Finite Group Databasebeta

← #13161235

Subgroups — #13161235

Order 1728.

880 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 142 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#11723988C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic, complemented
3C3×#63662C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5#11722036C211yesmaximal, characteristic
6#11722035C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic, complemented
8#10592499—31nomaximal
9#10591539—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12#88859—21no—
13#88859—21no—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#64156—31no—
22#64156—34no—
23#64156—64no—
24#64099—31no—
25#63872—34no—
26C32×#103—31no—
27#63662D311yescharacteristic, complemented
28#63648—31no—
29C8×C3⋊Dic3—31no—
30#63636—31no—
31C3×#835—34no—
32C3×#835D314yescomplemented
33#63564—34no—
34#63564—34no—
35#63553—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C32×C3⋊C8—21no—
49#4800—94no—
50#4800—91no—
51C2×C32×Q8—24no—
52C2×C32×Q8C2×D314yes—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C4⋊(C3⋊D3)—31no—
55C2×C4⋊(C3⋊D3)—34no—
56C2×C4⋊(C3⋊D3)—64no—
57C2×C32×C8—31no—
58C32×C4⋊C4—31no—
59#3752—61no—
60#3752—61no—
61#3752—64no—
62#3752—64no—
63#3752—68no—
64#3752—68no—
65C4⋊(C3⋊Dic3)—61no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67#3750—61no—
68C4×C3⋊Dic3—31no—
69C4×C3⋊Dic3—34no—
70C4×C3⋊Dic3—64no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic3⋊Dic3—64no—
77Dic3⋊Dic3—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3⋊(C3⋊D3)—41no—
80C2×C3⋊(C3⋊D3)—41no—
81C2×C3⋊(C3⋊D3)—41no—
82C33×C4—41no—
83C33×C4#4111yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—21no—
86C3⋊(C3⋊Dic3)—21no—
87C3×C3⋊Dic3—41no—
88C32×Dic3—41no—
89#947—94no—
90#947—91no—
91#947—184no—
92#890—94no—
93#890—91no—
94C3×#103—34no—
95C3×#103—91no—
96#835—34no—
97#835C3⋊D311yescharacteristic, complemented
98#821—94no—
99#821—91no—
100C8×Dic3—94no—
101C8×Dic3—91no—
102#809—94no—
103#809—91no—
104C22×C3⋊D3—68no—
105C22×C3⋊D3—64no—
106C22×C3⋊D3—61no—
107C32×Q8—21no—
108C32×Q8—21no—
109C32×Q8—24no—
110C32×Q8—24no—
111C32×Q8—44no—
112C32×Q8—44no—
113C2×C32×C4—28no—
114C2×C32×C4—24no—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—61no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—31no—
123C2×C3⋊Dic3—34no—
124C2×C3⋊Dic3—34no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—64no—
127C4⋊(C3⋊D3)—61no—
128C4⋊(C3⋊D3)—61no—
129C4⋊(C3⋊D3)—64no—
130C4⋊(C3⋊D3)—64no—
131C4⋊(C3⋊D3)—124no—
132C4⋊(C3⋊D3)—124no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—64no—
135C32×C8—61no—
136C3⋊(C3⋊C8)—21no—
137C3×C3⋊C8—24no—
138C3×C3⋊C8—64no—
139#488—271noSylow
140C2×C33#12411yescharacteristic
141C2×C33#13511yescharacteristic
142C2×C33#13511yescharacteristic
143C3⋊(C3⋊D3)—81no—
144C2×C3×Q8—24no—
145C2×C3×Q8D3214yes—
146C2×C3×Q8C2×C3⋊D311yescharacteristic
147C2×D12—94no—
148C2×D12—91no—
149C2×D12—184no—
150C2×C3×C8—34no—
151C2×C3×C8—91no—
152C3×C4⋊C4—34no—
153C3×C4⋊C4—91no—
154(C2×D3)⋊C4—184no—
155(C2×D3)⋊C4—184no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—181no—
158(C2×D3)⋊C4—188no—
159(C2×D3)⋊C4—188no—
160C4⋊Dic3—34no—
161C4⋊Dic3—181no—
162C4⋊Dic3—184no—
163C4⋊Dic3C2×C3⋊D311yescharacteristic
164Dic3⋊C4—181no—
165Dic3⋊C4—184no—
166C4×Dic3—94no—
167C4×Dic3—91no—
168C4×Dic3—184no—
169C2×C3⋊C8—34no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171C22×C32—24no—
172C22×C32D3×D414yes—
173C22×C32D3×D411yescharacteristic
174C2×C3⋊D3—128no—
175C2×C3⋊D3—128no—
176C2×C3⋊D3—128no—
177C2×C3⋊D3—124no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—124no—
