Finite Group Databasebeta

← #100126

Subgroups — #100126

Order 512.

720 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 114 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
1#21818C211yesmaximal, characteristic
2#14024C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7595C212yesmaximal
5#7017C212yesmaximal
6C22×C4⋊Q8C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C2×#431—21no—
9C2×#391C411yescharacteristic
10C2×#391C411yescharacteristic
11C2×#391C2211yescharacteristic
12#1984—24no—
13#1817—21no—
14#1817C2212yes—
15#1817C2212yes—
16C2×#324—21no—
17C2×#324—21no—
18#1441—21no—
19#1441—21no—
20#1436—21no—
21#1436—21no—
22C4⋊#95C411yescharacteristic
23C4×#95C2211yescharacteristic
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C23×Q8C2×C411yescharacteristic
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—42no—
29C2×C4⋊Q8—42no—
30C2×C4⋊Q8C814yescomplemented
31C2×C4×Q8—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8—42no—
34C2×C4×Q8—42no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42D411yescharacteristic
44C22×C42D411yescharacteristic
45C22×C42C2311yescharacteristic
46#431—42no—
47#431—42no—
48C2×C4⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C4×C8—22no—
52#391—22no—
53#391—22no—
54#391—42no—
55#391—42no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66#336—24no—
67#336—22no—
68#336—22no—
69#336—22no—
70#336D414yes—
71#324—42no—
72#324—42no—
73C22×Q8—22no—
74C22×Q8—22no—
75C22×Q8—22no—
76C22×Q8—21no—
77C22×Q8—22no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8C2×C812yes—
87C22×Q8C2×C812yes—
88C22×Q8M1612yes—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C22×C8—24no—
101C22×C8—42no—
102C4⋊Q8—48no—
103C4⋊Q8—48no—
104C4×Q8—44no—
105C4×Q8—44no—
106C4×Q8—48no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—22no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C2×C812yes—
142C2×C42C2×D412yes—
143C2×C42#4112yes—
144C4⋊C8—44no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149C4×C8—44no—
150#95—24no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—41no—
159#95—42no—
160#95—41no—
161C24C2×C22⋊C411yescharacteristic
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—24no—
165C2×Q8—24no—
166C2×Q8—24no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—82no—
179C2×Q8—82no—
180C2×Q8—82no—
181C2×Q8—82no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
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—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
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—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4C22⋊C812yes—
242C22×C4C22⋊C812yes—
243C22×C4#9912yes—
244C22×C4#9912yes—
245C22×C4#10012yes—
246C22×C4C4×D412yes—
247C22×C4#12011yescharacteristic
248C22×C4#12112yes—
249C22×C4#12311yescharacteristic
250C22×C4C22×C811yescharacteristic
251C22×C4C2×M1611yescharacteristic
252C22×C4#13112yes—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—44no—
256C2×C8—82no—
257C2×C8—82no—
258C2×C8—82no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C4⋊C4—82no—
265C42—22no—
266C42—22no—
267C42—24no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—42no—
274C42—82no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23#38611yescharacteristic
280C23C2×C22⋊C811yescharacteristic
281C23#40811yescharacteristic
282C23C2×#9911yescharacteristic
283C23C2×#9911yescharacteristic
284C23C2×#10011yescharacteristic
285C23#41311yescharacteristic
286Q8—48no—
287Q8—48no—
288Q8—84no—
289Q8—84no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4#32314yes—
362C2×C4C8×D412yes—
363C2×C4#43512yes—
364C2×C4#45612yes—
365C8—84no—
366C8—162no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22C2×#32311yescharacteristic
380C22#145911yescharacteristic
381C22#146811yescharacteristic
382C22#147111yescharacteristic
383C22#186911yescharacteristic
384C22#188411yescharacteristic
385C22#188611yescharacteristic
386C22#189811yescharacteristic
387C22#190011yescharacteristic
388C22#190911yescharacteristic
389C22#191211yescharacteristic
390C4—24no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—44no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—41no—
399C4—82no—
400C4—82no—
401C4—81no—
402C4—161no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2#895811yescharacteristic
408C2#907911yescharacteristic
409C2#910011yescharacteristic
410C2#944311yescharacteristic
411C2#944811yescharacteristic
412C2#945211yescharacteristic
413C2#946211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×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
30. C2×C4⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
#218182
#140242
#83032
#75952
#70172