Finite Group Databasebeta

← #98554

Subgroups — #98554

Order 512.

830 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2441C211yesmaximal, characteristic
2#8724C212yesmaximal, complemented
3#8559C211yesmaximal, characteristic
4#8559C211yesmaximal, characteristic
5#7641C212yesmaximal, complemented
6C22×#125C411yescharacteristic
7C22×#117C411yescharacteristic
8#2441—24no—
9#2441—24no—
10C2×#399—21no—
11C2×#396—21no—
12C2×#389—21no—
13C2×#389C411yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#387—21no—
16C2×#384C2211yescharacteristic
17C2×#380C411yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20#1797C2212yes—
21C2×#323—21no—
22C2×#323—21no—
23#1296—24no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C24×C4C2×C411yescharacteristic
28C2×#125—22no—
29C2×#125—22no—
30C2×#125—44no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—44no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C22⋊C4D411yescharacteristic
47C22×C22⋊C4D411yescharacteristic
48C22×C42—21no—
49C22×C42C2×C411yescharacteristic
50C22×C42C2311yescharacteristic
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55#399—28no—
56#396—28no—
57#389—28no—
58#389—22no—
59#389—22no—
60#389—22no—
61#389D414yes—
62#387—28no—
63#384—22no—
64#384D414yes—
65#380—22no—
66#380—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—22no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—22no—
75#336—22no—
76#323—42no—
77#323—42no—
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—42no—
88C23×C4—42no—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C22×C8—42no—
96C22×C8—42no—
97#125—48no—
98#125—44no—
99#125—44no—
100#117—48no—
101#117—44no—
102#117—44no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—82no—
143C2×C42—24no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—42no—
147C2×C42—44no—
148C2×C42—42no—
149C2×C42C2×D412yes—
150C2×C42C2×D412yes—
151C4⋊C8—44no—
152C4⋊C8—44no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—22no—
161#95—22no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24—41no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24#10112yes—
173C24C2×C22⋊C411yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
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—42no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
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#10112yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4#12012yes—
228C22×C4#12012yes—
229C22×C4#13512yes—
230C22×C4#13512yes—
231C22×C4AGL(1,ℤ/8)11yescharacteristic
232C22×C4AGL(1,ℤ/8)11yescharacteristic
233C22×C4#13711yescharacteristic
234C22×C4#13711yescharacteristic
235C2×C8—82no—
236C2×C8—82no—
237C2×C8—82no—
238C2×C8—82no—
239C2×C8—82no—
240C2×C8—82no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—48no—
247C4⋊C4—48no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4#10014yes—
258C22⋊C4—48no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—84no—
264C42—24no—
265C42—42no—
266C42—48no—
267C42—42no—
268C42—42no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23#37911yescharacteristic
297C23C2×#10111yescharacteristic
298C23C2×#10111yescharacteristic
299C23#41611yescharacteristic
300C23#41611yescharacteristic
301C23#41711yescharacteristic
302C23#41711yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4#41012yes—
357C2×C4C2×#10012yes—
358C2×C4#44912yes—
359C2×C4#44912yes—
360C2×C4#45212yes—
361C2×C4#45212yes—
362C8—162no—
363C8—162no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22#154411yescharacteristic
388C22#155211yescharacteristic
389C22#155211yescharacteristic
390C22#156111yescharacteristic
391C22#178211yescharacteristic
392C22#178311yescharacteristic
393C22#178411yescharacteristic
394C22#178811yescharacteristic
395C22#178811yescharacteristic
396C22#179511yescharacteristic
397C22#179611yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—44no—
406C4—82no—
407C4—81no—
408C4—82no—
409C4—82no—
410C4—81no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—82no—
416C2#834511yescharacteristic
417C2#837811yescharacteristic
418C2#838211yescharacteristic
419C2#838511yescharacteristic
420C2#839311yescharacteristic
421C2#852511yescharacteristic
422C2#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. #8724C2semidirect
5. #7641C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
#87242
#85592
#76412