Finite Group Databasebeta

← #80157

Subgroups — #80157

Order 384.

500 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#5806C211yesmaximal, characteristic, complemented
2C2×#932C211yesmaximal, characteristic, complemented
3C2×#916C211yesmaximal, characteristic, complemented
4#5631C211yesmaximal, characteristic, complemented
5C3×#450C211yesmaximal, characteristic, complemented
6#5172C211yesmaximal, characteristic, complemented
7#5167C211yesmaximal, characteristic, complemented
8#5162C211yesmaximal, characteristic, complemented
9#5052C211yesmaximal, characteristic, complemented
10#4787C211yesmaximal, characteristic, complemented
11#4785C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4778C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic, complemented
16#3267—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×#239C2211yescharacteristic, complemented
19C2×#237C2211yescharacteristic
20C2×#235C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24#953—21no—
25C2×#217—21no—
26#944C2211yescharacteristic, complemented
27C2×#216C2211yescharacteristic, complemented
28#939—21no—
29#938C2211yescharacteristic
30#936C2211yescharacteristic
31#935—21no—
32D4×Dic3C2211yescharacteristic
33C2×#214C2211yescharacteristic
34C2×#214C2211yescharacteristic
35#932—24no—
36C2×#213C2211yescharacteristic
37C2×#213C2211yescharacteristic
38C4×#67C2211yescharacteristic
39#924C2211yescharacteristic
40C2×#208C2211yescharacteristic
41#916—24no—
42C2×Dic12C2211yescharacteristic, complemented
43C2×#204C2211yescharacteristic, complemented
44C2×D3×C8C2211yescharacteristic, complemented
45#897C2211yescharacteristic, complemented
46#894—21no—
47#893C2211yescharacteristic, complemented
48D3×C4⋊C4—21no—
49#884—21no—
50#883—21no—
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic
53C3×C22⋊C8C2211yescharacteristic
54#835C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#815C2211yescharacteristic, complemented
59Dic3⋊C8C2211yescharacteristic, complemented
60#810C2211yescharacteristic, complemented
61#808C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64#547—31no—
65#546—31no—
66#497—31no—
67#470—31no—
68#465—31no—
69#463—31no—
70#461—31no—
71#452—31no—
72#452—31no—
73#450D311yescharacteristic, complemented
74#449—31no—
75#442—31no—
76#435—31no—
77C24×C3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—22no—
80C2×C3×D4—21no—
81C2×C3×D4—41no—
82C2×C3×D4C2311yescharacteristic
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67D411yescharacteristic
89C22×Dic3—21no—
90C22×Dic3—21no—
91#239—22no—
92#239—41no—
93#237—22no—
94#237—41no—
95#235—22no—
96#235—41no—
97C2×D12—21no—
98C2×D12C2311yescharacteristic
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3—21no—
102C2×C4×D3C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×SD16—22no—
106C3×SD16—41no—
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C3×C4⋊C4—21no—
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112#217—21no—
113#217—21no—
114#217—21no—
115#217—21no—
116#217—21no—
117#217D411yescharacteristic
118#216—22no—
119#214—22no—
120#214—22no—
121#213—22no—
122#213D414yescomplemented
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4D411yescharacteristic
126C4⋊Dic3—21no—
127C4⋊Dic3C2311yescharacteristic
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic3⋊C4—21no—
131Dic3⋊C4D411yescharacteristic
132C4×Dic3C2311yescharacteristic
133#208—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136Dic12—41no—
137#204—22no—
138D3×C8—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41—31no—
142C22×D4C2×D311yescharacteristic
143AGL(1,ℤ/8)—64no—
144#135—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16C2×D311yescharacteristic
151C2×SD16C2×D311yescharacteristic
152C2×D8—31no—
153C2×D8—31no—
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157#124—31no—
158#123—61no—
159#123—61no—
160#123—61no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8C2×D311yescharacteristic
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×D4—31no—
169C4×D4—31no—
170C4×D4—31no—
171C2×C4⋊C4—61no—
172C2×C22⋊C4—61no—
173#106—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102C2×D311yescharacteristic
180#102C2×D311yescharacteristic
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic
183#97—31no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—81no—
