Finite Group Databasebeta

← #480680

Subgroups — #480680

Order 512.

1080 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2502C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#11555C218yesmaximal, complemented
4#11512C212yesmaximal, complemented
5#11506C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407—22no—
9#2502—22no—
10#2502—22no—
11#2502C414yes—
12#2502C2214yes—
13C2×#398—22no—
14C2×#391C411yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#385—22no—
17C2×#384C411yescharacteristic
18C2×#384C2211yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20#1463—22no—
21#1463—22no—
22#1463C2214yes—
23#1463C2214yes—
24C2×#324—22no—
25C2×#324C2214yes—
26#1325C2218yescomplemented
27#1319C2218yescomplemented
28C23×Q8C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D412yes—
31C22×M16—42no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—42no—
34C2×C22⋊Q8—41no—
35C2×C22⋊Q8—41no—
36C2×C22⋊Q8C2×C414yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—42no—
41C22×C4⋊C4—41no—
42C22×C4⋊C4—41no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4D412yes—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55#407—24no—
56#407—24no—
57#407—42no—
58#407—42no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8—42no—
61#398—24no—
62#398—24no—
63#398—24no—
64#398—24no—
65#391—21no—
66#391—21no—
67#391—22no—
68#391C2×C412yes—
69#391C2×C414yes—
70#391C2312yes—
71#385—24no—
72#385—24no—
73#385—24no—
74#385—24no—
75#384—21no—
76#384—21no—
77#384—22no—
78#384C2×C412yes—
79#384C2×C414yes—
80#384C2312yes—
81C4×C22⋊C4—22no—
82C4×C22⋊C4C2×C414yes—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86#324—24no—
87#324—24no—
88#324—24no—
89#324C2318yes—
90C25C22⋊C411yescharacteristic
91C22×Q8—21no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×Q8—41no—
95C22×Q8—41no—
96C22×Q8C22×C412yes—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4—84no—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C412yes—
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C412yes—
115C23×C4C22⋊C412yes—
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C2×M16—44no—
122C2×M16—44no—
123C2×M16—88no—
124C22×C8—44no—
125C22⋊Q8—44no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—84no—
154C2×C4⋊C4—84no—
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2411yescharacteristic
166C2×C22⋊C4—28no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42C22×C412yes—
177C2×C42C22×C412yes—
178C22⋊C8—28no—
179C22⋊C8—44no—
180C22⋊C8—44no—
181C22⋊C8—44no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187C24—24no—
188C24—41no—
189C24—41no—
190C24—42no—
191C24#9912yes—
192C24#9911yescharacteristic
193C24#9911yescharacteristic
194C24C2×C22⋊C411yescharacteristic
195C24C2×C22⋊C412yes—
196C2×Q8—24no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—84no—
203C2×Q8—84no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—48no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D412yes—
269M16—88no—
270M16—88no—
271C2×C8—48no—
272C2×C8—88no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—48no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C22⋊C4—216no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C42—24no—
294C42—42no—
295C42—42no—
296C42—42no—
297C23—21no—
298C23—21no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—24no—
304C23—24no—
305C23—24no—
306C23—22no—
307C23—41no—
308C23—42no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—42no—
313C23—48no—
314C23—48no—
315C23#35614yes—
316C23C2×#9912yes—
317C23C2×#9911yescharacteristic
318C23C2×#9912yes—
319C23C2×#9911yescharacteristic
320C23C22×C22⋊C411yescharacteristic
321Q8—84no—
322Q8—84no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4#41312yes—
376C2×C4#56812yes—
377C8—88no—
378C22—24no—
379C22—24no—
380C22—21no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—48no—
395C22—44no—
396C22#212712yes—
397C22C2×#35612yes—
398C22C2×#35614yes—
399C22#287911yescharacteristic
400C22C22×#9911yescharacteristic
401C22C2×#41311yescharacteristic
402C4—22no—
403C4—22no—
404C4—44no—
405C4—44no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—82no—
412C4—82no—
413C2—21no—
414C2—21no—
415C2—22no—
416C2—22no—
417C2—24no—
418C2—42no—
419C2#1739211yescharacteristic
420C2C2×#212711yescharacteristic
421C2C22×#35611yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11555C2semidirect
4. #11512C2semidirect
5. #11506C2semidirect
26. #1325C22semidirect
27. #1319C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#14632
#115552
#115122
#115062