Finite Group Databasebeta

← #13155227

Subgroups — #13155227

Order 1728.

426 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#11723654C211yesmaximal, characteristic, complemented
2C3×#63810C211yesmaximal, characteristic, complemented
3#11722008C211yesmaximal, characteristic, complemented
4#11722007C211yesmaximal, characteristic, complemented
5C3⋊D3×C3⋊C16C211yesmaximal, characteristic, complemented
6C3×#63551C211yesmaximal, characteristic, complemented
7C3×#63546C211yesmaximal, characteristic, complemented
8#10591048—31nomaximal
9#10591032—31nomaximal
10#10590244—31nomaximal
11#89041—21no—
12C3×#3796—21no—
13C3×#3746C2211yescharacteristic
14C3×#3745C2211yescharacteristic
15C3×C8×C3⋊D3C2211yescharacteristic
16C32×Dic12C2211yescharacteristic
17C32×D24C2211yescharacteristic
18#88811—21no—
19C3×#3715—21no—
20C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
21C32×C3⋊C16C2211yescharacteristic
22C3×#919—31no—
23C3×#912—31no—
24#63810—31no—
25#63810D311yescharacteristic, complemented
26#63808—31no—
27#63807—31no—
28#63655—31no—
29#63654—31no—
30C16×C3⋊D3—31no—
31C3×#829—31no—
32C3×#826—31no—
33C3×Dic24—31no—
34C3×D48—31no—
35#63551D311yescharacteristic, complemented
36#63547—31no—
37#63547—31no—
38#63547—31no—
39#63547—31no—
40#63547—61no—
41#63547—61no—
42#63546D311yescharacteristic, complemented
43D3×C3⋊C16—31no—
44D3×C3⋊C16—31no—
45D3×C3⋊C16—61no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4×C3⋊D3D411yescharacteristic
48C3×#634—21no—
49C32×D12—21no—
50C32×Dic6—21no—
51#6295—41no—
52#6294—41no—
53C3×#626—41no—
54C3×D3×Dic3—41no—
55C33×C8C2311yescharacteristic
56C3×C3⋊(C3⋊C8)D411yescharacteristic
57#5194—91no—
58#4919—91no—
59#4912—91no—
60C3×#239—61no—
61C3×#237—61no—
62#3799—61no—
63#3796—21no—
64#3796—61no—
65#3795—61no—
66C32×Q16—31no—
67C32×D8—31no—
68#3746C2×D311yescharacteristic
69#3745C2×D311yescharacteristic
70C8×C3⋊D3C2×D311yescharacteristic
71C3×#215—61no—
72C3×#214—61no—
73C3×Dic12—21no—
74C3×Dic12—31no—
75C3×Dic12—31no—
76C3×Dic12—61no—
77C3×Dic12C2×D311yescharacteristic
78C3×Dic12C2×D311yescharacteristic
79C3×D24—21no—
80C3×D24—31no—
81C3×D24—31no—
82C3×D24—61no—
83C3×D24C2×D311yescharacteristic
84C3×D24C2×D311yescharacteristic
85C3×D3×C8—31no—
86C3×D3×C8—31no—
87C3×D3×C8—61no—
88#3717—61no—
89#3716—61no—
90#3715—21no—
91#3715—61no—
92C32×C16—31no—
93C3⋊(C3⋊C16)—31no—
94C3⋊(C3⋊C16)—31no—
95C3⋊(C3⋊C16)—31no—
96C3⋊(C3⋊C16)—61no—
97C3⋊(C3⋊C16)—61no—
98C3⋊(C3⋊C16)—61no—
99C3⋊(C3⋊C16)—121no—
100C3×C3⋊C16—21no—
101C3×C3⋊C16C2×D311yescharacteristic
102C3×C3⋊C16C2×D311yescharacteristic
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3D811yescharacteristic
105C2×C32×D3—41no—
106C33×C4C2×D411yescharacteristic
107C3×C3⋊Dic3—41no—
108C3×C3⋊Dic3D811yescharacteristic
109C32×Dic3—41no—
110C3×#135—91no—
111#919—31no—
112#919—91no—
113#912—31no—
114#912—91no—
115#904—91no—
116C3×Q32—91no—
117C3×D16—91no—
118#829—31no—
119#828—91no—
120#828—91no—
121#828—91no—
122#828—181no—
123#827—91no—
124#827—91no—
125#827—91no—
126#827—181no—
127#826—31no—
128C2×C3⋊C16—91no—
129Dic24—31no—
