Finite Group Databasebeta

← #13161471

Subgroups — #13161471

Order 1728.

414 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 71 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
2#11723649C211yesmaximal, characteristic, complemented
3C3×#63628C211yesmaximal, characteristic, complemented
4C3×#63627C211yesmaximal, characteristic, complemented
5C3×#63625C211yesmaximal, characteristic, complemented
6#11722069C211yesmaximal, characteristic, complemented
7#11722068C211yesmaximal, characteristic, complemented
8#10590246—31nomaximal
9#10590239—31nomaximal
10#10590238—31nomaximal
11#89041—21no—
12#89040—21no—
13C3×#3746C2211yescharacteristic
14C3×#3746C2211yescharacteristic
15C3×#3746C2211yescharacteristic
16C3×#3745C2211yescharacteristic
17C3×C8×C3⋊D3C2211yescharacteristic
18#88812—21no—
19#88811—21no—
20C33×C16C2211yescharacteristic
21C3×C3⋊(C3⋊C16)C2211yescharacteristic
22#63810—31no—
23#63808—31no—
24#63807—31no—
25#63802—31no—
26D3×Dic12—31no—
27D3×Dic12—31no—
28#63628D311yescharacteristic, complemented
29#63627D311yescharacteristic, complemented
30#63625D311yescharacteristic, complemented
31C3×Dic24—31no—
32C3×Dic24—31no—
33C3×Dic24—61no—
34C3×#800—31no—
35C3×#800—31no—
36C3×#800—61no—
37C3×#798—31no—
38C3×#798—31no—
39C3×#798—61no—
40#63551—31no—
41#63551—31no—
42#63550—31no—
43#63549—31no—
44#63548—31no—
45#63547—31no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4×C3⋊D3D411yescharacteristic
48C3×#634—21no—
49C3×#634—21no—
50C3×#634—21no—
51#6296—41no—
52#6295—41no—
53#6294—41no—
54#6294—41no—
55C33×C8C2311yescharacteristic
56C3×C3⋊(C3⋊C8)D411yescharacteristic
57#4918—91no—
58#4915—91no—
59#4909—91no—
60#3799—61no—
61#3797—61no—
62#3796—61no—
63#3795—61no—
64D3×Dic6—61no—
65D3×Dic6—61no—
66#3746C2×D311yescharacteristic
67#3746C2×D311yescharacteristic
68#3746C2×D311yescharacteristic
69#3745C2×D311yescharacteristic
70C8×C3⋊D3C2×D311yescharacteristic
71C3×Dic12—31no—
72C3×Dic12—31no—
73C3×Dic12—31no—
74C3×Dic12—31no—
75C3×Dic12—31no—
76C3×Dic12—31no—
77C3×Dic12—61no—
78C3×Dic12—61no—
79C3×Dic12—61no—
80C3×D24—31no—
81C3×D24—31no—
82C3×D24—61no—
83C3×D3×C8—31no—
84C3×D3×C8—31no—
85C3×D3×C8—61no—
86#3719—61no—
87#3719—61no—
88#3718—61no—
89#3717—61no—
90#3716—61no—
91#3715—61no—
92C32×C16—21no—
93C32×C16—21no—
94C32×C16—21no—
95C32×C16—41no—
96C32×C16C2×D311yescharacteristic
97C32×C16C2×D311yescharacteristic
98C32×C16C2×D311yescharacteristic
99C3⋊(C3⋊C16)C2×D311yescharacteristic
100C3×C3⋊C16—31no—
101C3×C3⋊C16—31no—
102C3×C3⋊C16—61no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3D811yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3×C3⋊Dic3—41no—
107C3×C3⋊Dic3—41no—
108C3×C3⋊Dic3—41no—
109C3×C3⋊Dic3D811yescharacteristic
110#919—91no—
111D3×Q16—91no—
112D3×Q16—91no—
113#912—91no—
114C2×Dic12—91no—
115#904—91no—
116C3×Q32—91no—
117C3×SD32—91no—
118C3×M32—91no—
119#829—91no—
120#829—91no—
121#828—91no—
122#827—91no—
123Dic24—31no—
124Dic24—31no—
125Dic24—61no—
