Finite Group Databasebeta

← #10714539

Subgroups — #10714539

Order 768.

482 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4696C211yesmaximal, characteristic
2C2×#4503C211yesmaximal, characteristic, complemented
3C2×#4503C211yesmaximal, characteristic, complemented
4C2×#4509C211yesmaximal, characteristic
5C2×#4509C211yesmaximal, characteristic
6#69770C211yesmaximal, characteristic, complemented
7C3×#1374C211yesmaximal, characteristic, complemented
8#13408—31nomaximal, Sylow
9C22×Dic12C2211yescharacteristic
10C22×D24C2211yescharacteristic
11C22×#204—21no—
12C2×#875—21no—
13C22×C3×C16C2211yescharacteristic
14C22×C3×C16C2211yescharacteristic
15C2×C3×C4×C8C2211yescharacteristic
16C2×#818C2211yescharacteristic
17C2×#818C2211yescharacteristic
18#4928—21no—
19#4696—22no—
20#4696—22no—
21#4551—21no—
22#4509—22no—
23#4509—22no—
24#4503—22no—
25#4503—22no—
26C2×#495—31no—
27C2×#358—31no—
28C2×#358—31no—
29C2×#357—31no—
30C2×#357—31no—
31#1942—31no—
32#1374D311yescharacteristic, complemented
33C22×D12—21no—
34C22×Dic6—21no—
35C22×C3×C8D411yescharacteristic
36C22×C3×C8C2311yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×(C2×D3)⋊C4—41no—
39C2×(C2×D3)⋊C4—41no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×Dic3⋊C4—41no—
43C2×Dic12—21no—
44C2×Dic12—21no—
45C2×Dic12—21no—
46C2×Dic12—21no—
47C2×Dic12—21no—
48C2×Dic12D412yes—
49C2×D24—21no—
50C2×D24—21no—
51C2×D24—21no—
52C2×D24—21no—
53C2×D24—21no—
54C2×D24D412yes—
55C2×#204—24no—
56C2×#204—42no—
57C2×#204—42no—
58#875—24no—
59#875—42no—
60C2×C3×C16—21no—
61C2×C3×C16—21no—
62C2×C3×C16—21no—
63C2×C3×C16—21no—
64C2×C3×C16—21no—
65C2×C3×C16—21no—
66C3×C4×C8—21no—
67C3×C4×C8—21no—
68C3⋊#95—41no—
69#818—21no—
70#818—21no—
71#818—21no—
72#818D412yes—
73C22×Q16—31no—
74C22×SD16—61no—
75C22×D8—31no—
76C2×#124—61no—
77C22×C16C2×D311yescharacteristic
78C22×C16C2×D311yescharacteristic
79#495—62no—
80#495—62no—
81C2×#107—31no—
82C2×#107—31no—
83C2×C4×C8C2×D311yescharacteristic
84#389—61no—
85#358—62no—
86#358—62no—
87#357—62no—
88#357—62no—
89#340—61no—
90C23×D3—41no—
91C22×C3×C4D811yescharacteristic
92C22×C3×C4D811yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×Dic3—41no—
95C22×Dic3—41no—
96C22×Dic3—41no—
97C2×D12—22no—
98C2×D12—41no—
99C2×D12—41no—
100C2×D12—41no—
101C2×D12—41no—
102C2×D12—41no—
103C2×Dic6—22no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C2×Dic6—41no—
107C2×Dic6—41no—
108C2×Dic6—41no—
109C2×C3×C8—21no—
110C2×C3×C8—21no—
111C2×C3×C8SD1612yes—
112C2×C3×C8C2×D411yescharacteristic
113C2×C3×C8C2×D411yescharacteristic
114C2×C3×C8C2×D411yescharacteristic
115C2×C3×C8#4111yescharacteristic
116C2×C3×C8#4111yescharacteristic
117C2×C3×C8#4111yescharacteristic
118C3×C42—21no—
119C3×C42—21no—
120(C2×D3)⋊C4—44no—
121(C2×D3)⋊C4—44no—
122C4⋊Dic3—22no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125C4⋊Dic3—41no—
126Dic3⋊C4—82no—
127Dic12—22no—
128Dic12—22no—
129Dic12—41no—
130Dic12—41no—
131D24—22no—
132D24—22no—
133D24—41no—
134D24—41no—
135#204—48no—
136C3×C16—41no—
137C3×C16—41no—
138C22×Q8—61no—
139C22×D4—61no—
140C2×Q16—32no—
141C2×Q16—61no—
142C2×Q16—61no—
143C2×Q16—61no—
144C2×Q16—61no—
145C2×Q16—61no—
146C2×SD16—64no—
147C2×SD16—122no—
148C2×SD16—122no—
149C2×D8—32no—
150C2×D8—61no—
151C2×D8—61no—
152C2×D8—61no—
153C2×D8—61no—
154C2×D8—61no—
155C22×C8D1211yescharacteristic
156C22×C8C22×D311yescharacteristic
157#124—64no—
158#124—122no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—121no—
162C2×C22⋊C4—121no—
163C2×C22⋊C4—121no—
164C2×C42D1211yescharacteristic
165C2×C16—21no—
166C2×C16—21no—
167C2×C16—21no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—21no—
171#107—32no—
172#107—61no—
173#107—61no—
174#107—61no—
175C4×C8—21no—
176C4×C8—21no—
177#95—121no—
178C23×C3C2×D811yescharacteristic
179C22×D3—42no—
180C22×D3—81no—
181C22×D3—81no—
182C22×D3—81no—
183C22×D3—81no—
184C22×D3—81no—
185C22×D3—81no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12411yescharacteristic
195C2×C3×C4C2×SD1611yescharacteristic
196C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207D12—42no—