130D48—31no—
131D3×C16—91no—
132D3×C16—91no—
133D3×C16—181no—
134C32×Q8—61no—
135C32×D4—61no—
136C4⋊(C3⋊D3)—21no—
137C4×C3⋊D3#6711yescharacteristic
138#634—21no—
139C3×#67—121no—
140C2×C3×Dic3—121no—
141C3×D12—21no—
142C3×D12—21no—
143C3×D12—41no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—121no—
147C3×D12—121no—
148C3×C4×D3—31no—
149C3×C4×D3—31no—
150C3×C4×D3—61no—
151C3×C4×D3—121no—
152C3×Dic6—21no—
153C3×Dic6—21no—
154C3×Dic6—41no—
155C3×Dic6—61no—
156C3×Dic6—61no—
157C3×Dic6—121no—
158#626—41no—
159#626—121no—
160#626—121no—
161#625—121no—
162#624—121no—
163D3×Dic3—41no—
164D3×Dic3—121no—
165D3×Dic3—121no—
166C32×C8—21no—
167C32×C8—21no—
168C32×C8—21no—
169C32×C8—41no—
170C32×C8C22×D311yescharacteristic
171C32×C8C22×D311yescharacteristic
172C32×C8C22×D311yescharacteristic
173C3⋊(C3⋊C8)#6711yescharacteristic
174C3×C3⋊C8—31no—
175C3×C3⋊C8—31no—
176C3×C3⋊C8—61no—
177#508—271noSylow
178C2×C33C2×D811yescharacteristic
179C3×C3⋊D3—21no—
180C3×C3⋊D3—81no—
181C32×D3—81no—
182C3×#41—181no—
183#239—61no—
184#239—181no—
185#237—61no—
186#237—181no—
187#235—181no—
188C3×Q16—31no—
189C3×Q16—31no—
190C3×Q16—61no—
191C3×Q16—91no—
192C3×SD16—181no—
193C3×D8—31no—
194C3×D8—31no—
195C3×D8—61no—
196C3×D8—91no—
197C2×C3×C8—91no—
198#215—61no—
199#215—181no—
200#214—61no—
201#214—181no—
202Dic12—31no—
203Dic12—31no—
204Dic12—61no—
205Dic12D3211yescharacteristic
206D24—31no—
207D24—31no—
208D24—61no—
209D24D3211yescharacteristic
210#204—181no—
211D3×C8—31no—
212D3×C8—31no—
213D3×C8—61no—
214C3×C16—31no—
215C3×C16—31no—
216C3×C16—61no—
217C3⋊C16—91no—
218C3⋊C16—91no—
219C3⋊C16—91no—
220C3⋊C16—181no—
221C3⋊C16—181no—
222C3⋊C16—181no—
223C3⋊C16—361no—
224C3⋊C16D3211yescharacteristic
225C22×C32—121no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3#21311yescharacteristic
228C2×C3×D3—31no—
229C2×C3×D3—31no—
230C2×C3×D3—41no—
231C2×C3×D3—41no—
232C2×C3×D3—61no—
233C2×C3×D3—81no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C2×C3×D3—241no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—41no—
241C32×C4—121no—
242C32×C4D3×D411yescharacteristic
243C32×C4D3×D411yescharacteristic
244C32×C4C2×#6711yescharacteristic
245C3⋊Dic3—41no—
246C3⋊Dic3#21311yescharacteristic
247C3×Dic3—31no—
248C3×Dic3—31no—
249C3×Dic3—41no—
250C3×Dic3—41no—
251C3×Dic3—61no—
252C3×Dic3—81no—
253C3×Dic3—121no—
254C3×Dic3—121no—
255C3×Dic3—241no—
256#135—91no—
257#135—271no—
258Q32—91no—
259SD32—271no—
260SD32—271no—
261D16—91no—
262C2×C16—271no—
263C33#50811yescharacteristic, complemented, Sylow
264C3×Q8—61no—
265C3×Q8—61no—
266C3×Q8—121no—
267C3×Q8—181no—
268C3×D4—61no—
269C3×D4—61no—
270C3×D4—121no—
271C3×D4—181no—
272C3×D4—361no—
273C2×C3×C4—91no—
274C2×C3×C4—361no—
275#67—121no—
276#67—361no—
277#67—361no—
278C2×Dic3—121no—
279C2×Dic3—361no—
280D12—21no—
281D12—61no—
282D12—61no—
283D12—121no—
284D12—121no—
285D12—361no—
286C4×D3—31no—
287C4×D3—31no—
288C4×D3—61no—
289C4×D3—121no—
290C4×D3—361no—
291C4×D3—361no—
292Dic6—21no—
293Dic6—61no—
294Dic6—61no—
295Dic6—121no—
296C3×C8—21no—
297C3×C8—21no—
298C3×C8—21no—
299C3×C8—41no—
