Finite Group Databasebeta

← #98512

Subgroups — #98512

Order 512.

892 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2442C211yesmaximal, characteristic
2#8719C212yesmaximal, complemented
3#8152C211yesmaximal, characteristic
4#8152C211yesmaximal, characteristic
5#7627C212yesmaximal, complemented
6C22×#125C411yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8#2442—24no—
9#2442—24no—
10C2×#400—21no—
11C2×#398—21no—
12C2×#389C411yescharacteristic
13C2×#389C2211yescharacteristic
14C2×#385—21no—
15C2×#382C2211yescharacteristic
16C2×#380—21no—
17C2×#380C411yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1749C2212yes—
20#1749C2212yes—
21C2×#323—21no—
22C2×#323—21no—
23#1295—24no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C24×C4—21no—
28C24×C4C2×C411yescharacteristic
29C2×#125—22no—
30C2×#125—22no—
31C2×#125—44no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—44no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C22⋊C4—21no—
45C22×C22⋊C4D411yescharacteristic
46C22×C22⋊C4D411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×#97—22no—
52C2×#97—22no—
53#400—28no—
54#398—28no—
55#389—22no—
56#389—22no—
57#389—22no—
58#389D414yes—
59#385—28no—
60#382—22no—
61#382D414yes—
62#380—28no—
63#380—22no—
64#380—22no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76#323—24no—
77#323—24no—
78C25C22⋊C411yescharacteristic
79C23×C4—22no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—24no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—44no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—82no—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C22×C8—42no—
101C22×C8—42no—
102#125—48no—
103#125—44no—
104#125—44no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—82no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4C22⋊C412yes—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C22⋊C412yes—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—42no—
142C2×C42—42no—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149#97—44no—
150#97—44no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—22no—
157#95—22no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—41no—
163C24—41no—
164C24—41no—
165C24—41no—
166C24#10112yes—
167C24#10412yes—
168C24#10412yes—
169C24C2×C22⋊C411yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—48no—
199C22×C4—48no—
200C22×C4—42no—
201C22×C4—48no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—48no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—84no—
227C22×C4—88no—
228C22×C4—88no—
229C22×C4#10112yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4#12012yes—
233C22×C4#12012yes—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—82no—
237C2×C8—82no—
238C2×C8—82no—
239C2×C8—82no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—48no—
246C4⋊C4—48no—
247C4⋊C4—48no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—84no—
252C4⋊C4—84no—
253C4⋊C4#10014yes—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—48no—
257C42—24no—
258C42—42no—
259C42—42no—
260C42—42no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23#33014yes—
288C23#37911yescharacteristic
289C23C2×#10111yescharacteristic
290C23C2×#10111yescharacteristic
291C23C2×#10411yescharacteristic
292C23C2×#10411yescharacteristic
293C23#42111yescharacteristic
294C23#42111yescharacteristic
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—816no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4#41012yes—
350C2×C4C2×#10012yes—
351C8—84no—
352C8—84no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—24no—
366C22—44no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22C2×#33011yescharacteristic
377C22#152811yescharacteristic
378C22#152811yescharacteristic
379C22#153411yescharacteristic
380C22#178211yescharacteristic
381C22#178311yescharacteristic
382C22#178411yescharacteristic
383C22#178911yescharacteristic
384C22#178911yescharacteristic
385C22#179511yescharacteristic
386C22#179611yescharacteristic
387C4—22no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—44no—
393C4—82no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—82no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—44no—
403C2#833511yescharacteristic
404C2#836211yescharacteristic
405C2#836311yescharacteristic
406C2#836711yescharacteristic
407C2#836911yescharacteristic
408C2#852511yescharacteristic
409C2#855511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ 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
2. #8719C2semidirect
5. #7627C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
#87192
#81522
#76272