Finite Group Databasebeta

← #13181470

Subgroups — #13181470

Order 1728.

1274 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2#11724493C211yesmaximal, characteristic, complemented
3C3⋊D3×#41⋊C3C212yesmaximal, complemented
4#11724491C211yesmaximal, characteristic, complemented
5#11724490C212yesmaximal, complemented
6#10596756C311yesmaximal, characteristic, complemented
7#5964⋊C3—34nomaximal
8C3×#3828—41nomaximal
9C32×#41⋊C3C2211yescharacteristic, complemented
10C32×#41⋊C3C2211yescharacteristic, complemented
11C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
12C3⋊D3×SL(2,𝔽3)C2212yescomplemented
13C3⋊(C3⋊(#41⋊C3))C2212yescomplemented
14C2×C32×#41C2×C311yescharacteristic, complemented
15#64367—31no—
16#64367C2×C311yescharacteristic, complemented
17#64366—31no—
18#64366C2×C311yescharacteristic, complemented
19#41×C3⋊D3—32no—
20#41×C3⋊D3C2×C312yescomplemented
21#64364C2×C312yescomplemented
22#64361—32no—
23C2×#3828—31no—
24C2×#3828—31no—
25C2×C3×#41⋊C3—24no—
26C2×C3×#41⋊C3D314yescomplemented
27#1010⋊C3—34no—
28D3×#41⋊C3—38no—
29C3⋊#994—34no—
30#1007⋊C3—38no—
31C2×C33×C4—41no—
32C3×#638—42no—
33C3×C4⋊(C3⋊D3)—41no—
34C3×C4×C3⋊D3—42no—
35C3×#634—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37#5964—34no—
38#5945—91no—
39C32×#41—31no—
40C32×#41—31no—
41C32×#41C22×C311yescharacteristic
42C32×#41C22×C311yescharacteristic
43C2×C32×Q8C22×C311yescharacteristic
44C2×C32×D4—31no—
45C22×C32×C4—31no—
46C2×#638—32no—
47C2×#638—32no—
48#3832—32no—
49#3832C22×C312yes—
50Q8×C3⋊D3—32no—
51Q8×C3⋊D3C22×C312yes—
52#3830—34no—
53#3830—32no—
54#3830—32no—
55D4×C3⋊D3—34no—
56D4×C3⋊D3—32no—
57D4×C3⋊D3—32no—
58#3828—32no—
59#3828—31no—
60#3828—32no—
61#3828—32no—
62#3828—31no—
63#3828—32no—
64#3828—32no—
65#3828—31no—
66#3828A411yescharacteristic
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4×C3⋊D3—32no—
70C2×C4×C3⋊D3—32no—
71C2×#634—31no—
72C2×#634—31no—
73C3×#235—124no—
74C3×#41⋊C3—24no—
75C3×#41⋊C3—24no—
76C3×#41⋊C3C2×D314yescomplemented
77C3×#41⋊C3C2×D314yescomplemented
78C2×C3×SL(2,𝔽3)—24no—
79C2×C3×SL(2,𝔽3)C2×D314yescomplemented
80D3×SL(2,𝔽3)—38no—
81C3⋊(#41⋊C3)—38no—
82C22×C33—41no—
83C2×C3×C3⋊D3—42no—
84C33×C4—41no—
85C33×C4—41no—
86C3×C3⋊Dic3—42no—
87C2×C3×#41C3×D314yescomplemented
88#1010—34no—
89#1010—94no—
90#1009—34no—
91#1009—94no—
92D3×#41—38no—
93D3×#41—98no—
94#1007—38no—
95#1004—98no—
96C2×#235—94no—
97C2×#235—94no—
982−1+4⋊C3—91no—
992−1+4⋊C3—92no—
100#994—91no—
101C2×#41⋊C3—24no—
102C2×#41⋊C3—92no—
103C2×#41⋊C3C3⋊D311yescharacteristic, complemented
104C23×C32—31no—
105C22×C3⋊D3—32no—
106C22×C3⋊D3—32no—
107C32×Q8—31no—
108C32×Q8C23×C311yescharacteristic
109C32×D4—31no—
110C32×D4—31no—
111C32×D4—32no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4—31no—
116C2×C32×C4—31no—
117C2×C32×C4—44no—
118C2×C32×C4—84no—
119C2×C32×C4C2×A411yescharacteristic
120#638—34no—
121#638—32no—
122#638—32no—
123#638—32no—
124#638C2×A412yes—
125C2×C3⋊Dic3—32no—
126C2×C3⋊Dic3—32no—
127C4⋊(C3⋊D3)—32no—
128C4⋊(C3⋊D3)—32no—
129C4⋊(C3⋊D3)—31no—
