Finite Group Databasebeta

← #80207

Subgroups — #80207

Order 384.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5817C211yesmaximal, characteristic, complemented
2C22×#215C211yesmaximal, characteristic, complemented
3C2×#919C211yesmaximal, characteristic, complemented
4#5630C211yesmaximal, characteristic, complemented
5C3×#451C211yesmaximal, characteristic, complemented
6#5191C211yesmaximal, characteristic, complemented
7#5187C211yesmaximal, characteristic, complemented
8#5185C211yesmaximal, characteristic, complemented
9#5048C211yesmaximal, characteristic, complemented
10#4819C211yesmaximal, characteristic, complemented
11#4810C211yesmaximal, characteristic, complemented
12#4807C211yesmaximal, characteristic, complemented
13#4788C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#67⋊C8C211yesmaximal, characteristic, complemented
16#3282—31nomaximal, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×#239C2211yescharacteristic, complemented
20C22×D12C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic, complemented
22C2×C3×Q16C2211yescharacteristic, complemented
23C3×C22⋊Q8C2211yescharacteristic
24#947—21no—
25#947C2211yescharacteristic
26#947C2211yescharacteristic, complemented
27#946—21no—
28#946C2211yescharacteristic, complemented
29Q8×Dic3C2211yescharacteristic, complemented
30C2×#215—24no—
31C2×#215C2211yescharacteristic
32C2×#215C2211yescharacteristic
33C2×#215C2211yescharacteristic
34C2×#215C2211yescharacteristic
35C2×#215C2211yescharacteristic, complemented
36#930—21no—
37#930C2211yescharacteristic
38C4×#67—21no—
39C4×#67C2211yescharacteristic
40C22×C3⋊C8C2211yescharacteristic
41#919—24no—
42C2×D24C2211yescharacteristic
43C2×D24C2211yescharacteristic
44C2×D3×C8C2211yescharacteristic, complemented
45#895—21no—
46#895C2211yescharacteristic
47#894—21no—
48#893C2211yescharacteristic
49#882—21no—
50D3×C22⋊C4—21no—
51C3×#103C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic
54#835C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#821C2211yescharacteristic
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59C8×Dic3C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#808C2211yescharacteristic, complemented
62C2×#135—31no—
63C22×SD16—31no—
64#548—31no—
65#546—31no—
66#495—31no—
67#465—31no—
68#465—31no—
69#464—31no—
70#463—31no—
71#452—31no—
72#451D311yescharacteristic, complemented
73#449—31no—
74#447—31no—
75C4×SD16—31no—
76C8×D4—31no—
77C23×D3—21no—
78C2×C3×Q8—21no—
79C2×C3×Q8—22no—
80C2×C3×Q8—41no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C22×C3×C4—21no—
85C22×C3×C4C2311yescharacteristic
86C2×#67—21no—
87C2×#67D411yescharacteristic
88#239—22no—
89#239—22no—
90#239—41no—
91#239—41no—
92C2×D12—22no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12—21no—
96C2×D12—21no—
97C2×D12—41no—
98C2×D12C2311yescharacteristic
99C2×D12C2311yescharacteristic
100C2×D12C2311yescharacteristic
101C2×C4×D3—21no—
102C2×C4×D3—21no—
103C2×C4×D3—21no—
104C2×C4×D3C2311yescharacteristic
105C3×Q16—22no—
106C3×Q16—41no—
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C3×C4⋊C4—21no—
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112#217D411yescharacteristic
113#215—22no—
114#215—22no—
115#215—22no—
116#215—22no—
117#215D414yescomplemented
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4D411yescharacteristic
127C4⋊Dic3—21no—
128C4⋊Dic3C2311yescharacteristic
129Dic3⋊C4—21no—
130Dic3⋊C4D411yescharacteristic
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133C2×C3⋊C8—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136D24—22no—
137D24—41no—
138D3×C8—22no—
139C2×#41—31no—
140C2×#41—31no—
141C22×Q8C2×D311yescharacteristic
142C22×D4—31no—
143#135—64no—
144C2×Q16C2×D311yescharacteristic, complemented
145C2×Q16C2×D311yescharacteristic, complemented
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—64no—
152C2×D8—31no—
153C2×D8—31no—
154C22×C8—31no—
155C22×C8—31no—
156#124—31no—
157#124—31no—
158#124—61no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—61no—
162C22⋊Q8C2×D311yescharacteristic
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C22⋊C4—61no—
173#106—31no—
174C4⋊C8—31no—
175#103—31no—
176#103C2×D311yescharacteristic, complemented
177#103C2×D311yescharacteristic, complemented
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic
183C4×C8—31no—
184C23×C3C2×D411yescharacteristic
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—81no—
191C22×D3C2×D411yescharacteristic
192C3×Q8—21no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×Q8—21no—
