Finite Group Databasebeta

← #10714529

Subgroups — #10714529

Order 768.

566 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5151C211yesmaximal, characteristic
2C2×C3×#357C211yesmaximal, characteristic, complemented
3C2×#4562C211yesmaximal, characteristic, complemented
4C2×#4509C211yesmaximal, characteristic, complemented
5#70611C211yesmaximal, characteristic, complemented
6C2×#4491C211yesmaximal, characteristic, complemented
7#67804C211yesmaximal, characteristic, complemented
8#13407—31nomaximal, Sylow
9C22×C3×D8C2211yescharacteristic
10C2×#940—21no—
11C22×#213—21no—
12C22×D24C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C2×C3×#107C2211yescharacteristic
15#5151—22no—
16#5151—22no—
17C2×#818C2211yescharacteristic
18C2×C8×Dic3C2211yescharacteristic
19C22×C3⋊C16C2211yescharacteristic
20#4972—21no—
21C3×#357—22no—
22#4562—22no—
23#4529—21no—
24#4509—22no—
25#4491—22no—
26C2×#493—31no—
27C2×#357—31no—
28C2×#357—31no—
29C2×#357—31no—
30C2×#357D311yescharacteristic, complemented
31#1942—31no—
32#1374—31no—
33C22×C3×D4—21no—
34C22×#67—41no—
35C22×#67—41no—
36C22×D12—21no—
37C2×C3×D8—21no—
38C2×C3×D8—21no—
39C2×C3×D8—21no—
40C2×C3×D8—21no—
41C2×C3×D8—21no—
42C2×C3×D8D412yes—
43C22×C3×C8C2311yescharacteristic
44C2×C3×C4⋊C4—21no—
45#940—24no—
46#940—42no—
47C2×#213—24no—
48C2×#213—42no—
49C2×#213—42no—
50C2×C4⋊Dic3—21no—
51C2×Dic3⋊C4—41no—
52C2×C4×Dic3D411yescharacteristic
53C22×C3⋊C8D411yescharacteristic
54C2×D24—21no—
55C2×D24—21no—
56C2×D24—21no—
57C2×D24—21no—
58C2×D24—21no—
59C2×D24D412yes—
60C2×C3×C16—21no—
61C2×C3×C16—21no—
62C2×C3×C16—21no—
63C3×#107—21no—
64C3×#107—21no—
65C3⋊#95—41no—
66#818—21no—
67#818D412yescomplemented
68C8×Dic3—21no—
69C8×Dic3—21no—
70C2×C3⋊C16—21no—
71C2×C3⋊C16—21no—
72C2×C3⋊C16—21no—
73C22×D8—31no—
74C22×D8—61no—
75C22×D8C2×D311yescharacteristic
76C2×#127—61no—
77C22×C16—31no—
78C22×C16C2×D311yescharacteristic
79#493—62no—
80#493—62no—
81C2×#107—31no—
82C2×#107C2×D311yescharacteristic
83C2×C4×C8—31no—
84#389—61no—
85#357—22no—
86#357—62no—
87#357—62no—
88#357—62no—
89#340—61no—
90C24×C3—41no—
91C23×D3—41no—
92C2×C3×D4—22no—
93C2×C3×D4—41no—
94C2×C3×D4—41no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C22×C3×C4—41no—
99C22×C3×C4C2×D411yescharacteristic
100C2×#67—44no—
101C2×#67—44no—
102C2×#67—82no—
103C2×#67—82no—
104C2×#67—82no—
105C2×#67—82no—
106C22×Dic3—41no—
107C22×Dic3D811yescharacteristic
108C22×Dic3D811yescharacteristic
109C2×D12—22no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C3×D8—22no—
116C3×D8—22no—
117C3×D8—41no—
118C3×D8—41no—
119C2×C3×C8C2×D411yescharacteristic
120C2×C3×C8C2×D411yescharacteristic
121C2×C3×C8C2×D411yescharacteristic
122C2×C3×C8#4111yescharacteristic
123C2×C3×C8#4111yescharacteristic
124C2×C3×C8#4111yescharacteristic
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127#213—48no—
128C4⋊Dic3—22no—
129C4⋊Dic3—41no—
130Dic3⋊C4—82no—
131C4×Dic3—21no—
132C4×Dic3—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8SD1612yes—
136D24—22no—
137D24—22no—
138D24—41no—
139D24—41no—
140C3×C16—41no—
141C3⋊C16—41no—
142C22×D4—21no—
143C22×D4—61no—
144C22×D4—121no—
145C22×D4—121no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—32no—
152C2×D8—64no—
153C2×D8—61no—
154C2×D8—61no—
155C2×D8—61no—
156C2×D8—61no—
157C2×D8—61no—
158C2×D8—122no—
159C2×D8—122no—
160C2×D8#6712yes—
161C22×C8—31no—
162C22×C8C22×D311yescharacteristic
163#127—64no—
164#127—122no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—121no—
168C2×C42—31no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—61no—
173C2×C16—61no—
174C2×C16—61no—
175#107—21no—
176#107—21no—
177#107—32no—
178#107—61no—
179C4×C8—61no—
180C4×C8—61no—
181#95—121no—
182C23×C3—42no—
183C23×C3—81no—
184C23×C3—81no—
185C23×C3—81no—
186C23×C3—81no—
187C23×C3—81no—
188C23×C3—81no—
189C23×C3C2×D811yescharacteristic
190C22×D3—42no—
191C22×D3—81no—
192C22×D3—81no—
193C22×D3—81no—
194C22×D3—81no—
195C22×D3—81no—
196C22×D3—81no—
197C3×D4—42no—
198C3×D4—42no—
199C3×D4—81no—
200C3×D4—81no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12411yescharacteristic
207C2×C3×C4C2×SD1611yescharacteristic
208C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
209C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
210#67—88no—