189C23×C3C2×D411yescharacteristic
190C22×D3—21no—
191C22×D3C2×D411yescharacteristic
192C3×Q8—21no—
193C3×Q8—21no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—42no—
198C3×D4C2×D412yes—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2411yescharacteristic
205#67—41no—
206#67—42no—
207#67—42no—
208#67—81no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—42no—
213C2×Dic3—81no—
214C2×Dic3C2×D411yescharacteristic
215C2×Dic3C2×D411yescharacteristic
216C2×Dic3C2×D411yescharacteristic
217D12—21no—
218D12—42no—
219D12C2×D412yes—
220C4×D3—21no—
221C4×D3—21no—
222C4×D3—42no—
223C4×D3—42no—
224C4×D3—42no—
225Dic6—21no—
226Dic6—21no—
227Dic6—21no—
228Dic6—21no—
229C3×C8—21no—
230C3×C8—21no—
231C3⋊C8—21no—
232C3⋊C8C2×D412yes—
233C24—21no—
234#41—62no—
235#41—62no—
236#41—62no—
237#41—121no—
238#41—121no—
239#41—121no—
240C2×Q8—31no—
241C2×Q8—31no—
242C2×Q8C22×D311yescharacteristic
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—41no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4C22×D311yescharacteristic
253C2×D4C22×D311yescharacteristic
254C22×C4—31no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4C22×D311yescharacteristic
261Q16—62no—
262Q16—121no—
263SD16—22no—
264SD16—41no—
265SD16—62no—
266SD16—62no—
267SD16—62no—
268D8—34no—
269D8—62no—
270M16—62no—
271C2×C8—31no—
272C2×C8—31no—
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⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4C22×D311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—31no—
286C22⋊C4—31no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C42—31no—
295C22×C3—22no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—84no—
305C22×C3C22×D411yescharacteristic
306C2×D3—21no—
307C2×D3—22no—
308C2×D3—21no—
309C2×D3—21no—
310C2×D3—41no—
311C2×D3—41no—
312C3×C4—21no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4C22×D411yescharacteristic
316C3×C4H(5,𝔽2)11yescharacteristic
317Dic3—21no—
318Dic3—21no—
319Dic3—21no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323C23—21no—
324C23—21no—
325C23—31no—
326C23—42no—
327C23—42no—
328C23—61no—
329C23—81no—
330C23D3×D411yescharacteristic
331Q8—21no—
332Q8—21no—
333Q8—61no—
334Q8—61no—
335Q8—61no—
336Q8—61no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—32no—
341D4—42no—
342D4—61no—
343D4—122no—
344D4—121no—
345D4—122no—
346D4—122no—
347D4—241no—
348D4D3×D412yes—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—122no—
362C2×C4—122no—
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—41no—
373C2×C3—42no—
374C2×C3—81no—
375C2×C3D4211yescharacteristic
376C2×C3#57511yescharacteristic
377C2×C3#57711yescharacteristic
378D3—41no—
379D3—42no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—61no—
390C22—61no—
391C22—61no—
392C22—62no—
393C22—84no—
394C22—121no—
395C22—121no—
396C22C2×D3×D411yescharacteristic
397C4—21no—
398C4—41no—
399C4—41no—
400C4—61no—
401C4—61no—
402C4—61no—
403C4—121no—
404C4—121no—
405C4—121no—
406C4C2×D3×D411yescharacteristic
407C4#100411yescharacteristic
408C3#326711yescharacteristic, complemented, Sylow
409C2—41no—
410C2—42no—
411C2—81no—
412C2—121no—
413C2—122no—
414C2#559011yescharacteristic
415C2#576211yescharacteristic
416C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
hypercenter#450D3
hypocenterC3#3267
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #5806C2semidirect
2. C2×#932C2semidirect
3. C2×#916C2semidirect
4. #5631C2semidirect
5. C3×#450C2semidirect
6. #5172C2semidirect
7. #5167C2semidirect
8. #5162C2semidirect
9. #5052C2semidirect
10. #4787C2semidirect
11. #4785C2semidirect
12. #4783C2semidirect
13. #4778C2semidirect
14. #4733C2semidirect
15. #4728C2semidirect
18. C2×#239C22semidirect
26. #944C22semidirect
27. C2×#216C22semidirect
42. C2×Dic12C22semidirect
43. C2×#204C22semidirect
44. C2×D3×C8C22semidirect
45. #897C22semidirect
47. #893C22semidirect
54. #835C22semidirect
55. #821C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #816C22semidirect
58. #815C22semidirect
59. Dic3⋊C8C22semidirect
60. #810C22semidirect
61. #808C22semidirect
73. #450D3semidirect
122. #213D4semidirect
408. C3#3267semidirect

Maximal subgroups

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

subgroupindex
#58062
C2×#9322
C2×#9162
#56312
C3×#4502
#51722
#51672
#51622
#50522
#47872
#47852
#47832
#47782
#47332
#47282
#32673