Finite Group Databasebeta

← #224200

Subgroups — #224200

Order 512.

737 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36566C211yesmaximal, characteristic
2#21346C212yesmaximal, complemented
3#9895C212yesmaximal, complemented
4#9893C212yesmaximal, complemented
5#9892C212yesmaximal, complemented
6#8579C211yesmaximal, characteristic
7#8579C212yesmaximal
8#8226C212yesmaximal
9#8224C211yesmaximal, characteristic
10#2864C2212yes—
11#2862C2211yescharacteristic
12#2589—22no—
13#2581C411yescharacteristic
14#2580C411yescharacteristic
15#2574—22no—
16#2465—22no—
17#2452C412yes—
18#2451C412yes—
19#2368—22no—
20#2351C411yescharacteristic
21#2349C411yescharacteristic
22C4×#124C2211yescharacteristic
23C4×#124C2212yes—
24C42×Q8C2211yescharacteristic
25#1985—22no—
26#1940—22no—
27#1916—22no—
28#1916—22no—
29#1767C2212yes—
30#1767C2212yes—
31#1767C2212yes—
32#1745C2212yes—
33#1745C2212yes—
34C4×#97C2212yes—
35C2×#329C2211yescharacteristic
36C2×#329C2212yes—
37C2×#329C2211yescharacteristic
38#1295C2212yes—
39#1295C2212yes—
40#1295C2212yes—
41#1295C2211yescharacteristic
42#1295C2212yes—
43#1295C2211yescharacteristic
44#1295C2212yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#124C2311yescharacteristic
47C2×C22⋊Q8—22no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D412yes—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C412yes—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×#97—22no—
58C2×#97C2312yes—
59C2×#97C2312yes—
60C2×#97C2312yes—
61C2×#97C2312yes—
62C2×C4×C8—22no—
63#400—21no—
64#399—21no—
65#398—22no—
66#396—22no—
67#395—21no—
68#395—22no—
69#393—21no—
70#393—22no—
71#391—22no—
72#391—22no—
73#391—22no—
74#391C2×C412yes—
75#391C2×C411yescharacteristic
76#390C2×C412yes—
77#390C2×C411yescharacteristic
78#389—22no—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#388—22no—
84#388—22no—
85#388—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#385—22no—
93#385—22no—
94#385—22no—
95#382—21no—
96#382C2×C411yescharacteristic
97#382C2×C411yescharacteristic
98#382C2312yes—
99#382C2311yescharacteristic
100#380—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C2×#95—21no—
111C43C2311yescharacteristic
112C43C2312yes—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#329—22no—
121#329—24no—
122#329Q814yescomplemented
123C22×Q8—22no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126C22×D4C22×C411yescharacteristic
127C23×C4—22no—
128C23×C4—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C4⋊Q8—82no—
134#124—22no—
135#124C2×Q812yes—
136#124#4112yes—
137C22⋊Q8—48no—
138C4×Q8—42no—
139C4×Q8—44no—
140C4×Q8—42no—
141C4×Q8—82no—
142C4×D4—42no—
143C4×D4—82no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2411yescharacteristic
179C4⋊C8—44no—
180C4⋊C8—44no—
181C4⋊C8—44no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—44no—
190#97C2×Q814yes—
191C4×C8—44no—
192#95—21no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—21no—
198#95—22no—
199#95—21no—
200#95—21no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95C22⋊C411yescharacteristic
207#95C22⋊C411yescharacteristic
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210C24—21no—
211C2×Q8—21no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—44no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8C4×Q812yes—
220C2×D4—21no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4C4×Q812yes—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C412yes—
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D412yes—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—42no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—82no—
277C22⋊C4—82no—
278C42—22no—
279C42—24no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—41no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—82no—
290C42C22×Q811yescharacteristic
291C42C2×#4111yescharacteristic
292C422−1+412yes—
293C23—22no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—42no—
298C23C22×C22⋊C411yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—44no—
302Q8—164no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
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—42no—
338C2×C4—42no—
339C2×C4—164no—
340C2×C4—162no—
341C2×C4#41312yes—
342C2×C4#42114yes—
343C2×C4C2×C4×Q811yescharacteristic
344C2×C4#51911yescharacteristic
345C2×C4#54812yes—
346C2×C4D4×Q812yes—
347C8—44no—
348C8—44no—
349C8—82no—
350C8—82no—
351C22—44no—
352C22—42no—
353C22—81no—
354C22—82no—
355C22—81no—
356C22Q8×C22⋊C411yescharacteristic
357C22C2×#41311yescharacteristic
358C22#288811yescharacteristic
359C22C2×#42112yes—
360C22#290112yes—
361C4—22no—
362C4—22no—
363C4—24no—
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—81no—
374C4—82no—
375C4—82no—
376C4—81no—
377C2—82no—
378C2#1767311yescharacteristic
379C2#1767912yes—
380C2#1768611yescharacteristic
381C2#2251511yescharacteristic
382C2#2252912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21346C2semidirect
3. #9895C2semidirect
4. #9893C2semidirect
5. #9892C2semidirect
122. #329Q8semidirect

Maximal subgroups

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

subgroupindex
#365662
#213462
#98952
#98932
#98922
#85792
#82262
#82242