208D12—42no—
209D12—81no—
210D12—81no—
211Dic6—42no—
212Dic6—42no—
213Dic6—81no—
214Dic6—81no—
215C3×C8—22no—
216C3×C8#12111yescharacteristic
217C3×C8#12111yescharacteristic
218C3×C8#12311yescharacteristic
219C3×C8#12311yescharacteristic
220C24—121no—
221C2×Q8—62no—
222C2×Q8—121no—
223C2×Q8—121no—
224C2×Q8—121no—
225C2×Q8—121no—
226C2×Q8—121no—
227C2×D4—62no—
228C2×D4—121no—
229C2×D4—121no—
230C2×D4—121no—
231C2×D4—121no—
232C2×D4—121no—
233C22×C4—121no—
234C22×C4—121no—
235C22×C4—121no—
236C22×C4D2411yescharacteristic
237C22×C4D2411yescharacteristic
238C22×C4C2×D1211yescharacteristic
239Q16—62no—
240Q16—62no—
241Q16—121no—
242Q16—121no—
243SD16—128no—
244D8—62no—
245D8—62no—
246D8—121no—
247D8—121no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8#20412yes—
251C2×C8#23511yescharacteristic
252C2×C8#23511yescharacteristic
253C2×C8D3×D411yescharacteristic
254C2×C8D3×D411yescharacteristic
255C2×C8D3×D411yescharacteristic
256C2×C8#23911yescharacteristic
257C4⋊C4—62no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—242no—
262C22⋊C4—124no—
263C22⋊C4—124no—
264C42—21no—
265C42—21no—
266C16—41no—
267C16—41no—
268C22×C3#44711yescharacteristic
269C22×C3#45911yescharacteristic
270C22×C3#48611yescharacteristic
271C22×C3#50811yescharacteristic
272C22×C3#50811yescharacteristic
273C22×C3#50911yescharacteristic
274C22×C3#51011yescharacteristic
275C2×D3—82no—
276C2×D3—82no—
277C2×D3—82no—
278C2×D3—161no—
279C2×D3—161no—
280C2×D3—161no—
281C2×D3—161no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4#39411yescharacteristic
285C3×C4#45011yescharacteristic
286C3×C4#46011yescharacteristic
287C3×C4#49011yescharacteristic
288Dic3—82no—
289Dic3—82no—
290Dic3—161no—
291C23—122no—
292C23—241no—
293C23—241no—
294C23—241no—
295C23—241no—
296C23—241no—
297C23—241no—
298C23C2×D2411yescharacteristic
299Q8—122no—
300Q8—122no—
301Q8—241no—
302Q8—241no—
303D4—122no—
304D4—122no—
305D4—241no—
306D4—241no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—241no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4—241no—
321C2×C4—241no—
322C2×C4#87511yescharacteristic
323C2×C4#88211yescharacteristic
324C2×C4#89511yescharacteristic
325C2×C4C2×#20411yescharacteristic
326C2×C4#90811yescharacteristic
327C2×C4#90811yescharacteristic
328C8—22no—
329C8#88411yescharacteristic
330C8#88411yescharacteristic
331C8#89411yescharacteristic
332C8#89411yescharacteristic
333C2×C3#204111yescharacteristic
334C2×C3#218611yescharacteristic
335C2×C3#218911yescharacteristic
336C2×C3#221011yescharacteristic
337C2×C3#221111yescharacteristic
338C2×C3#224211yescharacteristic
339C2×C3#224311yescharacteristic
340D3—162no—
341C22—242no—
342C22—242no—
343C22—242no—
344C22—481no—
345C22—481no—
346C22—481no—
347C22—481no—
348C22#469411yescharacteristic
349C22#473211yescharacteristic
350C22#484311yescharacteristic
351C22#490411yescharacteristic
352C22#490411yescharacteristic
353C22#490811yescharacteristic
354C22#490911yescharacteristic
355C4—41no—
356C4—41no—
357C4—242no—
358C4—242no—
359C4—481no—
360C4#467211yescharacteristic
361C4#471311yescharacteristic
362C4#473111yescharacteristic
363C4#484811yescharacteristic
364C3#1340811yescharacteristic, complemented, Sylow
365C2—482no—
366C2#6862911yescharacteristic
367C2#6905911yescharacteristic
368C2#6907911yescharacteristic
369C2#6910911yescharacteristic
370C2#6911111yescharacteristic
371C2#6928811yescharacteristic
372C2#6929411yescharacteristic
subgroupG/H
centerC23C2×D24
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#1374C2
socleC23×C3C2×D8
prime radicalC22×D24C22
hypercenter#1374D3
hypocenterC3#13408
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1374

Complements

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

NH ≅ G/Ntype
2. C2×#4503C2semidirect
3. C2×#4503C2semidirect
6. #69770C2semidirect
7. C3×#1374C2semidirect
32. #1374D3semidirect
364. C3#13408semidirect

Maximal subgroups

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

subgroupindex
C2×#46962
C2×#45032
C2×#45092
#697702
C3×#13742
#134083