Finite Group Databasebeta

← #98559

Subgroups — #98559

Order 512.

846 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2423C211yesmaximal, characteristic
2#8724C212yesmaximal, complemented
3#8559C211yesmaximal, characteristic
4#8559C211yesmaximal, characteristic
5#7643C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C22×C4×Q8C411yescharacteristic
8#2423—24no—
9#2423—24no—
10C2×#399—21no—
11C2×#398—21no—
12C2×#391C2211yescharacteristic
13C2×#389C411yescharacteristic
14C2×#388—21no—
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#1272—24no—
27C23×Q8C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—44no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—44no—
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×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4D411yescharacteristic
46C22×C22⋊C4D411yescharacteristic
47C22×C42—21no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2311yescharacteristic
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54#399—28no—
55#398—28no—
56#391—22no—
57#391D414yes—
58#389—22no—
59#389—22no—
60#388—28no—
61#387—28no—
62#384—22no—
63#384D414yes—
64#380—22no—
65#380—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C2×#95—21no—
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
79C22×Q8—24no—
80C22×Q8—21no—
81C22×Q8—42no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8C22⋊C412yes—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—41no—
92C23×C4—41no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C22×C8—42no—
102C22×C8—42no—
103C22⋊Q8—48no—
104C22⋊Q8—44no—
105C22⋊Q8—44no—
106C4×Q8—48no—
107C4×Q8—44no—
108C4×Q8—44no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—82no—
141C2×C22⋊C4SD1612yes—
142C2×C22⋊C4Q1612yes—
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—24no—
161#95—22no—
162#95—22no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24#10312yes—
174C24C2×C22⋊C411yescharacteristic
175C2×Q8—24no—
176C2×Q8—22no—
177C2×Q8—48no—
178C2×Q8—48no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8#10014yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—84no—
233C22×C4—84no—
234C22×C4#10312yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4#12012yes—
238C22×C4#12012yes—
239C22×C4C2×SD1611yescharacteristic
240C22×C4C2×Q1611yescharacteristic
241C22×C4#13512yes—
242C22×C4AGL(1,ℤ/8)11yescharacteristic
243C22×C4#13712yes—
244C22×C4#13711yescharacteristic
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—48no—
256C4⋊C4—48no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—84no—
262C4⋊C4—84no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—48no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—84no—
269C42—24no—
270C42—42no—
271C42—42no—
272C42—48no—
273C42—42no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23#37911yescharacteristic
302C23#41311yescharacteristic
303C23#41311yescharacteristic
304C23C2×#10311yescharacteristic
305C23#41611yescharacteristic
306C23#41711yescharacteristic
307C23#41911yescharacteristic
308Q8—44no—
309Q8—48no—
310Q8—44no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4#41012yes—
365C2×C4C2×#10012yes—
366C2×C4#44812yes—
367C2×C4#44912yes—
368C2×C4#45112yes—
369C2×C4#45212yes—
370C8—162no—
371C8—162no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22#154211yescharacteristic
396C22#155011yescharacteristic
397C22#155411yescharacteristic
398C22#155911yescharacteristic
399C22#178211yescharacteristic
400C22#178311yescharacteristic
401C22#178511yescharacteristic
402C22#178711yescharacteristic
403C22#178811yescharacteristic
404C22#179511yescharacteristic
405C22#179611yescharacteristic
406C4—22no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—44no—
413C4—42no—
414C4—82no—
415C4—81no—
416C4—81no—
417C4—82no—
418C4—82no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—82no—
424C2#834311yescharacteristic
425C2#838011yescharacteristic
426C2#838211yescharacteristic
427C2#838511yescharacteristic
428C2#839111yescharacteristic
429C2#852311yescharacteristic
430C2#855311yescharacteristic
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. #7643C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24232
#87242
#85592
#76432