Finite Group Databasebeta

← #98539

Subgroups — #98539

Order 512.

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

normal subgroups only

This group has 878 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×#2492C211yesmaximal, characteristic
2#8559C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#8340C212yesmaximal, complemented
5#7643C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C22×C22⋊Q8C411yescharacteristic
8#2492—24no—
9#2492—24no—
10C2×#400—21no—
11C2×#393—21no—
12C2×#391—21no—
13C2×#391C411yescharacteristic
14C2×#391C2211yescharacteristic
15C2×#388—21no—
16C2×#380C411yescharacteristic
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1797C2212yes—
20#1797C2212yes—
21C2×#332—24no—
22C2×#323—21no—
23C2×#323—21no—
24#1280—21no—
25#1280—21no—
26#1272—24no—
27C23×Q8—21no—
28C23×Q8C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—44no—
32C2×C4⋊Q8D414yes—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—44no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C22⋊C4D411yescharacteristic
44C22×C22⋊C4D411yescharacteristic
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2311yescharacteristic
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52#400—28no—
53#393—28no—
54#391—28no—
55#391—22no—
56#391—22no—
57#391—22no—
58#391D414yes—
59#388—28no—
60#380—22no—
61#380—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C2×#95—21no—
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#332—48no—
77#323—42no—
78#323—42no—
79C25C22⋊C411yescharacteristic
80C22×Q8—21no—
81C22×Q8—24no—
82C22×Q8—42no—
83C22×Q8—44no—
84C22×Q8—42no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8—42no—
88C22×Q8—41no—
89C22×Q8—41no—
90C22×Q8—82no—
91C22×Q8C22⋊C412yes—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C22×C8—42no—
109C22×C8—42no—
110C4⋊Q8—28no—
111C4⋊Q8—44no—
112C4⋊Q8—48no—
113C22⋊Q8—48no—
114C22⋊Q8—44no—
115C22⋊Q8—44no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C22⋊C412yes—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4—82no—
140C2×C22⋊C4SD1612yes—
141C2×C22⋊C4Q1612yes—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—24no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—82no—
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C4⋊C8—44no—
151C4⋊C8—44no—
152C22⋊C8—42no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—22no—
163#95—22no—
164C24—21no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24—41no—
174C24#10312yes—
175C24C2×C22⋊C411yescharacteristic
176C2×Q8—24no—
177C2×Q8—48no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—44no—
184C2×Q8—44no—
185C2×Q8—84no—
186C2×Q8—84no—
187C2×Q8—84no—
188C2×Q8—88no—
189C2×Q8—84no—
190C2×Q8—84no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—84no—
240C22×C4—84no—
241C22×C4#10312yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4#12012yes—
245C22×C4#12012yes—
246C22×C4C2×SD1611yescharacteristic
247C22×C4C2×Q1611yescharacteristic
248C22×C4#13512yes—
249C22×C4AGL(1,ℤ/8)11yescharacteristic
250C22×C4#13712yes—
251C22×C4#13711yescharacteristic
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C2×C8—82no—
256C2×C8—82no—
257C2×C8—82no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—48no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—48no—
265C4⋊C4—42no—
266C4⋊C4#9914yes—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—48no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—84no—
273C42—24no—
274C42—42no—
275C42—42no—
276C42—42no—
277C42—84no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23#37911yescharacteristic
306C23#41311yescharacteristic
307C23#41311yescharacteristic
308C23C2×#10311yescharacteristic
309C23#41611yescharacteristic
310C23#41711yescharacteristic
311C23#41911yescharacteristic
312Q8—88no—
313Q8—88no—
314Q8—84no—
315Q8—84no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4C2×#9912yes—
370C2×C4#41312yes—
371C2×C4#44812yes—
372C2×C4#44912yes—
373C2×C4#45112yes—
374C2×C4#45212yes—
375C8—162no—
376C8—162no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22#154211yescharacteristic
401C22#155011yescharacteristic
402C22#155411yescharacteristic
403C22#155911yescharacteristic
404C22#178111yescharacteristic
405C22#178511yescharacteristic
406C22#178511yescharacteristic
407C22#178711yescharacteristic
408C22#178811yescharacteristic
409C22#179511yescharacteristic
410C22#179611yescharacteristic
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—44no—
418C4—42no—
419C4—82no—
420C4—81no—
421C4—82no—
422C4—82no—
423C4—81no—
424C2—21no—
425C2—21no—
426C2—21no—
427C2—21no—
428C2—82no—
429C2#834311yescharacteristic
430C2#837911yescharacteristic
431C2#838111yescharacteristic
432C2#838911yescharacteristic
433C2#839511yescharacteristic
434C2#852311yescharacteristic
435C2#855211yescharacteristic
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
4. #8340C2semidirect
5. #7643C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24922
#85592
#83402
#76432