Finite Group Databasebeta

← #10705771

Subgroups — #10705771

Order 768.

768 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 87 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#82403C211yesmaximal, characteristic
2C3×#1465C211yesmaximal, characteristic, complemented
3C4×#833C211yesmaximal, characteristic, complemented
4#67456C214yesmaximal, complemented
5#11779—31nomaximal, Sylow
6C23×D12C411yescharacteristic
7C2×C3×C4×D4C2211yescharacteristic
8C2×#937—21no—
9C2×#930—21no—
10C3×#407—21no—
11C2×#833—21no—
12#5212C411yescharacteristic
13#4991C2212yes—
14#4960C412yes—
15#4955—21no—
16#4939—21no—
17C4×#208—21no—
18C3×#324C2212yescomplemented
19C3×#323—21no—
20#4530C2212yescomplemented
21#2408—31no—
22C4×#100—31no—
23#1465D311yescharacteristic, complemented
24#1345—34no—
25C24×D3—21no—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4C2×C411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C22×#67—22no—
30C22×D12—21no—
31C22×D12—44no—
32C22×D12—41no—
33C22×D12—41no—
34C22×D12C2×C412yes—
35C2×C3×M16—41no—
36C3×C4×D4—41no—
37C3×C4×D4—41no—
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×C3×C22⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×#217—21no—
43#937—44no—
44#930—44no—
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×C4⋊Dic3—21no—
49C2×#208—22no—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—41no—
52#833—24no—
53C3⋊#95C2×C412yes—
54#803—41no—
55C4×C3⋊C8—41no—
56C23×D4—31no—
57C2×#121—61no—
58C2×#120—61no—
59C2×C4×D4C2×D311yescharacteristic
60C2×#100—61no—
61#407—21no—
62C4×M16—61no—
63#394—32no—
64#392—32no—
65#390—61no—
66#380—31no—
67#379—61no—
68#343—32no—
69#324C2×D312yescomplemented
70#323—21no—
71C24×C3C22⋊C411yescharacteristic
72C24×C3C22⋊C411yescharacteristic
73C23×D3—22no—
74C23×D3—44no—
75C23×D3—42no—
76C23×D3—42no—
77C23×D3—42no—
78C23×D3—82no—
79C2×C3×D4—22no—
80C2×C3×D4—22no—
81C2×C3×D4—41no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C2×#67—44no—
94C2×#67—44no—
95C2×#67—82no—
96C22×Dic3—22no—
97C2×D12—24no—
98C2×D12—42no—
99C2×D12—42no—
100C2×D12—48no—
101C2×D12—44no—
102C2×D12—41no—
103C2×D12—41no—
104C2×D12—44no—
105C2×D12—44no—
106C3×M16—82no—
107C2×C3×C8—42no—
108C3×C4⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—41no—
111C3×C42—21no—
112C3×C42—21no—
113#217—42no—
114(C2×D3)⋊C4—44no—
115(C2×D3)⋊C4—44no—
116(C2×D3)⋊C4—82no—
117C4⋊Dic3—42no—
118#208—24no—
119#208—42no—
120C2×C3⋊C8—42no—
121C25—61no—
122C22×D4—32no—
123C22×D4—62no—
124C22×D4—61no—
125C22×D4—121no—
126C22×D4—121no—
127C22×D4—124no—
128C22×D4D1211yescharacteristic
129C23×C4C4×D311yescharacteristic
130C23×C4#6711yescharacteristic
131C2×M16—41no—
132C2×M16—62no—
133#121—124no—
134#120—124no—
135C4×D4—41no—
136C4×D4—41no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4C22×D311yescharacteristic
139C2×C22⋊C4—61no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4C4×D311yescharacteristic
144C2×C22⋊C4#6711yescharacteristic
145C2×C42D1211yescharacteristic
146#100—64no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149#97—121no—
150C4×C8—121no—
151#95—32no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3#10412yes—
162C23×C3C2×C22⋊C411yescharacteristic
163C22×D3—24no—
164C22×D3—42no—
165C22×D3—42no—
166C22×D3—42no—
167C22×D3—48no—
168C22×D3—42no—
169C22×D3—42no—
170C22×D3—42no—
171C22×D3—88no—
172C22×D3—88no—
173C22×D3—84no—
174C22×D3—84no—
