Finite Group Databasebeta

← #10692894

Subgroups — #10692894

Order 768.

690 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 103 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#82378C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic, complemented
3C3×#1466C211yesmaximal, characteristic, complemented
4#70818C212yesmaximal, complemented
5#67749C212yesmaximal, complemented
6#9457—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#940C412yes—
9C2×#930—21no—
10C2×#930—21no—
11C2×#874C2211yescharacteristic
12C3×#432C2212yes—
13#5207—21no—
14C2×#830—21no—
15#5011—24no—
16#4988C412yes—
17#4960—21no—
18#4960—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4926C2212yes—
21C3×#323—21no—
22C3×#323—21no—
23#4485—24no—
24#2973—31no—
25#2541—31no—
26#1555—32no—
27#1466D311yescharacteristic, complemented
28#1285—32no—
29C22×C3×D4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×#67—22no—
33C22×#67—22no—
34C22×D12—21no—
35C22×D12—21no—
36C22×D12C2×C412yes—
37C3×C4×D4—22no—
38C3×C4×D4—22no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C22⋊C4D411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43#940—44no—
44#930—44no—
45#930—44no—
46C2×(C2×D3)⋊C4—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×C4⋊Dic3—21no—
49C2×C4×Dic3C2×C412yes—
50C2×#208—22no—
51C22×C3⋊C8—22no—
52#874—22no—
53#874—22no—
54C3×C4⋊C8—22no—
55C3×C22⋊C8—22no—
56C3×#97—22no—
57C3⋊#95—21no—
58#830—22no—
59#830—22no—
60#830—22no—
61C4⋊(C3⋊C8)D414yescomplemented
62#803—22no—
63C2×#127—32no—
64C2×#127—31no—
65C2×#121—61no—
66C2×#121—61no—
67C2×C4×D4C2×D311yescharacteristic
68#435—64no—
69#432—32no—
70#432C2×D312yes—
71#407—61no—
72C2×C22⋊C8—61no—
73#392—61no—
74#392—61no—
75#390—32no—
76C4×C4⋊C4—31no—
77#331—64no—
78#323—21no—
79#323—21no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C22⋊C411yescharacteristic
82C23×D3—22no—
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—41no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C2×#67—44no—
98C2×#67—44no—
99C2×#67—44no—
100C2×#67—44no—
101C2×#67—82no—
102C2×#67—82no—
103C22×Dic3—22no—
104C2×D12—24no—
105C2×D12—42no—
106C2×D12—42no—
107C2×D12—42no—
108C2×D12—42no—
109C2×D12—41no—
110C2×D12—42no—
111C2×D12—41no—
112C2×D12—82no—
113C2×C3×C8—42no—
114C3×C4⋊C4—22no—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—22no—
117C3×C42C2×D412yes—
118C3×C42#4112yes—
119(C2×D3)⋊C4—44no—
120(C2×D3)⋊C4—44no—
121C4⋊Dic3—24no—
122C4×Dic3—42no—
123#208—82no—
124C2×C3⋊C8—24no—
125C2×C3⋊C8—82no—
126C22×D4—32no—
127C22×D4—62no—
128C22×D4—62no—
129C22×D4—61no—
130C22×D4—61no—
131C22×D4C4×D311yescharacteristic
132C23×C4D1211yescharacteristic
133C23×C4D1211yescharacteristic
134C2×M16—62no—
135C22×C8—62no—
136#127—62no—
137#127—62no—
138#127—124no—
139#121—124no—
140#121—124no—
141C4×D4—22no—
142C4×D4—22no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C22⋊C4—62no—
146C2×C22⋊C4—62no—
147C2×C22⋊C4#6711yescharacteristic
148C2×C22⋊C4#6711yescharacteristic
149C2×C42—32no—
150C2×C42C22×D311yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—34no—
153C22⋊C8—22no—
154C22⋊C8—62no—
155C22⋊C8—62no—
156C22⋊C8—62no—
157#97—22no—
158#97—62no—
159#95—61no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3#10012yes—
170C23×C3C2×C22⋊C411yescharacteristic
171C22×D3—42no—
172C22×D3—42no—
173C22×D3—42no—
