214yesmaximal, complemented
6#7775C214yesmaximal, complemented
7#7641C211yesmaximal, characteristic
8#7576C211yesmaximal, characteristic
9#7573C211yesmaximal, characteristic
10#2857C2211yescharacteristic
11#2810—22no—
12#2810—22no—
13#2810—22no—
14#2810C2214yes—
15C2×#400—21no—
16C2×#400—21no—
17C2×#400—21no—
18C2×#389C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387—21no—
22C2×#387—21no—
23C2×#387—21no—
24#2100C2212yes—
25#2100C2212yes—
26#2100C2212yes—
27#2088—22no—
28#2087—22no—
29#2058C2212yes—
30#2057C2212yes—
31#2043C2212yes—
32#1933—22no—
33#1872—21no—
34#1872—21no—
35#1853—21no—
36#1852—21no—
37#1824—21no—
38#1823—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1271C2214yes—
43#1271C2214yes—
44#1271C2214yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C22⋊Q8—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58C2×#107—22no—
59C2×#106—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×#103—22no—
63C2×#103—22no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C4×C8—21no—
68#400—24no—
69#400—24no—
70#400—24no—
71#400—24no—
72#400—24no—
73#400—24no—
74#391—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389D412yes—
84#389D412yes—
85#389D412yes—
86#389C2312yes—
87#389C2312yes—
88#389C2312yes—
89#387—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#387—24no—
94#387—24no—
95#384C2311yescharacteristic
96#380—21no—
97#380—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#336C2312yes—
117#336C2312yes—
118#336C2312yes—
119C22×Q8—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C4⋊Q8—81no—
133C22⋊Q8—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4—81no—
164C2×C4⋊C4—81no—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C22⋊C4—41no—
172C2×C42—24no—
173C2×C42—81no—
174C2×C42—82no—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2411yescharacteristic
179#107—82no—
180#106—82no—
181C4⋊C8—81no—
182C4⋊C8—81no—
183#103—44no—
184#103—82no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187C22⋊C8—42no—
188C4×C8—81no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—41no—
203#95—41no—
204C24#12011yescharacteristic
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#13512yes—
245C22×C4#13512yes—
246C22×C4#13712yes—
247C22×C4#13712yes—
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—82no—
289C22⋊C4—82no—
290C42—41no—
291C42—41no—
292C42—41no—
293C42—41no—
294C42—84no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#45212yes—
300C23#45212yes—
301C23#45412yes—
302C23C2×#12011yescharacteristic
303Q8—84no—
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—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4#45212yes—
353C2×C4#45212yes—
354C2×C4C2×#13511yescharacteristic
355C2×C4C2×#13511yescharacteristic
356C2×C4C2×#13711yescharacteristic
357C2×C4C2×#13711yescharacteristic
358C2×C4#57512yes—
359C8—82no—
360C8—82no—
361C8—82no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22#163814yes—
369C22C2×#45211yescharacteristic
370C22C2×#45211yescharacteristic
371C22#300511yescharacteristic
372C22C2×#45411yescharacteristic
373C22#302011yescharacteristic
374C22#302011yescharacteristic
375C22#302611yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—41no—
384C4—82no—
385C4—82no—
386C4—82no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4—161no—
391C4—161no—
392C2—24no—
393C2#1828911yescharacteristic
394C2#1828911yescharacteristic
395C2#1832011yescharacteristic
396C2C2×#163811yescharacteristic
397C2#1876311yescharacteristic
398C2#1878111yescharacteristic
399C2#1900311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #7832C2semidirect
6. #7775C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28102
#299692
#299682
#293292
#78322
#77752
#76412
#75762
#75732