180C2×C3⋊D3—121no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—121no—
183C32×C4—24no—
184C32×C4—24no—
185C32×C4—48no—
186C32×C4—44no—
187C32×C4—41no—
188C32×C4—121no—
189C32×C4#23514yes—
190C32×C4#23511yescharacteristic
191C32×C4#23714yes—
192C32×C4#23711yescharacteristic
193C3⋊Dic3—41no—
194C3⋊Dic3—61no—
195C3⋊Dic3—61no—
196C3⋊Dic3—64no—
197C3⋊Dic3—64no—
198C3⋊Dic3—124no—
199C3⋊Dic3—124no—
200C3×Dic3—44no—
201C3×Dic3—124no—
202#125—271no—
203#124—271no—
204#103—31no—
205#103—91no—
206#102—271no—
207#102—271no—
208C4×C8—271no—
209C33#48811yescharacteristic, complemented, Sylow
210C22×D3—188no—
211C22×D3—184no—
212C22×D3—181no—
213C3×Q8—24no—
214C3×Q8—24no—
215C3×Q8—21no—
216C3×Q8—21no—
217C3×Q8—44no—
218C3×Q8—44no—
219C2×C3×C4—28no—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—64no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—64no—
229C2×Dic3—94no—
230C2×Dic3—94no—
231C2×Dic3—91no—
232C2×Dic3—91no—
233C2×Dic3—184no—
234C2×Dic3—184no—
235D12—184no—
236D12—184no—
237D12—181no—
238D12—181no—
239D12—364no—
240D12—364no—
241C3×C8—64no—
242C3×C8—181no—
243C3⋊C8—21no—
244C3⋊C8—64no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32#88514yes—
249C2×C32#88511yescharacteristic
250C2×C32#91614yes—
251C2×C32#91611yescharacteristic
252C2×C32#91914yes—
253C2×C32#91911yescharacteristic
254C3⋊D3—248no—
255C3⋊D3—244no—
256C3⋊D3—241no—
257C2×Q8D3×C3⋊D311yescharacteristic
258C2×D4—271no—
259C2×C8—31no—
260C2×C8—91no—
261C4⋊C4—31no—
262C4⋊C4—91no—
263C4⋊C4—541no—
264C4⋊C4—541no—
265C22⋊C4—541no—
266C22⋊C4—541no—
267C42—271no—
268C22×C3—24no—
269C22×C3#381114yes—
270C22×C3D4×C3⋊D311yescharacteristic
271C2×D3—368no—
272C2×D3—368no—
273C2×D3—368no—
274C2×D3—361no—
275C2×D3—361no—
276C2×D3—361no—
277C2×D3—364no—
278C2×D3—364no—
279C2×D3—364no—
280C3×C4—24no—
281C3×C4—24no—
282C3×C4—48no—
283C3×C4—41no—
284C3×C4—44no—
285C3×C4—124no—
286C3×C4—361no—
287C3×C4#380414yes—
288C3×C4#380414yes—
289C3×C4#382811yescharacteristic
290C3×C4#383011yescharacteristic
291Dic3—41no—
292Dic3—124no—
293Dic3—181no—
294Dic3—181no—
295Dic3—184no—
296Dic3—184no—
297Dic3—364no—
298Dic3—364no—
299C32—24no—
300C32#480014yescomplemented
301C32#480011yescharacteristic, complemented
302C23—541no—
303Q8—21no—
304Q8—21no—
305D4—541no—
306D4—541no—
307C2×C4—21no—
308C2×C4—61no—
309C2×C4—181no—
310C2×C4—271no—
311C2×C4—271no—
312C2×C4C2×D3×C3⋊D311yescharacteristic
313C8—61no—
314C8—181no—
315C2×C3—24no—
316C2×C3—24no—
317C2×C3—24no—
318C2×C3#6395014yes—
319C2×C3#6395014yes—
320C2×C3#6395314yes—
321C2×C3#6409411yescharacteristic
322C2×C3#6412511yescharacteristic
323C2×C3#6412811yescharacteristic
324D3—728no—
325D3—724no—
326D3—721no—
327C22—1081no—
328C22—1081no—
329C22—1081no—
330C22#8903911yescharacteristic
331C4—41no—
332C4—121no—
333C4—361no—
334C4—541no—
335C4—541no—
336C4#8902811yescharacteristic
337C4#8903111yescharacteristic
338C3—24no—
339C3#1059153914yescomplemented
340C3#1059249911yescharacteristic, complemented
341C2—2161no—
342C2#1172358711yescharacteristic
343C2#1172358911yescharacteristic
344C2#1172359211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#488
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11723450C2semidirect
3. C3×#63662C2semidirect
4. C32×#835C2semidirect
7. #11722018C2semidirect
27. #63662D3semidirect
32. C3×#835D3semidirect
97. #835C3⋊D3semidirect
209. C33#488semidirect
300. C32#4800semidirect
301. C32#4800semidirect
339. C3#10591539semidirect
340. C3#10592499semidirect

Maximal subgroups

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

subgroupindex
#117239882
#117234502
C3×#636622
C32×#8352
#117220362
#117220352
#117220182
#105924993
#105915393