130C4⋊(C3⋊D3)C2×A411yescharacteristic
131C4×C3⋊D3—34no—
132C4×C3⋊D3—32no—
133C4×C3⋊D3—32no—
134C4×C3⋊D3—32no—
135C4×C3⋊D3C2×A412yes—
136#634—31no—
137#634—32no—
138#634—32no—
139#634C2×A411yescharacteristic
140C3×#67—128no—
141C3×D12—124no—
142C3×C4×D3—128no—
143C3×Dic6—124no—
144C3×SL(2,𝔽3)—24no—
145C3×SL(2,𝔽3)C22×D314yes—
146#585—91noSylow
147C2×C33—41no—
148C2×C33—81no—
149C3×C3⋊D3—82no—
150C3×#41—34no—
151C3×#41—34no—
152C3×#41—361no—
153C3×#41C2×C3×D314yes—
154C3×#41C2×C3×D314yes—
155C2×C3×Q8C2×C3×D314yes—
156C2×C3×D4—34no—
157C22×C3×C4—34no—
158C2×#67—98no—
159C2×#67—98no—
160#239—38no—
161#239—98no—
162D3×Q8—38no—
163D3×Q8—98no—
164#237—916no—
165#237—98no—
166#237—98no—
167D3×D4—98no—
168D3×D4—916no—
169D3×D4—98no—
170#235—34no—
171#235—94no—
172#235—98no—
173#235—94no—
174#235—98no—
175#235—98no—
176#235—94no—
177#235—98no—
178#235—98no—
179C2×D12—94no—
180C2×D12—94no—
181C2×C4×D3—98no—
182C2×C4×D3—98no—
183C2×Dic6—94no—
184C2×Dic6—94no—
185#41⋊C3—24no—
186#41⋊C3—24no—
187#41⋊C3—92no—
188#41⋊C3C2×C3⋊D311yescharacteristic, complemented
189#41⋊C3C2×C3⋊D311yescharacteristic, complemented
190C2×SL(2,𝔽3)—24no—
191C2×SL(2,𝔽3)—92no—
192C2×SL(2,𝔽3)C2×C3⋊D311yescharacteristic, complemented
193C22×C32—31no—
194C22×C32—31no—
195C22×C32—44no—
196C22×C32—84no—
197C22×C32—121no—
198C22×C32C22×A411yescharacteristic
199C2×C3⋊D3—32no—
200C2×C3⋊D3—122no—
201C2×C3⋊D3—122no—
202C2×C3⋊D3C22×A412yes—
203C2×C3×D3—128no—
204C32×C4—31no—
205C32×C4—31no—
206C32×C4—44no—
207C32×C4—44no—
208C32×C4—84no—
209C32×C4—84no—
210C32×C4C22×A411yescharacteristic
211C32×C4C22×A411yescharacteristic
212C3⋊Dic3—32no—
213C3⋊Dic3C22×A412yes—
214C3×Dic3—128no—
2152−1+4—91no—
2162−1+4—92no—
2172−1+4—271no—
218H(5,𝔽2)—91no—
219H(5,𝔽2)—272no—
220H(5,𝔽2)—271no—
221C2×#41—92no—
222C2×#41—271no—
223C2×#41—271no—
224C2×#41—272no—
225C2×#41C3×C3⋊D311yescharacteristic, complemented
226C33—41noSylow
227C23×C3—34no—
228C22×D3—98no—
229C22×D3—98no—
230C3×Q8—34no—
231C3×Q8—361no—
232C3×Q8C22×C3×D314yes—
233C3×D4—34no—
234C3×D4—34no—
235C3×D4—38no—
236C3×D4—361no—
237C3×D4—362no—
238C2×C3×C4—34no—
239C2×C3×C4—34no—
240C2×C3×C4—34no—
241C2×C3×C4—34no—
242C2×C3×C4—34no—
243C2×C3×C4—41no—
244C2×C3×C4—84no—
245C2×C3×C4—362no—
246C2×C3×C4D3×A414yes—
247#67—38no—
248#67—98no—
249#67—98no—
250#67—916no—
251#67—98no—
252C2×Dic3—98no—
253C2×Dic3—98no—
254D12—34no—
255D12—98no—
256D12—98no—
257D12—94no—
258C4×D3—38no—
259C4×D3—98no—
260C4×D3—98no—
261C4×D3—916no—
262C4×D3—98no—
263Dic6—34no—
264Dic6—98no—
265Dic6—94no—
266Dic6—98no—
267SL(2,𝔽3)—24no—
268SL(2,𝔽3)C22×C3⋊D311yescharacteristic
269C2×C32—21no—
270C2×C32—44no—
271C2×C32—61no—
272C2×C32—61no—
273C2×C32—84no—
274C2×C32—88no—
275C2×C32—84no—
276C2×C32C23×A411yescharacteristic
277C3⋊D3—22no—
278C3⋊D3—62no—
279C3×D3—248no—
280#41—31no—
281#41—31no—
282#41—91no—
283#41—92no—
284#41—272no—
285#41—272no—
286#41—271no—
287#41—271no—
288#41—272no—
289#41—272no—
290#41—272no—
291#41—274no—
292#41—272no—
293#41—272no—
294#41—271no—
295#41—272no—