175C3×D4—44no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—22no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—81no—
189C2×C3×C4#9911yescharacteristic
190C2×C3×C4#9911yescharacteristic
191C2×C3×C4AGL(1,ℤ/8)12yes—
192#67—88no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196D12—44no—
197D12—44no—
198D12—44no—
199D12—44no—
200C3×C8—82no—
201C3⋊C8—44no—
202C24—62no—
203C24—122no—
204C24—122no—
205C24—124no—
206C24—122no—
207C24—242no—
208C24(C2×D3)⋊C411yescharacteristic
209C24(C2×D3)⋊C411yescharacteristic
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—41no—
213C2×D4—64no—
214C2×D4—122no—
215C2×D4—122no—
216C2×D4—128no—
217C2×D4—124no—
218C2×D4—124no—
219C2×D4—124no—
220C2×D4—124no—
221C2×D4—124no—
222C2×D4—121no—
223C2×D4—121no—
224C2×D4—242no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—62no—
232C22×C4(C2×D3)⋊C411yescharacteristic
233C22×C4(C2×D3)⋊C411yescharacteristic
234C22×C4C2×C4×D311yescharacteristic
235C22×C4C2×D1211yescharacteristic
236C22×C4C2×#6711yescharacteristic
237M16—64no—
238M16—82no—
239M16—122no—
240C2×C8—42no—
241C2×C8—122no—
242C4⋊C4—41no—
243C4⋊C4—122no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—124no—
247C22⋊C4—122no—
248C22⋊C4—124no—
249C22⋊C4—242no—
250C42—21no—
251C42—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—42no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—81no—
264C22×C3C2×#9911yescharacteristic
265C22×C3#41811yescharacteristic
266C22×C3C2×#10411yescharacteristic
267C2×D3—44no—
268C2×D3—44no—
269C2×D3—44no—
270C2×D3—82no—
271C2×D3—82no—
272C2×D3—816no—
273C2×D3—82no—
274C2×D3—82no—
275C3×C4—21no—
276C3×C4—22no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4AΣL(1,𝔽16)12yes—
281Dic3—82no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—64no—
292C23—122no—
293C23—122no—
294C23—122no—
295C23—128no—
296C23—122no—
297C23—122no—
298C23—122no—
299C23—244no—
300C23—248no—
301C23—248no—
302C23—244no—
303C23#82512yes—
304C23C2×(C2×D3)⋊C411yescharacteristic
305D4—44no—
306D4—124no—
307D4—124no—
308D4—124no—
309D4—124no—
310D4—248no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4#80611yescharacteristic
328C2×C4#80611yescharacteristic
329C2×C4#90812yes—
330C8—82no—
331C8—124no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—42no—
335C2×C3—81no—
336C2×C3#150711yescharacteristic
337C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
338C2×C3#213411yescharacteristic
339D3—84no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—81no—
352C22—124no—
353C22—124no—
354C22—124no—
355C22—242no—
356C22—242no—
357C22—2416no—
358C22—242no—
359C22—242no—
360C22C2×#80611yescharacteristic
361C22#513111yescharacteristic
362C22C2×#82511yescharacteristic
363C4—22no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—242no—
369C4#447112yes—
370C3#1177911yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—42no—
374C2—81no—
375C2—244no—
376C2#6997711yescharacteristic
377C2C2×#447111yescharacteristic
378C2#7014911yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1465C2
socleC22×C3C2×#99
prime radical#82403C2
cosocleC2×C3×C4⋊C4C23
hypercenter#1465D3
hypocenterC3#11779
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. C3×#1465C2semidirect
3. C4×#833C2semidirect
4. #67456C2semidirect
18. C3×#324C22semidirect
20. #4530C22semidirect
23. #1465D3semidirect
69. #324C2×D3semidirect
370. C3#11779semidirect

Maximal subgroups

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

subgroupindex
#824032
C3×#14652
C4×#8332
#674562
#117793