Finite Group Databasebeta

← #83989

Subgroups — #83989

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#137C211yesmaximal, characteristic, complemented
2#5831C211yesmaximal, characteristic
3Q8×#67C211yesmaximal, characteristic, complemented
4C2×D3×Q16C211yesmaximal, characteristic, complemented
5#5239C211yesmaximal, characteristic, complemented
6#5228C211yesmaximal, characteristic, complemented
7#5186C211yesmaximal, characteristic, complemented
8#5184C211yesmaximal, characteristic, complemented
9#5182C211yesmaximal, characteristic, complemented
10#5171C211yesmaximal, characteristic, complemented
11#5170C211yesmaximal, characteristic, complemented
12#5167C211yesmaximal, characteristic, complemented
13#5166C211yesmaximal, characteristic, complemented
14#5137C211yesmaximal, characteristic, complemented
15#5127C211yesmaximal, characteristic, complemented
16#3270—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C2×D3×Q8C2211yescharacteristic
20C2×D3×Q8C2211yescharacteristic
21C3×#137—24no—
22C2×C3×Q16C2211yescharacteristic
23C2×C3×Q16C2211yescharacteristic
24C2×C3×SD16C2211yescharacteristic
25C2×C3×SD16C2211yescharacteristic
26C2×C3×M16C2211yescharacteristic
27#946—21no—
28#946C2211yescharacteristic
29Q8×Dic3C2211yescharacteristic
30Q8×Dic3C2211yescharacteristic
31#944—21no—
32#944C2211yescharacteristic
33C2×#216C2211yescharacteristic
34C2×#216C2211yescharacteristic
35C2×#216C2211yescharacteristic
36#938—21no—
37#938C2211yescharacteristic
38#936—21no—
39#936C2211yescharacteristic
40C2×#214C2211yescharacteristic
41C4×#67—21no—
42C4×#67—21no—
43C4×#67C2211yescharacteristic
44#924—21no—
45#924—21no—
46#924C2211yescharacteristic
47#924C2211yescharacteristic
48D3×Q16—24no—
49C2×Dic12C2211yescharacteristic
50C2×D3×C8C2211yescharacteristic
51#835C2211yescharacteristic
52#835C2211yescharacteristic
53#835C2211yescharacteristic
54#832C2211yescharacteristic
55#830C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57#817C2211yescharacteristic
58#816C2211yescharacteristic
59#816C2211yescharacteristic
60#815C2211yescharacteristic
61Dic3⋊C8C2211yescharacteristic
62C2×#137D311yescharacteristic, complemented
63C22×Q16—31no—
64D4×Q8—31no—
65#548—31no—
66#497—31no—
67#470—31no—
68#465—31no—
69#464—31no—
70#462—31no—
71#452—31no—
72#451—31no—
73#451—31no—
74#448—31no—
75#441—31no—
76#435—31no—
77C3×#41—22no—
78C3×#41—41no—
79C2×C3×Q8—21no—
80C2×C3×Q8—22no—
81C2×C3×Q8—41no—
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4—21no—
86C2×C3×D4C2311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C2×#67—21no—
91C2×#67D411yescharacteristic
92D3×Q8—22no—
93D3×Q8—22no—
94D3×Q8—41no—
95D3×Q8—41no—
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3C2311yescharacteristic
99C2×Dic6—21no—
100C2×Dic6—21no—
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C3×Q16—22no—
104C3×Q16D414yescomplemented
105C3×SD16—22no—
106C3×SD16—22no—
107C3×M16—22no—
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110#217—21no—
111#217—21no—
112#217—21no—
113#217—21no—
114#217—21no—
115#217D411yescharacteristic
116#216—22no—
117#216—22no—
118#216—41no—
119#214—41no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4D411yescharacteristic
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3C2311yescharacteristic
125C4⋊Dic3C2311yescharacteristic
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4—21no—
130Dic3⋊C4—21no—
131Dic3⋊C4D411yescharacteristic
132C4×Dic3—21no—
133C4×Dic3—21no—
134C4×Dic3C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137Dic12—22no—
138D3×C8—22no—
139C2×#41C2×D311yescharacteristic
140C22×Q8—31no—
141C22×Q8—31no—
142C22×Q8C2×D311yescharacteristic
143#137—24no—
144C2×Q16—31no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×Q16—31no—
148C2×Q16—64no—
149C2×Q16C2×D311yescharacteristic
150C2×Q16C2×D311yescharacteristic
151C2×SD16—31no—
152C2×SD16C2×D311yescharacteristic
153C2×SD16C2×D311yescharacteristic
154C2×M16C2×D311yescharacteristic
155C22×C8—31no—
156C4⋊Q8—31no—
157C4⋊Q8—61no—
158#124—31no—
159#124—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166#121—31no—
167#121—61no—
168C4×Q8—31no—
169C4×Q8—31no—
170C4×D4—31no—