126Dic24—91no—
127#800—31no—
128#800—31no—
129#800—61no—
130#800—91no—
131#798—31no—
132#798—31no—
133#798—61no—
134C4⋊(C3⋊D3)—21no—
135C4×C3⋊D3D1211yescharacteristic
136#634—21no—
137#634—21no—
138#634—21no—
139C3×D12—61no—
140C3×D12—61no—
141C3×D12—121no—
142C3×C4×D3—31no—
143C3×C4×D3—31no—
144C3×C4×D3—61no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—121no—
152C3×Dic6—121no—
153C3×Dic6—121no—
154#627—121no—
155#627—121no—
156#627—121no—
157#626—121no—
158#626—121no—
159#625—121no—
160#624—121no—
161#624—121no—
162D3×Dic3—121no—
163D3×Dic3—121no—
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)D1211yescharacteristic
174C3×C3⋊C8—31no—
175C3×C3⋊C8—31no—
176C3×C3⋊C8—61no—
177#510—271noSylow
178C2×C33C2×D811yescharacteristic
179C3×C3⋊D3—21no—
180C3×C3⋊D3—81no—
181#239—181no—
182D3×Q8—181no—
183D3×Q8—181no—
184#237—181no—
185#235—181no—
186C2×Dic6—181no—
187C3×Q16—91no—
188C3×Q16—91no—
189C3×Q16—91no—
190C3×D8—91no—
191C2×C3×C8—91no—
192#216—181no—
193#216—181no—
194#215—181no—
195#214—181no—
196Dic12—31no—
197Dic12—31no—
198Dic12—31no—
199Dic12—31no—
200Dic12—31no—
201Dic12—31no—
202Dic12—61no—
203Dic12—61no—
204Dic12—61no—
205Dic12—181no—
206D24—31no—
207D24—31no—
208D24—61no—
209#204—181no—
210D3×C8—31no—
211D3×C8—31no—
212D3×C8—61no—
213C3×C16—21no—
214C3×C16—21no—
215C3×C16—21no—
216C3×C16—41no—
217C3×C16—91no—
218C3×C16D3211yescharacteristic
219C3×C16D3211yescharacteristic
220C3×C16D3211yescharacteristic
221C3⋊C16—31no—
222C3⋊C16—31no—
223C3⋊C16—61no—
224C2×C3⋊D3—41no—
225C2×C3⋊D3D2411yescharacteristic
226C2×C3×D3—31no—
227C2×C3×D3—31no—
228C2×C3×D3—61no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—241no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—41no—
236C32×C4C2×D1211yescharacteristic
237C32×C4D3×D411yescharacteristic
238C32×C4D3×D411yescharacteristic
239C3⋊Dic3—41no—
240C3⋊Dic3—41no—
241C3⋊Dic3—41no—
242C3⋊Dic3D2411yescharacteristic
243C3×Dic3—31no—
244C3×Dic3—31no—
245C3×Dic3—61no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C3×Dic3—121no—
249C3×Dic3—121no—
250C3×Dic3—121no—
251C3×Dic3—121no—
252C3×Dic3—241no—
253C3×Dic3—241no—
254C3×Dic3—241no—
255#135—271no—
256C2×Q16—271no—
257Q32—91no—
258Q32—271no—
259SD32—91no—
260SD32—271no—
261M32—91no—
262C33#51011yescharacteristic, complemented, Sylow
263C3×Q8—181no—
264C3×Q8—181no—
265C3×Q8—181no—
266C3×D4—181no—
267C2×C3×C4—91no—
268#67—361no—
269#67—361no—
270C2×Dic3—361no—
271C2×Dic3—361no—
272D12—61no—
273D12—61no—
274D12—121no—
275D12—361no—
276C4×D3—31no—
277C4×D3—31no—
278C4×D3—61no—
279C4×D3—361no—
280C4×D3—361no—
281C4×D3—361no—
282C4×D3—361no—
283Dic6—61no—
284Dic6—61no—
285Dic6—61no—
286Dic6—61no—
287Dic6—61no—
288Dic6—61no—
289Dic6—121no—
290Dic6—121no—
291Dic6—121no—
292Dic6—361no—
293Dic6—361no—
294Dic6—361no—
295C3×C8—21no—
296C3×C8—21no—
297C3×C8—21no—
298C3×C8—41no—
299C3×C8—91no—
300C3×C8C2×D3211yescharacteristic
301C3×C8C2×D3211yescharacteristic
302C3×C8C2×D3211yescharacteristic
