Finite Group Databasebeta

← #13159990

Subgroups — #13159990

Order 1728.

840 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 156 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×C3×#3744C211yesmaximal, characteristic
2#11723439C211yesmaximal, characteristic
3Dic3×#634C211yesmaximal, characteristic, complemented
4C2×#88800C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic
8#10592544—31nomaximal
9#10590859—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×#3744—22no—
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19#88803—21no—
20#88803—21no—
21Dic3×C3⋊Dic3—21no—
22#88800—22no—
23C2×#3746—31no—
24C2×#3744D311yescharacteristic
25#64084—31no—
26C4×#634—31no—
27C2×C3×#204—34no—
28#63860—34no—
29Dic3×Dic6—34no—
30C2×#3719—34no—
31C32×C4⋊C8—31no—
32#63648—31no—
33#63643—31no—
34#63643—34no—
35#63643—64no—
36C3×Dic3⋊C8D314yescomplemented
37#63563—34no—
38C22×C3×C3⋊D3—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×#634—21no—
48C3×#634D412yes—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51C33×C8—21no—
52C32×C3⋊C8—21no—
53#5166—91no—
54#4845—94no—
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×#634—31no—
57C2×#634—34no—
58C2×#634—64no—
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—34no—
61C2×C3×Dic6—34no—
62C2×C32×C8—24no—
63C2×C32×C8—31no—
64C2×C32×C8C2×D314yes—
65C2×C32×C8C2×D311yescharacteristic
66C32×C42—31no—
67#3752—61no—
68#3752—61no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72#3750—61no—
73C4×C3⋊Dic3—61no—
74#3746—62no—
75#3744—22no—
76C3×C4×Dic3C2×D314yes—
77C2×C3×C3⋊C8C2×D314yes—
78C3×#204—68no—
79Dic3⋊Dic3—64no—
80(C2×D3)⋊Dic3—64no—
81(C2×D3)⋊Dic3—64no—
82Dic32—64no—
83#3719—68no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—41no—
86C2×C3×C3⋊D3—41no—
87C2×C3×C3⋊D3—41no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—41no—
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—22no—
93C32×Dic3—21no—
94C32×Dic3—21no—
95C2×C3×SD16—91no—
96Q8×Dic3—91no—
97C2×#216—91no—
98#936—91no—
99C2×Dic12—94no—
100C2×#204—34no—
101#875—94no—
102C4×Dic6—94no—
103C3×C4⋊C8—34no—
104#832—91no—
105#821—94no—
106#816—94no—
107#816—91no—
108#816—184no—
109Dic3⋊C8C3⋊D311yescharacteristic, complemented
110C22×C3⋊D3—21no—
111C22×C3×D3—64no—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—124no—
123C2×C3⋊Dic3—124no—
124C4⋊(C3⋊D3)—21no—
125C4⋊(C3⋊D3)—21no—
126#634—21no—
127#634—68no—
128#634—61no—
129#634—64no—
130#634—61no—
131#634—64no—
132#634—124no—
133#634#6712yes—
134C2×C3×Dic3—64no—
135C2×C3×Dic3D1214yes—
136C2×C3×Dic3D1214yes—
137C3×D12—64no—
138C3×D12—64no—
139C3×Dic6—38no—
140C3×Dic6—64no—
141C32×C8—21no—
142C32×C8—24no—
143C32×C8—44no—
144C32×C8—61no—
145C3×C3⋊C8—24no—
146#464—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13511yescharacteristic
149C2×C33#13711yescharacteristic
150C3×C3⋊D3—81no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C2×D12—34no—
154C2×Dic6—34no—
155C2×Dic6—94no—
156C2×Dic6—91no—
157C2×Dic6—184no—
158C3×SD16—182no—
159C2×C3×C8—24no—
160C2×C3×C8—34no—
161C2×C3×C8D3214yes—
162C2×C3×C8C2×C3⋊D311yescharacteristic
163C3×C42—34no—
164#217—181no—
165#217—181no—
166#216—182no—
167(C2×D3)⋊C4—184no—
168(C2×D3)⋊C4—184no—
169C4⋊Dic3—94no—
170C4⋊Dic3—91no—
171C4⋊Dic3—184no—
172C4⋊Dic3—181no—
173Dic3⋊C4—184no—
174C4×Dic3—184no—
175C4×Dic3—181no—
176C4×Dic3C2×C3⋊D311yescharacteristic
177C2×C3⋊C8C2×C3⋊D311yescharacteristic
178Dic12—188no—
179#204—68no—
180C22×C32—24no—
181C22×C32C2×D1214yes—
182C22×C32D3×D411yescharacteristic
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C2×C3×D3—124no—
187C2×C3×D3—124no—
188C2×C3×D3—124no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—61no—