197C3×Q8—42no—
198C3×Q8C2×D412yes—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—42no—
206C2×C3×C4C2×D411yescharacteristic
207C2×C3×C4C2411yescharacteristic
208#67—22no—
209#67—42no—
210C2×Dic3—21no—
211C2×Dic3C2×D411yescharacteristic
212C2×Dic3C2×D411yescharacteristic
213C2×Dic3C2×D411yescharacteristic
214D12—21no—
215D12—21no—
216D12—21no—
217D12—21no—
218D12—21no—
219D12—42no—
220D12—42no—
221D12—42no—
222D12—81no—
223D12C2×D412yes—
224C4×D3—21no—
225C4×D3—21no—
226C4×D3—42no—
227C4×D3—42no—
228C4×D3—81no—
229C3×C8—21no—
230C3×C8—21no—
231C3⋊C8—21no—
232C3⋊C8C2×D412yes—
233C24—61no—
234#41—62no—
235#41—62no—
236#41—121no—
237#41—121no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8C22×D311yescharacteristic
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—62no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—121no—
255C22×C4—21no—
256C22×C4—31no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4C22×D311yescharacteristic
261Q16—22no—
262Q16—41no—
263SD16—34no—
264SD16—62no—
265SD16—62no—
266SD16—62no—
267SD16—62no—
268D8—62no—
269D8—121no—
270C2×C8—31no—
271C2×C8—31no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—31no—
278C4⋊C4—31no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4C22×D311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—31no—
284C22⋊C4—31no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C42—31no—
294C42—61no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3#13512yescomplemented
298C22×C3C22×D411yescharacteristic
299C2×D3—21no—
300C2×D3—22no—
301C2×D3—21no—
302C2×D3—21no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—84no—
312C3×C4—22no—
313C3×C4—21no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4C22×D411yescharacteristic
318C3×C4H(5,𝔽2)11yescharacteristic
319Dic3—21no—
320Dic3—21no—
321Dic3—21no—
322Dic3—41no—
323C23—31no—
324C23—61no—
325C23—61no—
326C23—61no—
327C23—122no—
328C23—122no—
329C23—241no—
330C23D3×D411yescharacteristic
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—21no—
336Q8—42no—
337Q8D3×D412yes—
338D4—32no—
339D4—62no—
340D4—61no—
341D4—61no—
342D4—61no—
343D4—61no—
344D4—61no—
345D4—122no—
346D4—122no—
347D4—122no—
348D4—122no—
349D4—241no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—42no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—241no—
366C2×C4D3×D411yescharacteristic
367C2×C4C23×D311yescharacteristic
368C8—21no—
369C8—21no—
370C8—32no—
371C8—61no—
372C2×C3—22no—
373C2×C3D4211yescharacteristic
374C2×C3C2×#13511yescharacteristic
375C2×C3#57711yescharacteristic
376D3—41no—
377D3—42no—
378D3—81no—
379D3—81no—
380C22—21no—
381C22—21no—
382C22—61no—
383C22—61no—
384C22—61no—
385C22—62no—
386C22—121no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—244no—
395C22#91912yescomplemented
396C22C2×D3×D411yescharacteristic
397C4—21no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—61no—
403C4—61no—
404C4—61no—
405C4—121no—
406C4C2×D3×D411yescharacteristic
407C4#100411yescharacteristic
408C3#328211yescharacteristic, complemented, Sylow
409C2—22no—
410C2—121no—
411C2—122no—
412C2—241no—
413C2—241no—
414C2#559011yescharacteristic
415C2C2×#91911yescharacteristic
416C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
hypercenter#451D3
hypocenterC3#3282
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #5817C2semidirect
2. C22×#215C2semidirect
3. C2×#919C2semidirect
4. #5630C2semidirect
5. C3×#451C2semidirect
6. #5191C2semidirect
7. #5187C2semidirect
8. #5185C2semidirect
9. #5048C2semidirect
10. #4819C2semidirect
11. #4810C2semidirect
12. #4807C2semidirect
13. #4788C2semidirect
14. #4732C2semidirect
15. #67⋊C8C2semidirect
19. C2×#239C22semidirect
21. C2×C3×Q16C22semidirect
22. C2×C3×Q16C22semidirect
26. #947C22semidirect
28. #946C22semidirect
29. Q8×Dic3C22semidirect
35. C2×#215C22semidirect
44. C2×D3×C8C22semidirect
51. C3×#103C22semidirect
52. C3×#103C22semidirect
54. #835C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
59. C8×Dic3C22semidirect
60. #809C22semidirect
61. #808C22semidirect
72. #451D3semidirect
117. #215D4semidirect
144. C2×Q16C2×D3semidirect
145. C2×Q16C2×D3semidirect
176. #103C2×D3semidirect
177. #103C2×D3semidirect
297. C22×C3#135semidirect
395. C22#919semidirect
408. C3#3282semidirect

Maximal subgroups

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

subgroupindex
#58172
C22×#2152
C2×#9192
#56302
C3×#4512
#51912
#51872
#51852
#50482
#48192
#48102
#48072
#47882
#47322
#67⋊C82
#32823