Finite Group Databasebeta

← #13159988

Subgroups — #13159988

Order 1728.

824 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723436C211yesmaximal, characteristic
3#11723410C211yesmaximal, characteristic, complemented
4C2×#88798C211yesmaximal, characteristic, complemented
5#11722184C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic
8#10592497—31nomaximal
9#10590796—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3744—22no—
17C32×C4⋊Dic3C2211yescharacteristic
18#88807—21no—
19#88803—21no—
20#88803—21no—
21Dic3×C3⋊Dic3—21no—
22#88798—22no—
23#64156—31no—
24C2×#3756—31no—
25C2×#3744D311yescharacteristic
26#64096—31no—
27C2×C3×#204—34no—
28#63863—34no—
29#63843—34no—
30C2×#3718—34no—
31C32×#103—31no—
32#63641—31no—
33#63641—34no—
34#63641—64no—
35#63636—31no—
36C3×#816D314yescomplemented
37#63562—34no—
38C22×C3×C3⋊D3—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×#634—21no—
46C3×#634D412yes—
47C2×C32×Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C33×C8—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53#5166—91no—
54#4798—94no—
55C2×C32×Q8—31no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×#634C2×D311yescharacteristic
58C2×C3×D12—34no—
59C2×C3×Dic6—34no—
60C2×C3×Dic6C2×D314yes—
61C2×C32×C8—24no—
62C2×C32×C8C2×D314yes—
63C2×C32×C8C2×D311yescharacteristic
64C32×C4⋊C4—31no—
65#3756—62no—
66#3752—61no—
67#3752—61no—
68#3750—61no—
69C4×C3⋊Dic3—31no—
70C4×C3⋊Dic3—34no—
71C4×C3⋊Dic3—61no—
72C4×C3⋊Dic3—64no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76#3744—22no—
77C3×C4⋊Dic3C2×D314yes—
78C3×#204—68no—
79Dic3⋊Dic3—64no—
80(C2×D3)⋊Dic3—64no—
81(C2×D3)⋊Dic3—64no—
82Dic32—64no—
83#3718—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)—21no—
91C3⋊(C3⋊Dic3)—21no—
92C3×C3⋊Dic3—22no—
93C32×Dic3—41no—
94C32×Dic3—41no—
95C2×C3×SD16—91no—
96#947—94no—
97Q8×Dic3—91no—
98C2×#216—94no—
99C2×#216—91no—
100#936—91no—
101C2×#204—34no—
102#887—94no—
103C3×#103—34no—
104#832—91no—
105#816C3⋊D311yescharacteristic, complemented
106Dic3⋊C8—94no—
107Dic3⋊C8—91no—
108Dic3⋊C8—184no—
109#809—94no—
110C22×C3⋊D3—21no—
111C22×C3×D3—64no—
112C32×Q8—61no—
113C32×Q8—61no—
114C2×C32×C4—24no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—34no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—64no—
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)—21no—
128#634—21no—
129#634#6712yes—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—64no—
133C3×D12—64no—
134C3×D12—64no—
135C3×Dic6—24no—
136C3×Dic6—24no—
137C3×Dic6—38no—
138C3×Dic6—64no—
139C32×C8—21no—
140C32×C8—24no—
141C32×C8—44no—
142C3⋊(C3⋊C8)—61no—
143C3⋊(C3⋊C8)—64no—
144C3⋊(C3⋊C8)—124no—
145#464—271noSylow
146C2×C33#12111yescharacteristic
147C2×C33#13511yescharacteristic
148C2×C33#13711yescharacteristic
149C3×C3⋊D3—81no—
150C2×C3×Q8—34no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C2×D12—34no—
154C2×Dic6—34no—
155C2×Dic6C2×C3⋊D311yescharacteristic
156C3×SD16—182no—
157C2×C3×C8—24no—
158C2×C3×C8D3214yes—
159C2×C3×C8C2×C3⋊D311yescharacteristic
160C3×C4⋊C4—34no—
161#217—181no—
162#217—181no—
163#216—182no—
164#216—188no—
165(C2×D3)⋊C4—184no—
166(C2×D3)⋊C4—184no—
167C4⋊Dic3—181no—
168C4⋊Dic3C2×C3⋊D311yescharacteristic
169Dic3⋊C4—184no—
170C4×Dic3—94no—
171C4×Dic3—91no—
172C4×Dic3—184no—
173C4×Dic3—181no—
174C4×Dic3—184no—
175C2×C3⋊C8—94no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—184no—
178#204—68no—
179C22×C32—24no—
180C22×C32D3×D414yes—
181C22×C32D3×D411yescharacteristic
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—124no—
186C2×C3×D3—124no—