300C3×C8—91no—
301C3×C8C2×D3211yescharacteristic
302C3×C8C2×D3211yescharacteristic
303C3×C8C2×D3211yescharacteristic
304C3⋊C8—31no—
305C3⋊C8—31no—
306C3⋊C8—61no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—41no—
311C2×C32—241no—
312C2×C32D3×D811yescharacteristic
313C2×C32D3×D811yescharacteristic
314C2×C32C2×#21311yescharacteristic
315C3⋊D3—21no—
316C3⋊D3—81no—
317C3×D3—62no—
318C3×D3—61no—
319C3×D3—61no—
320C3×D3—81no—
321C3×D3—81no—
322C3×D3—161no—
323C3×D3—241no—
324C3×D3—241no—
325C3×D3—481no—
326#41—181no—
327#41—541no—
328Q16—31no—
329Q16—91no—
330SD16—181no—
331SD16—541no—
332D8—31no—
333D8—91no—
334C2×C8—91no—
335C16—31no—
336C16—271no—
337C22×C3—91no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3—241no—
341C22×C3—361no—
342C2×D3—31no—
343C2×D3—31no—
344C2×D3—41no—
345C2×D3—61no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—241no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—41no—
353C3×C4—91no—
354C3×C4—121no—
355C3×C4—121no—
356C3×C4—241no—
357C3×C4—361no—
358C3×C4#380211yescharacteristic
359C3×C4D3×#6711yescharacteristic
360C3×C4D3×#6711yescharacteristic
361Dic3—31no—
362Dic3—31no—
363Dic3—41no—
364Dic3—61no—
365Dic3—121no—
366Dic3—121no—
367Dic3—241no—
368C32—21no—
369C32—21no—
370C32—21no—
371C32—41no—
372C32#491211yescharacteristic, complemented
373C32#491911yescharacteristic, complemented
374C32#519411yescharacteristic, complemented
375Q8—61no—
376Q8—181no—
377D4—61no—
378D4—181no—
379D4—361no—
380D4—1081no—
381C2×C4—91no—
382C2×C4—361no—
383C2×C4—1081no—
384C8—91no—
385C8D3311yescharacteristic
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—41no—
390C2×C3—181no—
391C2×C3—241no—
392C2×C3—241no—
393C2×C3—481no—
394C2×C3—721no—
395C2×C3#6379711yescharacteristic
396C2×C3D3×#21311yescharacteristic
397C2×C3D3×#21311yescharacteristic
398D3—62no—
399D3—61no—
400D3—61no—
401D3—81no—
402D3—241no—
403D3—241no—
404D3—481no—
405C22—91no—
406C22—121no—
407C22—361no—
408C4—91no—
409C4—121no—
410C4—361no—
411C4#8895311yescharacteristic
412C3—21no—
413C3—21no—
414C3—21no—
415C3—41no—
416C3#1059024411yescharacteristic, complemented
417C3#1059103211yescharacteristic, complemented
418C3#1059104811yescharacteristic, complemented
419C2—181no—
420C2—241no—
421C2—721no—
422C2#1172283111yescharacteristic
subgroupG/H
centerC2#11722831
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
hypocenterC33#508
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11723654C2semidirect
2. C3×#63810C2semidirect
3. #11722008C2semidirect
4. #11722007C2semidirect
5. C3⋊D3×C3⋊C16C2semidirect
6. C3×#63551C2semidirect
7. C3×#63546C2semidirect
25. #63810D3semidirect
35. #63551D3semidirect
42. #63546D3semidirect
263. C33#508semidirect
372. C32#4912semidirect
373. C32#4919semidirect
374. C32#5194semidirect
416. C3#10590244semidirect
417. C3#10591032semidirect
418. C3#10591048semidirect

Maximal subgroups

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

subgroupindex
#117236542
C3×#638102
#117220082
#117220072
C3⋊D3×C3⋊C162
C3×#635512
C3×#635462
#105910483
#105910323
#105902443