Finite Group Databasebeta

← #213913

Subgroups — #213913

Order 512.

1652 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1652 conjugacy classes of subgroups — too many to draw. Only its 206 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#35666C211yesmaximal, characteristic
2#20593C212yesmaximal
3#9513C218yesmaximal, complemented
4#8753C212yesmaximal, complemented
5#8551C212yesmaximal, complemented
6C22×2−1+4C2211yescharacteristic
7C2×#536C411yescharacteristic
8C2×#536C411yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10C2×#413—22no—
11C22×#101—22no—
12#2879C2214yes—
13#2434—22no—
14#2434—22no—
15#2432—22no—
16#2432—22no—
17#2406C414yes—
18#2406C2214yes—
19#2310C2211yescharacteristic
20#2297C411yescharacteristic
21#2297C411yescharacteristic
22#1785—22no—
23#1785C2214yes—
24#1784—22no—
25#1784C2214yes—
26#1463C2218yes—
27#1306C2218yes—
28C2×2−1+4—44no—
29C2×2−1+4—44no—
30C2×2−1+4—44no—
31C22×#41C2×C411yescharacteristic
32C22×#41C2×C411yescharacteristic
33C22×#41C2×C412yes—
34C22×#41D412yes—
35C22×#41D412yes—
36C22×#41C2312yes—
37C23×Q8C2×C411yescharacteristic
38C23×Q8D412yes—
39C23×Q8D412yes—
40C23×Q8C2311yescharacteristic
41C22×M16—24no—
42#536—44no—
43#536—44no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—44no—
49C2×#124—21no—
50C2×#124—21no—
51C2×#123C2×C414yes—
52C2×#123C2×C414yes—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2312yes—
59C22×C42C2×C411yescharacteristic
60#413—48no—
61C2×#101—24no—
62C2×#101—24no—
63C2×#101—24no—
64C2×#101—48no—
65C2×#100—24no—
66#408—48no—
67#407—24no—
68#407C2318yes—
69#395—24no—
70#395—24no—
71#394—24no—
72#394—24no—
73#391—21no—
74#391—21no—
75#391—22no—
76#391—22no—
77#390—21no—
78#390—21no—
79#386C2×C414yes—
80#386C2×C414yes—
81#385—24no—
82#385—24no—
83#385—24no—
84#385—24no—
85#381C2×C412yes—
86#381C2312yes—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89#324—48no—
902−1+4—416no—
912−1+4—432no—
922−1+4—416no—
93C2×#41—24no—
94C2×#41—24no—
95C2×#41—28no—
96C2×#41—28no—
97C2×#41—42no—
98C2×#41—42no—
99C2×#41—44no—
100C2×#41—42no—
101C2×#41—42no—
102C2×#41—44no—
103C2×#41—44no—
104C2×#41—44no—
105C2×#41—44no—
106C2×#41—44no—
107C2×#41—44no—
108C2×#41—44no—
109C2×#41—44no—
110C2×#41—44no—
111C22×Q8—28no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—24no—
115C22×Q8—44no—
116C22×Q8—42no—
117C22×Q8—44no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22⋊C412yes—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C2×D412yes—
129C22×Q8C2×D412yes—
130C22×Q8C2×D412yes—
131C22×D4C22⋊C411yescharacteristic
132C22×D4C22⋊C412yes—
133C22×D4C22⋊C411yescharacteristic
134C22×D4C22×C412yes—
135C22×D4C22×C412yes—
136C22×D4C2×D412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2411yescharacteristic
148C2×M16—28no—
149C2×M16—44no—
150C2×M16—48no—
151C2×M16—48no—
152C22×C8—44no—
153C4⋊Q8—48no—
154C4⋊Q8—48no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157#124—44no—
158#124—44no—
159#123—48no—
160#123—48no—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163#117—42no—
164#117—42no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—44no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4C22×C412yes—
178C2×C22⋊C4C22×C412yes—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—42no—
184#101—48no—
185#101—48no—
186#101—48no—
187#100—48no—
188C22⋊C8—216no—
189C22⋊C8—48no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194C24C2×C22⋊C412yes—
195C24C2×C22⋊C412yes—
196C24C23×C411yescharacteristic
197#41—416no—
198#41—416no—
199#41—416no—
200#41—416no—
201#41—48no—
202#41—48no—
203#41—48no—
204#41—48no—
205#41—432no—
206#41—432no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—416no—
225C2×Q8—416no—
226C2×Q8—48no—
227C2×Q8—48no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—432no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—84no—
236C2×Q8—84no—
237C2×Q8#9914yes—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4#9914yes—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4H(5,𝔽2)14yes—
298M16—48no—
299M16—48no—
300M16—88no—
301C2×C8—48no—
302C2×C8—88no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—42no—
319C23—22no—
320C23—21no—
321C23—21no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—42no—
327C23—44no—
328C23—44no—
329C23C2×#9914yes—
330C23C22×C22⋊C411yescharacteristic
331C23#51812yes—
332Q8—44no—
333Q8—48no—
334Q8—48no—
335Q8—48no—
336Q8—48no—
337Q8—44no—
338Q8—84no—
339Q8—84no—
340D4—44no—
341D4—48no—
342D4—48no—
343D4—44no—
344D4—48no—
345D4—48no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—28no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—44no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—84no—
384C2×C4C2×#9912yes—
385C2×C4C2×#9912yes—
386C2×C4C2×#9914yes—
387C2×C4#53414yes—
388C8—88no—
389C22—22no—
390C22—24no—
391C22—22no—
392C22—24no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—44no—
397C22—44no—
398C22—44no—
399C22—42no—
400C22—82no—
401C22—82no—
402C22#230811yescharacteristic
403C22C22×#9911yescharacteristic
404C22C22×#9911yescharacteristic
405C4—24no—
406C4—24no—
407C4—22no—
408C4—22no—
409C4—44no—
410C4—42no—
411C4—42no—
412C4—82no—
413C4—82no—
414C4—82no—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—44no—
419C2—82no—
420C2#2174611yescharacteristic
421C2#2178012yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×2−1+4C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9513C2semidirect
4. #8753C2semidirect
5. #8551C2semidirect

Maximal subgroups

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

subgroupindex
#356662
#205932
#95132
#87532
#85512