187C2×C3×D3—124no—
188C32×C4—24no—
189C32×C4—24no—
190C32×C4—121no—
191C32×C4—121no—
192C32×C4#23514yes—
193C32×C4D3×D414yes—
194C32×C4#23711yescharacteristic
195C32×C4C2×#6711yescharacteristic
196C3⋊Dic3—22no—
197C3⋊Dic3—68no—
198C3⋊Dic3—61no—
199C3⋊Dic3—64no—
200C3⋊Dic3—61no—
201C3⋊Dic3—64no—
202C3⋊Dic3—124no—
203C3×Dic3—44no—
204C3×Dic3—44no—
205C3×Dic3—68no—
206C2×Q16—271no—
207C2×SD16—91no—
208#124—271no—
209C4×Q8—271no—
210C4⋊C8—271no—
211#103—31no—
212#102—271no—
213C33#46411yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—64no—
216C3×Q8—64no—
217C3×Q8—64no—
218C3×Q8—92no—
219C3×Q8—181no—
220C3×D4—181no—
221C3×D4—181no—
222C2×C3×C4—24no—
223C2×C3×C4—64no—
224C2×C3×C4—64no—
225C2×C3×C4—181no—
226C2×C3×C4C2×D3214yes—
227C2×C3×C4C22×C3⋊D311yescharacteristic
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—64no—
231C2×Dic3—94no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—184no—
236C2×Dic3—184no—
237D12—64no—
238D12—64no—
239Dic6—21no—
240Dic6—21no—
241Dic6—38no—
242Dic6—64no—
243C3×C8—24no—
244C3×C8—21no—
245C3×C8—44no—
246C3⋊C8—184no—
247C3⋊C8—181no—
248C3⋊C8—364no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32#88414yes—
253C2×C32#91511yescharacteristic
254C2×C32#91614yes—
255C2×C32#91611yescharacteristic
256C2×C32#91814yes—
257C2×C32#93911yescharacteristic
258C3⋊D3—81no—
259C3×D3—244no—
260C2×Q8—31no—
261C2×Q8—91no—
262C2×D4—91no—
263Q16—542no—
264SD16—182no—
265C2×C8—271no—
266C2×C8D3×C3⋊D311yescharacteristic
267C4⋊C4—31no—
268C4⋊C4—541no—
269C22⋊C4—541no—
270C22⋊C4—541no—
271C42—271no—
272C42—541no—
273C22×C3—24no—
274C22×C3—361no—
275C22×C3—361no—
276C22×C3—361no—
277C22×C3#380214yes—
278C22×C3D4×C3⋊D311yescharacteristic
279C2×D3—124no—
280C2×D3—124no—
281C2×D3—124no—
282C3×C4—24no—
283C3×C4—24no—
284C3×C4—124no—
285C3×C4—124no—
286C3×C4—182no—
287C3×C4#379514yes—
288C3×C4D3×#6714yes—
289C3×C4#382811yescharacteristic
290C3×C4D4×C3⋊D311yescharacteristic
291Dic3—41no—
292Dic3—41no—
293Dic3—68no—
294Dic3—188no—
295Dic3—184no—
296Dic3—184no—
297Dic3—181no—
298Dic3—181no—
299Dic3—364no—
300C32—24no—
301C32#479814yescomplemented
302C32#516611yescharacteristic, complemented
303C23—181no—
304Q8—61no—
305Q8—61no—
306Q8—92no—
307Q8—181no—
308D4—181no—
309D4—181no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—181no—
313C2×C4—271no—
314C2×C4—271no—
315C2×C4C2×D3×C3⋊D311yescharacteristic
316C8—21no—
317C8—541no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—721no—
322C2×C3#6380914yes—
323C2×C3#6381114yes—
324C2×C3#6388514yes—
325C2×C3#6409311yescharacteristic
326C2×C3#6412511yescharacteristic
327C2×C3#6412711yescharacteristic
328D3—244no—
329C22—361no—
330C22—361no—
331C22—361no—
332C22#8902611yescharacteristic
333C4—121no—
334C4—121no—
335C4—182no—
336C4—541no—
337C4—541no—
338C4#8902111yescharacteristic
339C4C3⋊D3×#6711yescharacteristic
340C3—24no—
341C3#1059079614yescomplemented
342C3#1059249711yescharacteristic, complemented
343C2—721no—
344C2#1172334211yescharacteristic
345C2#1172334611yescharacteristic
346C2#1172351911yescharacteristic
subgroupG/H
centerC22#89026
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. #11723410C2semidirect
4. C2×#88798C2semidirect
5. #11722184C2semidirect
6. C32×#816C2semidirect
36. C3×#816D3semidirect
105. #816C3⋊D3semidirect
213. C33#464semidirect
301. C32#4798semidirect
302. C32#5166semidirect
341. C3#10590796semidirect
342. C3#10592497semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
#117234362
#117234102
C2×#887982
#117221842
C32×#8162
#117220332
#105924973
#105907963