Finite Group Databasebeta

← #97600

Subgroups — #97600

Order 512.

902 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 120 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×#2448C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic, complemented
3#8311C212yesmaximal, complemented
4C4×#324C211yesmaximal, characteristic, complemented
5#7572C212yesmaximal
6C22×#124C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8#2448—24no—
9#2448—24no—
10C2×#399—21no—
11C2×#397—21no—
12C2×#391C411yescharacteristic
13C2×#390C411yescharacteristic
14C2×#390C2211yescharacteristic
15C2×#389C411yescharacteristic
16C2×#388—21no—
17C2×#388—21no—
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×Q8⋊C8—24no—
22C2×#324—21no—
23C2×#324—21no—
24#1280—21no—
25#1280—21no—
26#1270—24no—
27C23×Q8C2×C411yescharacteristic
28C23×D4C2×C411yescharacteristic
29C2×#124—22no—
30C2×#124—22no—
31C2×#124—44no—
32C2×C4×Q8—22no—
33C2×C4×Q8—44no—
34C2×C4×Q8D414yes—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C42—21no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42C2311yescharacteristic
51C2×C4⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C4×C8—22no—
55#399—28no—
56#397—28no—
57#391—22no—
58#391—22no—
59#390—22no—
60#390—22no—
61#390—22no—
62#390D414yes—
63#389—22no—
64#389—22no—
65#388—28no—
66#388—28no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76Q8⋊C8—48no—
77#324—24no—
78#324—42no—
79C25—21no—
80C22×Q8—24no—
81C22×Q8—21no—
82C22×Q8—22no—
83C22×Q8—22no—
84C22×Q8—42no—
85C22×Q8C22⋊C412yes—
86C22×D4—21no—
87C22×D4—22no—
88C22×D4—22no—
89C22×D4—42no—
90C22×D4—42no—
91C22×D4C22⋊C412yes—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C22×C8—42no—
103C22×C8—42no—
104#124—48no—
105#124—44no—
106#124—44no—
107C4×Q8—28no—
108C4×Q8—44no—
109C4×Q8—48no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—82no—
125C2×C4⋊C4C22⋊C412yes—
126C2×C4⋊C4C22⋊C412yes—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—82no—
141C2×C22⋊C4—82no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—24no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—42no—
150C2×C42—44no—
151C2×C42—42no—
152C2×C42C2×D412yes—
153C2×C42C2×D412yes—
154C4⋊C8—44no—
155C22⋊C8—24no—
156C22⋊C8—42no—
157C22⋊C8—42no—
158C22⋊C8—42no—
159C4×C8—44no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—22no—
164#95—22no—
165C24—22no—
166C24—42no—
167C24—42no—
168C24—42no—
169C24—42no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24—41no—
174C24—41no—
175C24—41no—
176C24C2×C22⋊C411yescharacteristic
177C2×Q8—24no—
178C2×Q8—22no—
179C2×Q8—24no—
180C2×Q8—24no—
181C2×Q8—48no—
182C2×Q8—48no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8#9914yes—
186C2×D4—24no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—84no—
193C2×D4—84no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4#10412yes—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4#12012yes—
242C22×C4#12012yes—
243C22×C4#13112yes—
244C22×C4#13112yes—
245C22×C4#13512yes—
246C22×C4AGL(1,ℤ/8)11yescharacteristic
247C22×C4#13711yescharacteristic
248C2×C8—44no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C4⋊C4—24no—
255C4⋊C4—44no—
256C4⋊C4—48no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—84no—
263C22⋊C4—48no—
264C22⋊C4—48no—
265C22⋊C4—48no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—84no—
269C22⋊C4—84no—
270C42—24no—
271C42—42no—
272C42—48no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42—42no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—82no—
289C23—82no—
290C23—82no—
291C23—82no—
292C23—82no—
293C23—82no—
294C23—82no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23—82no—
299C23—82no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23#37911yescharacteristic
305C23#40811yescharacteristic
306C23#40811yescharacteristic
307C23#41611yescharacteristic
308C23#41711yescharacteristic
309C23C2×#10411yescharacteristic
310C23#42111yescharacteristic
311Q8—28no—
312Q8—48no—
313Q8—44no—
314D4—84no—
315D4—84no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4#40812yes—
368C2×C4#40812yes—
369C2×C4C2×#9912yes—
370C2×C4#41312yes—
371C2×C4#44912yes—
372C2×C4#45212yes—
373C8—84no—
374C8—162no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—162no—
395C22—162no—
396C22—162no—
397C22—162no—
398C22#148211yescharacteristic
399C22#148311yescharacteristic
400C22#149411yescharacteristic
401C22#149511yescharacteristic
402C22#178011yescharacteristic
403C22#178111yescharacteristic
404C22#178511yescharacteristic
405C22#178811yescharacteristic
406C22#178911yescharacteristic
407C22#179511yescharacteristic
408C22#179611yescharacteristic
409C4—22no—
410C4—22no—
411C4—22no—
412C4—44no—
413C4—42no—
414C4—42no—
415C4—81no—
416C4—81no—
417C4—82no—
418C4—81no—
419C4—82no—
420C4#146814yes—
421C2—21no—
422C2—21no—
423C2—21no—
424C2—21no—
425C2—162no—
426C2#830411yescharacteristic
427C2#831511yescharacteristic
428C2#832511yescharacteristic
429C2#836111yescharacteristic
430C2#836811yescharacteristic
431C2#837711yescharacteristic
432C2#838611yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ 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
2. #8558C2semidirect
3. #8311C2semidirect
4. C4×#324C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24482
#85582
#83112
C4×#3242
#75722