303C3⋊C8—31no—
304C3⋊C8—31no—
305C3⋊C8—61no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—41no—
310C2×C32C2×D2411yescharacteristic
311C2×C32D3×D811yescharacteristic
312C2×C32D3×D811yescharacteristic
313C3⋊D3—21no—
314C3⋊D3—81no—
315C3×D3—61no—
316C3×D3—61no—
317C3×D3—121no—
318C3×D3—241no—
319C3×D3—241no—
320C3×D3—481no—
321#41—541no—
322C2×Q8—541no—
323Q16—91no—
324Q16—91no—
325Q16—91no—
326Q16—541no—
327SD16—541no—
328D8—91no—
329C2×C8—91no—
330C16—91no—
331C16#111311yescharacteristic
332C22×C3—91no—
333C22×C3—361no—
334C2×D3—31no—
335C2×D3—31no—
336C2×D3—61no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—241no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—41no—
344C3×C4—91no—
345C3×C4—361no—
346C3×C4—361no—
347C3×C4—361no—
348C3×C4D3×D1211yescharacteristic
349C3×C4D3×D1211yescharacteristic
350C3×C4#380211yescharacteristic
351Dic3—31no—
352Dic3—31no—
353Dic3—61no—
354Dic3—121no—
355Dic3—121no—
356Dic3—121no—
357Dic3—121no—
358Dic3—121no—
359Dic3—121no—
360Dic3—241no—
361Dic3—241no—
362Dic3—241no—
363C32—21no—
364C32—21no—
365C32—21no—
366C32—41no—
367C32#490911yescharacteristic, complemented
368C32#491511yescharacteristic, complemented
369C32#491811yescharacteristic, complemented
370Q8—181no—
371Q8—181no—
372Q8—181no—
373Q8—1081no—
374D4—181no—
375D4—1081no—
376C2×C4—91no—
377C2×C4—1081no—
378C2×C4—1081no—
379C8—91no—
380C8C2×#111311yescharacteristic
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—41no—
385C2×C3—181no—
386C2×C3—721no—
387C2×C3D3×D2411yescharacteristic
388C2×C3D3×D2411yescharacteristic
389C2×C3#6379711yescharacteristic
390D3—61no—
391D3—61no—
392D3—121no—
393D3—241no—
394D3—241no—
395D3—481no—
396C22—91no—
397C22—361no—
398C4—91no—
399C4—361no—
400C4—361no—
401C4—361no—
402C4C4⋊#111311yescharacteristic
403C3—21no—
404C3—21no—
405C3—21no—
406C3—41no—
407C3#1059023811yescharacteristic, complemented
408C3#1059023911yescharacteristic, complemented
409C3#1059024611yescharacteristic, complemented
410C2—181no—
411C2—721no—
412C2#1172364711yescharacteristic
subgroupG/H
centerC2#11723647
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×C16C22
socleC2×C33C2×D8
prime radical#11723654C2
hypercenterC16#1113
hypocenterC33#510
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16

Complements

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

NH ≅ G/Ntype
2. #11723649C2semidirect
3. C3×#63628C2semidirect
4. C3×#63627C2semidirect
5. C3×#63625C2semidirect
6. #11722069C2semidirect
7. #11722068C2semidirect
28. #63628D3semidirect
29. #63627D3semidirect
30. #63625D3semidirect
262. C33#510semidirect
367. C32#4909semidirect
368. C32#4915semidirect
369. C32#4918semidirect
407. C3#10590238semidirect
408. C3#10590239semidirect
409. C3#10590246semidirect

Maximal subgroups

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

subgroupindex
#117236542
#117236492
C3×#636282
C3×#636272
C3×#636252
#117220692
#117220682
#105902463
#105902393
#105902383