211#67—88no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—42no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C2×Dic3D1612yes—
220C2×Dic3SD3212yes—
221D12—42no—
222D12—42no—
223D12—81no—
224D12—81no—
225C3×C8#12111yescharacteristic
226C3×C8#12111yescharacteristic
227C3×C8#12311yescharacteristic
228C3×C8#12311yescharacteristic
229C3⋊C8—22no—
230C24—41no—
231C24—121no—
232C2×D4—22no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—62no—
239C2×D4—124no—
240C2×D4—124no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—121no—
244C2×D4—121no—
245C2×D4—121no—
246C2×D4—242no—
247C2×D4—242no—
248C2×D4—242no—
249C2×D4—242no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—41no—
253C22×C4—121no—
254C22×C4D3×D411yescharacteristic
255D8—22no—
256D8—22no—
257D8—41no—
258D8—41no—
259D8—62no—
260D8—62no—
261D8—128no—
262D8—121no—
263D8—121no—
264C2×C8—32no—
265C2×C8—61no—
266C2×C8—61no—
267C2×C8#23511yescharacteristic
268C2×C8#23511yescharacteristic
269C2×C8D3×D411yescharacteristic
270C2×C8D3×D411yescharacteristic
271C2×C8#23711yescharacteristic
272C2×C8C2×#6711yescharacteristic
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—62no—
276C4⋊C4—121no—
277C4⋊C4—242no—
278C42—61no—
279C42—61no—
280C16—41no—
281C16—121no—
282C22×C3—82no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3—161no—
286C22×C3—161no—
287C22×C3—161no—
288C22×C3—161no—
289C22×C3#44711yescharacteristic
290C22×C3#45911yescharacteristic
291C22×C3#48611yescharacteristic
292C22×C3C2×D1611yescharacteristic
293C22×C3C2×SD3211yescharacteristic
294C22×C3#50911yescharacteristic
295C22×C3#50911yescharacteristic
296C2×D3—82no—
297C2×D3—82no—
298C2×D3—82no—
299C2×D3—161no—
300C2×D3—161no—
301C2×D3—161no—
302C2×D3—161no—
303C3×C4—161no—
304C3×C4#39411yescharacteristic
305C3×C4#45011yescharacteristic
306C3×C4#46011yescharacteristic
307C3×C4#49011yescharacteristic
308Dic3—22no—
309Dic3—22no—
310Dic3—82no—
311C23—42no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—122no—
319C23—241no—
320C23—241no—
321C23—241no—
322C23—241no—
323C23—241no—
324C23—241no—
325C23D3×D811yescharacteristic
326D4—42no—
327D4—42no—
328D4—81no—
329D4—81no—
330D4—122no—
331D4—122no—
332D4—248no—
333D4—248no—
334D4—241no—
335D4—241no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—122no—
346C2×C4—241no—
347C2×C4—241no—
348C2×C4#88411yescharacteristic
349C2×C4#88511yescharacteristic
350C2×C4#91111yescharacteristic
351C2×C4D3×SD1611yescharacteristic
352C2×C4#91411yescharacteristic
353C2×C4#93711yescharacteristic
354C8—62no—
355C8#88311yescharacteristic
356C8#88411yescharacteristic
357C8#92911yescharacteristic
358C8#93911yescharacteristic
359C2×C3—162no—
360C2×C3#204111yescharacteristic
361C2×C3#218411yescharacteristic
362C2×C3#218711yescharacteristic
363C2×C3#220611yescharacteristic
364C2×C3#220711yescharacteristic
365C2×C3#224011yescharacteristic
366C2×C3#224411yescharacteristic
367D3—162no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22—161no—
375C22—242no—
376C22—242no—
377C22—242no—
378C22—481no—
379C22—481no—
380C22—481no—
381C22—481no—
382C22#476011yescharacteristic
383C22#478611yescharacteristic
384C22D3×D1611yescharacteristic
385C22#491111yescharacteristic
386C22D3×SD3211yescharacteristic
387C22#491411yescharacteristic
388C22#515611yescharacteristic
389C4—62no—
390C4—62no—
391C4—161no—
392C4—242no—
393C4#476211yescharacteristic
394C4#476411yescharacteristic
395C4#495911yescharacteristic
396C4#516911yescharacteristic
397C3#1340711yescharacteristic, complemented, Sylow
398C2—162no—
399C2—482no—
400C2#6918311yescharacteristic
401C2#6918611yescharacteristic
402C2#6920211yescharacteristic
403C2#6920411yescharacteristic
404C2#7021111yescharacteristic
405C2#7146711yescharacteristic
406C2#7148511yescharacteristic
subgroupG/H
centerC23D3×D8
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC2×C3×#357C2
socleC23×C3C2×D8
prime radicalC2×#5151C2
hypercenterC2×#357D3
hypocenterC3#13407
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×C3×#357C2semidirect
3. C2×#4562C2semidirect
4. C2×#4509C2semidirect
5. #70611C2semidirect
6. C2×#4491C2semidirect
7. #67804C2semidirect
30. C2×#357D3semidirect
67. #818D4semidirect
397. C3#13407semidirect

Maximal subgroups

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

subgroupindex
C2×#51512
C2×C3×#3572
C2×#45622
C2×#45092
#706112
C2×#44912
#678042
#134073