Finite Group Databasebeta

← #129795

Subgroups — #129795

Order 512.

972 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 116 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×#2396C211yesmaximal, characteristic
2C2×#1352C212yesmaximal
3C2×#1351C212yesmaximal
4#10726C211yesmaximal, characteristic
5#8300C211yesmaximal, characteristic
6C22×C4×Q8C411yescharacteristic, complemented
7C22×#117C2211yescharacteristic
8#2396—24no—
9#2396—24no—
10C2×#400C411yescharacteristic, complemented
11C2×#399C2211yescharacteristic
12C2×#398—21no—
13C2×#396—21no—
14C2×#395C411yescharacteristic, complemented
15C2×#393C2211yescharacteristic
16C2×#381—21no—
17C2×C4×C4⋊C4—21no—
18C2×C4×C4⋊C4C411yescharacteristic, complemented
19C2×C4×C22⋊C4—21no—
20#1817—21no—
21C2×#342—21no—
22C2×#342C2212yes—
23C2×#324—21no—
24C2×#323C2212yes—
25#1352—24no—
26#1351—24no—
27C23×Q8C2×C411yescharacteristic
28C24×C4D411yescharacteristic
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—42no—
32C2×C4×Q8—42no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—42no—
36C2×#117—42no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C22⋊C4—22no—
47C22×C22⋊C4D411yescharacteristic
48C22×C22⋊C4C2311yescharacteristic
49C22×C42—21no—
50C22×C42—21no—
51C22×C42C2×C411yescharacteristic
52C22×C42D411yescharacteristic
53C2×C22⋊C8—22no—
54#400—22no—
55#400—22no—
56#399—22no—
57#399D414yes—
58#398—28no—
59#396—28no—
60#395—22no—
61#395—22no—
62#393—22no—
63#393D414yes—
64#381—24no—
65#381—42no—
66C4×C4⋊C4—28no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C22⋊C4—28no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—42no—
75C2×#95C2×C411yescharacteristic
76C2×#95C2×C411yescharacteristic
77#342—22no—
78#342—22no—
79#342—42no—
80#336—42no—
81#324—42no—
82#323—22no—
83#323—22no—
84C25C2×D411yescharacteristic
85C22×Q8—21no—
86C22×Q8—44no—
87C22×Q8—41no—
88C22×Q8—41no—
89C22×Q8C22⋊C412yes—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—42no—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4SD1612yes—
106C23×C4Q1612yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C22×C8—42no—
110C4×Q8—48no—
111C4×Q8—48no—
112#117—48no—
113#117—48no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C22⋊C412yes—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4C2×D412yes—
150C2×C22⋊C4C2×D412yes—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—41no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—42no—
164C22⋊C8—42no—
165C22⋊C8—42no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—21no—
170#95—21no—
171#95—24no—
172#95—48no—
173#95C22⋊C412yes—
174#95C22⋊C412yes—
175C24—21no—
176C24—21no—
177C24—21no—
178C24—21no—
179C24—42no—
180C24—42no—
181C24C2×C22⋊C411yescharacteristic
182C24#12012yes—
183C24C2×SD1611yescharacteristic
184C24C2×Q1611yescharacteristic
185C24#13712yes—
186C2×Q8—44no—
187C2×Q8—42no—
188C2×Q8—44no—
189C2×Q8—42no—
190C2×Q8—48no—
191C2×Q8—44no—
192C2×Q8—44no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—44no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—84no—
250C22×C4—84no—
251C22×C4#9912yes—
252C22×C4#9912yes—
253C22×C4#10312yes—
254C22×C4#10312yes—
255C22×C4#10412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4#12012yes—
259C2×C8—82no—
260C2×C8—84no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—48no—
264C4⋊C4—48no—
265C4⋊C4—48no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—48no—
269C4⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—44no—
273C22⋊C4—48no—
274C22⋊C4—48no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—82no—
278C22⋊C4—82no—
279C42—24no—
280C42—24no—
281C42—44no—
282C42—48no—
283C42—48no—
284C23—22no—
285C23—22no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—44no—
299C23—42no—
300C23—42no—
301C23—82no—
302C23—81no—
303C23—81no—
304C23—82no—
305C23#37911yescharacteristic
306C23C2×#9911yescharacteristic
307C23C2×#9911yescharacteristic
308C23C2×#10311yescharacteristic
309C23#41911yescharacteristic
310C23C2×#10411yescharacteristic
311C23#42111yescharacteristic
312C23#44812yes—
313C23#45112yes—
314Q8—44no—
315Q8—44no—
316Q8—48no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—48no—
333C2×C4—48no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—84no—
371C2×C4—88no—
372C2×C4—164no—
373C2×C4—162no—
374C2×C4—162no—
375C2×C4#32814yes—
376C8—162no—
377C22—22no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—82no—
395C22—82no—
396C22—81no—
397C22—81no—
398C22C2×#32811yescharacteristic
399C22#150811yescharacteristic
400C22#151111yescharacteristic
401C22#152011yescharacteristic
402C22#178111yescharacteristic
403C22#178711yescharacteristic
404C22#178911yescharacteristic
405C22#212012yes—
406C22#212612yes—
407C4—24no—
408C4—44no—
409C4—42no—
410C4—44no—
411C4—41no—
412C4—41no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—81no—
417C4—82no—
418C4—164no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—42no—
424C2—42no—
425C2#832911yescharacteristic
426C2#1171512yes—
427C2#1181012yes—
428C2#1268911yescharacteristic
429C2#1269511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C22×C4×Q8C4semidirect
10. C2×#400C4semidirect
14. C2×#395C4semidirect
18. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
C2×#13522
C2×#13512
#107262
#83002