Finite Group Databasebeta

← #13155259

Subgroups — #13155259

Order 1728.

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

normal subgroups only

This group has 414 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
2C3×#63802C211yesmaximal, characteristic, complemented
3#11722013C211yesmaximal, characteristic, complemented
4#11722008C211yesmaximal, characteristic, complemented
5#11722002C211yesmaximal, characteristic, complemented
6C3×#63551C211yesmaximal, characteristic, complemented
7C3×#63549C211yesmaximal, characteristic, complemented
8#10591047—31nomaximal
9#10591045—31nomaximal
10#10590246—31nomaximal
11#89041—21no—
12C3×#3797—21no—
13C3×#3746C2211yescharacteristic
14C3×#3745C2211yescharacteristic
15C3×C8×C3⋊D3C2211yescharacteristic
16C32×Dic12C2211yescharacteristic
17C32×Dic12C2211yescharacteristic
18#88811—21no—
19C3×#3718—21no—
20C3⋊(C3⋊(C3⋊C16))C2211yescharacteristic
21C32×C3⋊C16C2211yescharacteristic
22C3×D3×Q16—31no—
23C3×D3×Q16—31no—
24#63810—31no—
25#63808—31no—
26#63807—31no—
27#63802D311yescharacteristic, complemented
28#63656—31no—
29#63655—31no—
30#63625—31no—
31C3×#829—31no—
32C3×#828—31no—
33C3×Dic24—31no—
34C3×#800—31no—
35#63551D311yescharacteristic, complemented
36#63550—31no—
37#63550—31no—
38#63550—61no—
39#63549D311yescharacteristic, complemented
40#63547—31no—
41#63547—31no—
42#63547—61no—
43#63543—31no—
44#63543—31no—
45#63543—61no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4×C3⋊D3D411yescharacteristic
48C3×#634—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51#6295—41no—
52#6294—41no—
53C3×#627—41no—
54C3×#624—41no—
55C33×C8C2311yescharacteristic
56C3×C3⋊(C3⋊C8)D411yescharacteristic
57#5179—91no—
58#4918—91no—
59#4915—91no—
60C3×D3×Q8—61no—
61C3×D3×Q8—61no—
62#3799—61no—
63#3797—21no—
64#3796—61no—
65#3795—61no—
66C32×Q16—31no—
67C32×Q16—31no—
68#3746C2×D311yescharacteristic
69#3745C2×D311yescharacteristic
70C8×C3⋊D3C2×D311yescharacteristic
71C3×#216—61no—
72C3×#216—61no—
73C3×Dic12—21no—
74C3×Dic12—21no—
75C3×Dic12—31no—
76C3×Dic12—31no—
77C3×Dic12—61no—
78C3×Dic12C2×D311yescharacteristic
79C3×Dic12C2×D311yescharacteristic
80C3×Dic12C2×D311yescharacteristic
81C3×Dic12C2×D311yescharacteristic
82C3×D24—31no—
83C3×D24—31no—
84C3×D24—61no—
85C3×D3×C8—31no—
86C3×D3×C8—31no—
87C3×D3×C8—61no—
88#3718—21no—
89#3717—61no—
90#3716—61no—
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
105C33×C4C2×D411yescharacteristic
106C3×C3⋊Dic3—41no—
107C3×C3⋊Dic3D811yescharacteristic
108C32×Dic3—41no—
109C32×Dic3—41no—
110C2×C3×Q16—91no—
111#919—91no—
112D3×Q16—31no—
113D3×Q16—31no—
114#912—91no—
115#904—91no—
116C3×Q32—91no—
117C3×SD32—91no—
118#829—31no—
119#829—91no—
120#829—91no—
121#829—91no—
122#829—181no—
123#828—31no—
124#828—91no—
125#828—91no—
126#828—181no—
127#827—91no—
128#812—91no—
129Dic24—31no—
130#800—31no—
131#798—91no—
132#798—91no—
133#798—181no—
134C32×Q8—61no—
135C32×Q8—61no—
136C4⋊(C3⋊D3)—21no—
137C4×C3⋊D3#6711yescharacteristic
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×C4×D3—121no—
146C3×C4×D3—121no—
147C3×Dic6—21no—
148C3×Dic6—21no—
149C3×Dic6—21no—
150C3×Dic6—21no—
151C3×Dic6—41no—
152C3×Dic6—41no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3×Dic6—121no—
156C3×Dic6—121no—
157C3×Dic6—121no—
158#627—41no—
159#626—121no—
160#626—121no—
161#625—121no—
162#624—41no—
163#624—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)#6711yescharacteristic
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—
181C2×C3×Q8—181no—
182#239—181no—
183D3×Q8—61no—
184D3×Q8—61no—
185#237—181no—
186#235—181no—
187C3×Q16—31no—
188C3×Q16—31no—
189C3×Q16—31no—
190C3×Q16—31no—
191C3×Q16—61no—
192C3×Q16—61no—
193C3×Q16—91no—
194C3×Q16—181no—
195C3×D8—91no—
196C2×C3×C8—91no—
197#216—61no—
198#216—61no—
199#215—181no—
200#214—181no—
201Dic12—31no—
202Dic12—31no—
203Dic12—61no—
204Dic12D3211yescharacteristic
205Dic12D3211yescharacteristic
206D24—31no—