192C32×C4—61no—
193C32×C4D3×D414yes—
194C32×C4#23711yescharacteristic
195C32×C4#23914yes—
196C32×C4C2×#6711yescharacteristic
197C3⋊Dic3—22no—
198C3⋊Dic3—128no—
199C3⋊Dic3—124no—
200C3⋊Dic3—121no—
201C3⋊Dic3—121no—
202C3⋊Dic3—124no—
203C3⋊Dic3—244no—
204C3×Dic3—24no—
205C3×Dic3—24no—
206C3×Dic3—68no—
207C2×Q16—271no—
208C2×SD16—91no—
209#124—271no—
210C4×Q8—271no—
211C4⋊C8—31no—
212#103—271no—
213#102—271no—
214C33#46411yescharacteristic, complemented, Sylow
215C23×C3—181no—
216C22×D3—64no—
217C3×Q8—92no—
218C3×Q8—181no—
219C3×D4—181no—
220C3×D4—181no—
221C2×C3×C4—24no—
222C2×C3×C4—34no—
223C2×C3×C4—34no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—64no—
228C2×Dic3—184no—
229C2×Dic3—184no—
230C2×Dic3—181no—
231C2×Dic3—181no—
232C2×Dic3—364no—
233C2×Dic3—364no—
234C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
235C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
236D12—64no—
237D12—64no—
238Dic6—38no—
239Dic6—64no—
240Dic6—188no—
241Dic6—184no—
242Dic6—184no—
243Dic6—181no—
244Dic6—181no—
245Dic6—364no—
246C3×C8—24no—
247C3×C8—21no—
248C3×C8—44no—
249C3×C8—64no—
250C3⋊C8—21no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32#89514yes—
255C2×C32#90414yes—
256C2×C32#90914yes—
257C2×C32#91511yescharacteristic
258C2×C32#91611yescharacteristic
259C2×C32#93911yescharacteristic
260C3⋊D3—81no—
261C3×D3—244no—
262C2×Q8—91no—
263C2×Q8—271no—
264C2×D4—91no—
265Q16—542no—
266SD16—182no—
267C2×C8—31no—
268C2×C8D3×C3⋊D311yescharacteristic
269C4⋊C4—271no—
270C4⋊C4—541no—
271C22⋊C4—541no—
272C22⋊C4—541no—
273C42—31no—
274C42—541no—
275C22×C3—24no—
276C22×C3—361no—
277C22×C3—361no—
278C22×C3—361no—
279C22×C3D3×D1214yes—
280C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
281C2×D3—124no—
282C2×D3—124no—
283C2×D3—124no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—64no—
287C3×C4—64no—
288C3×C4—182no—
289C3×C4#379614yes—
290C3×C4D3×#6714yes—
291C3×C4D4×C3⋊D311yescharacteristic
292C3×C4#383211yescharacteristic
293Dic3—21no—
294Dic3—21no—
295Dic3—68no—
296Dic3—368no—
297Dic3—364no—
298Dic3—361no—
299Dic3—361no—
300Dic3—364no—
301Dic3—724no—
302C32—24no—
303C32#484514yescomplemented
304C32#516611yescharacteristic, complemented
305C23—181no—
306Q8—92no—
307Q8—181no—
308Q8—541no—
309Q8—541no—
310D4—181no—
311D4—181no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—181no—
315C2×C4—541no—
316C2×C4—541no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—21no—
319C8—61no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—721no—
324C2×C3#6380014yes—
325C2×C3#6380614yes—
326C2×C3#6388614yes—
327C2×C3#6410411yescharacteristic
328C2×C3#6411311yescharacteristic
329C2×C3#6411811yescharacteristic
330D3—244no—
331C22—361no—
332C22—361no—
333C22—361no—
334C22D3×C4⋊(C3⋊D3)11yescharacteristic
335C4—61no—
336C4—61no—
337C4—182no—
338C4—1081no—
339C4—1081no—
340C4#8901711yescharacteristic
341C4C3⋊D3×#6711yescharacteristic
342C3—24no—
343C3#1059085914yescomplemented
344C3#1059254411yescharacteristic, complemented
345C2—721no—
346C2#1172332611yescharacteristic
347C2#1172333511yescharacteristic
348C2#1172348211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#464
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. Dic3×#634C2semidirect
4. C2×#88800C2semidirect
5. #11722186C2semidirect
6. C32×Dic3⋊C8C2semidirect
36. C3×Dic3⋊C8D3semidirect
109. Dic3⋊C8C3⋊D3semidirect
214. C33#464semidirect
303. C32#4845semidirect
304. C32#5166semidirect
343. C3#10590859semidirect
344. C3#10592544semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
#117234392
Dic3×#6342
C2×#888002
#117221862
C32×Dic3⋊C82
#117220342
#105925443
#105908593