171C4×D4—61no—
172C4×D4—61no—
173#106—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#103—31no—
179#103—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97—31no—
184C23×C3—21no—
185C23×C3C2×D411yescharacteristic
186C22×D3C2×D411yescharacteristic
187C3×Q8—21no—
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—42no—
192C3×Q8C2×D412yes—
193C3×Q8C2×D412yes—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—42no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2411yescharacteristic
206#67—22no—
207#67—42no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3C2×D411yescharacteristic
213C2×Dic3C2×D411yescharacteristic
214C2×Dic3C2×D411yescharacteristic
215C4×D3—21no—
216C4×D3—21no—
217C4×D3—42no—
218C4×D3—42no—
219Dic6—21no—
220Dic6—21no—
221Dic6—21no—
222Dic6—21no—
223Dic6—42no—
224Dic6—42no—
225C3×C8—21no—
226C3×C8C2×D412yes—
227C3⋊C8—21no—
228C3⋊C8—21no—
229#41—22no—
230#41—41no—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×Q8—31no—
234C2×Q8—31no—
235C2×Q8—41no—
236C2×Q8—61no—
237C2×Q8—61no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8—121no—
241C2×Q8—121no—
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×D4—21no—
246C2×D4—31no—
247C2×D4—61no—
248C2×D4C22×D311yescharacteristic
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—31no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4C22×D311yescharacteristic
255Q16—22no—
256Q16—62no—
257Q16—62no—
258Q16—62no—
259Q16—121no—
260Q16#6714yescomplemented
261SD16—22no—
262SD16—22no—
263SD16—121no—
264M16—22no—
265C2×C8—31no—
266C2×C8—31no—
267C2×C8—62no—
268C2×C8C22×D311yescharacteristic
269C2×C8C22×D311yescharacteristic
270C4⋊C4—31no—
271C4⋊C4—31no—
272C4⋊C4—31no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C22⋊C4—31no—
281C22⋊C4—31no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C42—31no—
289C42—61no—
290C42—61no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3C22×D411yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3#13712yescomplemented
301C3×C4—22no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4C22×D411yescharacteristic
306C3×C4H(5,𝔽2)11yescharacteristic
307Dic3—21no—
308Dic3—21no—
309Dic3—21no—
310Dic3—41no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314C23—21no—
315C23—31no—
316C23D3×D411yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—42no—
322Q8—61no—
323Q8—61no—
324Q8—61no—
325Q8—61no—
326Q8—122no—
327Q8—122no—
328Q8C2×#6712yes—
329Q8C2×#6712yes—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—62no—
334D4—122no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4D3×D411yescharacteristic
354C2×C4C23×D311yescharacteristic
355C8—21no—
356C8—61no—
357C8—61no—
358C8C2×#6712yes—
359C2×C3—41no—
360C2×C3—81no—
361C2×C3D4211yescharacteristic
362C2×C3C2×#13711yescharacteristic
363C2×C3#57811yescharacteristic
364D3—22no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—32no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—61no—
373C22—61no—
374C22C2×D3×D411yescharacteristic
375C4—21no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—61no—
380C4—61no—
381C4—61no—
382C4—121no—
383C4—121no—
384C4—121no—
385C4—121no—
386C4#100411yescharacteristic
387C4C22×#6711yescharacteristic
388C3#327011yescharacteristic, complemented, Sylow
389C2—41no—
390C2—62no—
391C2—81no—
392C2D3×#13711yescharacteristic
393C2#577911yescharacteristic
394C2D4×#6711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#137C2
socleC22×C3C22×D4
prime radical#5831C2
hypercenterC2×#137D3
hypocenterC3#3270
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×C3×#137C2semidirect
3. Q8×#67C2semidirect
4. C2×D3×Q16C2semidirect
5. #5239C2semidirect
6. #5228C2semidirect
7. #5186C2semidirect
8. #5184C2semidirect
9. #5182C2semidirect
10. #5171C2semidirect
11. #5170C2semidirect
12. #5167C2semidirect
13. #5166C2semidirect
14. #5137C2semidirect
15. #5127C2semidirect
62. C2×#137D3semidirect
104. C3×Q16D4semidirect
260. Q16#67semidirect
300. C2×D3#137semidirect
388. C3#3270semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1372
#58312
Q8×#672
C2×D3×Q162
#52392
#52282
#51862
#51842
#51822
#51712
#51702
#51672
#51662
#51372
#51272
#32703