207D24—31no—
208D24—61no—
209#204—181no—
210D3×C8—31no—
211D3×C8—31no—
212D3×C8—61no—
213C3×C16—31no—
214C3×C16—31no—
215C3×C16—61no—
216C3⋊C16—91no—
217C3⋊C16—91no—
218C3⋊C16—91no—
219C3⋊C16—181no—
220C3⋊C16—181no—
221C3⋊C16—181no—
222C3⋊C16—361no—
223C3⋊C16D3211yescharacteristic
224C2×C3⋊D3—41no—
225C2×C3⋊D3#21311yescharacteristic
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×C4—121no—
237C32×C4—121no—
238C32×C4D3×D411yescharacteristic
239C32×C4D3×D411yescharacteristic
240C32×C4C2×#6711yescharacteristic
241C3⋊Dic3—41no—
242C3⋊Dic3#21311yescharacteristic
243C3×Dic3—31no—
244C3×Dic3—31no—
245C3×Dic3—41no—
246C3×Dic3—41no—
247C3×Dic3—41no—
248C3×Dic3—41no—
249C3×Dic3—61no—
250C3×Dic3—81no—
251C3×Dic3—81no—
252C3×Dic3—121no—
253C3×Dic3—121no—
254C3×Dic3—241no—
255#135—271no—
256C2×Q16—91no—
257Q32—91no—
258Q32—271no—
259SD32—91no—
260SD32—271no—
261M32—271no—
262C33#51011yescharacteristic, complemented, Sylow
263C3×Q8—61no—
264C3×Q8—61no—
265C3×Q8—61no—
266C3×Q8—61no—
267C3×Q8—121no—
268C3×Q8—121no—
269C3×Q8—181no—
270C3×Q8—361no—
271C3×D4—181no—
272C2×C3×C4—91no—
273C2×C3×C4—361no—
274#67—361no—
275#67—361no—
276C2×Dic3—361no—
277D12—61no—
278D12—61no—
279D12—121no—
280D12—361no—
281C4×D3—31no—
282C4×D3—31no—
283C4×D3—61no—
284C4×D3—121no—
285C4×D3—121no—
286C4×D3—361no—
287C4×D3—361no—
288Dic6—21no—
289Dic6—21no—
290Dic6—61no—
291Dic6—61no—
292Dic6—121no—
293Dic6—121no—
294Dic6—121no—
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×C32D3×D811yescharacteristic
311C2×C32D3×D811yescharacteristic
312C2×C32C2×#21311yescharacteristic
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—181no—
323Q16—31no—
324Q16—31no—
325Q16—91no—
326Q16—181no—
327SD16—541no—
328D8—91no—
329C2×C8—91no—
330C16—31no—
331C16—271no—
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—121no—
346C3×C4—121no—
347C3×C4—121no—
348C3×C4—121no—
349C3×C4—241no—
350C3×C4—241no—
351C3×C4—361no—
352C3×C4#380211yescharacteristic
353C3×C4D3×#6711yescharacteristic
354C3×C4D3×#6711yescharacteristic
355Dic3—31no—
356Dic3—31no—
357Dic3—41no—
358Dic3—41no—
359Dic3—61no—
360Dic3—121no—
361Dic3—121no—
362Dic3—241no—
363C32—21no—
364C32—21no—
365C32—21no—
366C32—41no—
367C32#491511yescharacteristic, complemented
368C32#491811yescharacteristic, complemented
369C32#517911yescharacteristic, complemented
370Q8—61no—
371Q8—61no—
372Q8—181no—
373Q8—361no—
374D4—181no—
375D4—1081no—
376C2×C4—91no—
377C2×C4—361no—
378C2×C4—1081no—
379C8—91no—
380C8D3311yescharacteristic
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—41no—
385C2×C3—181no—
386C2×C3—721no—
387C2×C3#6379711yescharacteristic
388C2×C3D3×#21311yescharacteristic
389C2×C3D3×#21311yescharacteristic
390D3—61no—
391D3—61no—
392D3—121no—
393D3—241no—
394D3—241no—
395D3—481no—
396C22—91no—
397C22—361no—
398C4—91no—
399C4—121no—
400C4—121no—
401C4—361no—
402C4#8895311yescharacteristic
403C3—21no—
404C3—21no—
405C3—21no—
406C3—41no—
407C3#1059024611yescharacteristic, complemented
408C3#1059104511yescharacteristic, complemented
409C3#1059104711yescharacteristic, complemented
410C2—181no—
411C2—721no—
412C2#1172283111yescharacteristic
subgroupG/H
centerC2#11722831
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
prime radical#11723654C2
hypocenterC33#510
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
2. C3×#63802C2semidirect
3. #11722013C2semidirect
4. #11722008C2semidirect
5. #11722002C2semidirect
6. C3×#63551C2semidirect
7. C3×#63549C2semidirect
27. #63802D3semidirect
35. #63551D3semidirect
39. #63549D3semidirect
262. C33#510semidirect
367. C32#4915semidirect
368. C32#4918semidirect
369. C32#5179semidirect
407. C3#10590246semidirect
408. C3#10591045semidirect
409. C3#10591047semidirect

Maximal subgroups

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

subgroupindex
#117236542
C3×#638022
#117220132
#117220082
#117220022
C3×#635512
C3×#635492
#105910473
#105910453
#105902463