296#41C2×C3×C3⋊D311yescharacteristic
297#41C2×C3×C3⋊D311yescharacteristic
298C2×Q8—92no—
299C2×Q8—271no—
300C2×Q8—271no—
301C2×Q8—272no—
302C2×Q8C2×C3×C3⋊D311yescharacteristic
303C2×D4—31no—
304C2×D4—271no—
305C2×D4—271no—
306C2×D4—272no—
307C2×D4—274no—
308C2×D4—272no—
309C2×D4—272no—
310C2×D4—272no—
311C22×C4—31no—
312C22×C4—272no—
313C22×C4—272no—
314C22×C3—34no—
315C22×C3—34no—
316C22×C3—41no—
317C22×C3—84no—
318C22×C3—124no—
319C22×C3—362no—
320C22×C3C2×D3×A414yes—
321C2×D3—38no—
322C2×D3—98no—
323C2×D3—368no—
324C2×D3—368no—
325C3×C4—34no—
326C3×C4—34no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—84no—
330C3×C4—84no—
331C3×C4—362no—
332C3×C4C2×D3×A414yes—
333C3×C4C2×D3×A414yes—
334Dic3—38no—
335Dic3—98no—
336C32—44no—
337C32—84no—
338C32#594511yescharacteristic, complemented
339C23—31no—
340C23—272no—
341C23—272no—
342Q8—31no—
343Q8—91no—
344Q8—272no—
345Q8—272no—
346Q8—271no—
347Q8C22×C3×C3⋊D311yescharacteristic
348D4—31no—
349D4—31no—
350D4—32no—
351D4—92no—
352D4—91no—
353D4—272no—
354D4—272no—
355D4—272no—
356D4—272no—
357D4—274no—
358D4—272no—
359D4—271no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—92no—
366C2×C4—272no—
367C2×C4—272no—
368C2×C4—274no—
369C2×C4—272no—
370C2×C4—272no—
371C2×C4—272no—
372C2×C4A4×C3⋊D311yescharacteristic
373C2×C3—24no—
374C2×C3—41no—
375C2×C3—64no—
376C2×C3—64no—
377C2×C3—81no—
378C2×C3—88no—
379C2×C3—84no—
380C2×C3—722no—
381C2×C3C22×D3×A414yes—
382D3—68no—
383D3—188no—
384C22—31no—
385C22—31no—
386C22—92no—
387C22—121no—
388C22—272no—
389C22—1082no—
390C22—1082no—
391C22C2×A4×C3⋊D311yescharacteristic
392C4—31no—
393C4—31no—
394C4—92no—
395C4—272no—
396C4C2×A4×C3⋊D311yescharacteristic
397C4C2×A4×C3⋊D311yescharacteristic
398C3—41no—
399C3—84no—
400C3#5964⋊C314yescomplemented
401C2—21no—
402C2—61no—
403C2—61no—
404C2—182no—
405C2—542no—
406C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
centerC4C2×A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C22×A4×C3⋊D3
FittingC2×C32×#41C2×C3
socleC2×C32C23×A4
hypercenterC2×C4A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. #11724493C2semidirect
3. C3⋊D3×#41⋊C3C2semidirect
4. #11724491C2semidirect
5. #11724490C2semidirect
6. #10596756C3semidirect
9. C32×#41⋊C3C22semidirect
10. C32×#41⋊C3C22semidirect
11. C2×C32×SL(2,𝔽3)C22semidirect
12. C3⋊D3×SL(2,𝔽3)C22semidirect
13. C3⋊(C3⋊(#41⋊C3))C22semidirect
14. C2×C32×#41C2×C3semidirect
16. #64367C2×C3semidirect
18. #64366C2×C3semidirect
20. #41×C3⋊D3C2×C3semidirect
21. #64364C2×C3semidirect
26. C2×C3×#41⋊C3D3semidirect
76. C3×#41⋊C3C2×D3semidirect
77. C3×#41⋊C3C2×D3semidirect
79. C2×C3×SL(2,𝔽3)C2×D3semidirect
87. C2×C3×#41C3×D3semidirect
103. C2×#41⋊C3C3⋊D3semidirect
188. #41⋊C3C2×C3⋊D3semidirect
189. #41⋊C3C2×C3⋊D3semidirect
192. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
225. C2×#41C3×C3⋊D3semidirect
338. C32#5945semidirect
400. C3#5964⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
#117244932
C3⋊D3×#41⋊C32
#117244912
#117244902
#105967563
#5964⋊C33
C3×#38284