174C22×D3—42no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—42no—
178C3×D4—44no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—22no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4C4×D412yes—
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12112yes—
194C2×C3×C4#12311yescharacteristic
195#67—88no—
196#67—88no—
197C2×Dic3—24no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200D12—84no—
201D12—82no—
202D12—84no—
203D12—82no—
204C3×C8—82no—
205C3⋊C8—44no—
206C24—62no—
207C24(C2×D3)⋊C411yescharacteristic
208C24(C2×D3)⋊C411yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C2×D4—64no—
213C2×D4—122no—
214C2×D4—122no—
215C2×D4—122no—
216C2×D4—122no—
217C2×D4—124no—
218C2×D4—124no—
219C2×D4—124no—
220C2×D4—124no—
221C2×D4—121no—
222C2×D4—121no—
223C2×D4—122no—
224C2×D4—242no—
225C2×D4—242no—
226C2×D4—242no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—62no—
234C22×C4(C2×D3)⋊C411yescharacteristic
235C22×C4(C2×D3)⋊C411yescharacteristic
236C22×C4C2×C4×D311yescharacteristic
237C22×C4C2×D1211yescharacteristic
238C22×C4C2×#6711yescharacteristic
239M16—242no—
240C2×C8—42no—
241C2×C8—64no—
242C2×C8—242no—
243C4⋊C4—22no—
244C4⋊C4—64no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—124no—
248C22⋊C4—124no—
249C42—122no—
250C42D3×D412yes—
251C42#23712yes—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3#38611yescharacteristic
265C22×C3C2×#10011yescharacteristic
266C22×C3#41311yescharacteristic
267C2×D3—82no—
268C2×D3—82no—
269C2×D3—82no—
270C2×D3—82no—
271C2×D3—82no—
272C2×D3—82no—
273C2×D3—82no—
274C3×C4—21no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4#45312yes—
280Dic3—44no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—122no—
291C23—122no—
292C23—122no—
293C23—122no—
294C23—122no—
295C23—122no—
296C23—122no—
297C23#82312yes—
298C23C2×(C2×D3)⋊C411yescharacteristic
299D4—44no—
300D4—244no—
301D4—242no—
302D4—244no—
303D4—248no—
304D4—248no—
305D4—242no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—64no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4#67⋊C412yes—
322C2×C4#88211yescharacteristic
323C2×C4#88611yescharacteristic
324C2×C4#93912yes—
325C8—82no—
326C8—124no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—42no—
330C2×C3—81no—
331C2×C3#188711yescharacteristic
332C2×C3#190011yescharacteristic
333C2×C3#191011yescharacteristic
334D3—162no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—242no—
348C22—242no—
349C22—242no—
350C22—242no—
351C22—242no—
352C22—242no—
353C22—242no—
354C22#495811yescharacteristic
355C22C2×#82311yescharacteristic
356C22#513211yescharacteristic
357C4—22no—
358C4—22no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—124no—
363C4#474712yes—
364C3#945711yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—482no—
370C2#6998511yescharacteristic
371C2#7004711yescharacteristic
372C2#7007011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1466C2
socleC22×C3#386
prime radical#82378C2
cosocleC2×C3×C42C23
hypercenter#1466D3
hypocenterC3#9457
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
2. #82302C2semidirect
3. C3×#1466C2semidirect
4. #70818C2semidirect
5. #67749C2semidirect
27. #1466D3semidirect
61. C4⋊(C3⋊C8)D4semidirect
364. C3#9457semidirect

Maximal subgroups

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

subgroupindex
#823782
#823022
C3×#14662
#708182
#677492
#94573