Finite Group Databasebeta

← #13159994

Subgroups — #13159994

Order 1728.

824 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 154 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×C32×#204C211yesmaximal, characteristic
2#11723434C211yesmaximal, characteristic
3#11723397C211yesmaximal, characteristic, complemented
4C2×#88798C211yesmaximal, characteristic, complemented
5#11722184C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic
8#10592865—31nomaximal
9#10590796—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×#204—22no—
18Dic3⋊(C3⋊Dic3)—21no—
19(C2×D3)⋊(C3⋊Dic3)—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22#88798—22no—
23C2×C32×SD16—31no—
24Q8×C3⋊Dic3—31no—
25C2×#3756—31no—
26#64145—31no—
27C2×C3×#204D314yes—
28#63863—34no—
29#63843—34no—
30C2×#3718—34no—
31#63659—31no—
32#63643D311yescharacteristic, complemented
33#63641—31no—
34#63641—34no—
35#63641—64no—
36C3×#816—34no—
37#63562—34no—
38C22×C32×D3—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C3×#634—21no—
45C3×#634—21no—
46C2×C32×Dic3—21no—
47C32×D12—21no—
48C32×D12—21no—
49C32×Dic6—21no—
50C32×Dic6D412yes—
51C33×C8—21no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53#5166—94no—
54#4798—91no—
55C2×C32×Q8—31no—
56C2×C32×D4—31no—
57C2×#634C2×D311yescharacteristic
58C2×C3×D12C2×D314yes—
59C2×C3×Dic6—34no—
60C2×C3×Dic6C2×D314yes—
61C32×SD16—62no—
62C2×C32×C8—24no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C2×D311yescharacteristic
65#3757—61no—
66#3757—61no—
67#3756—62no—
68C4⋊(C3⋊Dic3)—61no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—61no—
73C4×C3⋊Dic3—64no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C3×C4⋊Dic3—34no—
78C3×#204—28no—
79Dic3⋊Dic3—64no—
80(C2×D3)⋊Dic3—64no—
81(C2×D3)⋊Dic3—64no—
82Dic32—64no—
83#3718—68no—
84C22×C33C2×D411yescharacteristic
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—21no—
91C3⋊(C3⋊Dic3)—21no—
92C3×C3⋊Dic3—41no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—22no—
95C2×C3×SD16—34no—
96#947—91no—
97Q8×Dic3—94no—
98C2×#216—94no—
99C2×#216—91no—
100#936—94no—
101C2×#204C3⋊D311yescharacteristic
102#887—91no—
103C3×#103—91no—
104#832—94no—
105#816—34no—
106Dic3⋊C8—94no—
107Dic3⋊C8—91no—
108Dic3⋊C8—184no—
109#809—91no—
110C23×C32—61no—
111C22×C3×D3—24no—
112C32×Q8—32no—
113C32×Q8—61no—
114C32×D4—61no—
115C32×D4—61no—
116C2×C32×C4—24no—
117C2×C32×C4—61no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—31no—
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—34no—
125C2×C3⋊Dic3—34no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—64no—
128#634—21no—
129#634—21no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—64no—
132C2×C3×Dic3—64no—
133C3×D12—24no—
134C3×D12—24no—
135C3×Dic6—24no—
136C3×Dic6—64no—
137C3×Dic6—64no—
138C3×Dic6#6718yes—
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
149C32×D3—81no—
150C2×C3×Q8—34no—
151C2×C3×Q8—91no—
152C2×C3×D4—34no—
153C2×D12C2×C3⋊D311yescharacteristic
154C2×Dic6—34no—
155C2×Dic6C2×C3⋊D311yescharacteristic
156C3×SD16—68no—
157C2×C3×C8—24no—
158C2×C3×C8D3214yes—
159C2×C3×C8C2×C3⋊D311yescharacteristic
160C3×C4⋊C4—91no—
161#217—184no—
162#217—184no—
163#216—188no—
164#216—182no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—181no—
167C4⋊Dic3—34no—
168C4⋊Dic3—184no—
169Dic3⋊C4—181no—
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—22no—
179C22×C32—24no—
180C22×C32—121no—
181C22×C32—121no—
182C22×C32—121no—
183C22×C32D3×D414yes—
184C22×C32D3×D411yescharacteristic
185C2×C3×D3—44no—
186C2×C3×D3—44no—
187C2×C3×D3—44no—
188C32×C4—24no—
189C32×C4—24no—
190C32×C4—62no—
191C32×C4#23511yescharacteristic
192C32×C4D3×D411yescharacteristic
193C32×C4#23714yes—
194C32×C4C2×#6714yes—
195C3⋊Dic3—41no—
196C3⋊Dic3—41no—
197C3⋊Dic3—68no—
198C3⋊Dic3—64no—
199C3⋊Dic3—61no—
200C3⋊Dic3—61no—
201C3⋊Dic3—64no—
202C3⋊Dic3—124no—
203C3×Dic3—28no—
204C3×Dic3—124no—
205C3×Dic3—124no—
206C2×Q16—271no—
207C2×SD16—31no—
208#124—271no—
209C4×Q8—271no—
210C4⋊C8—271no—
211#103—91no—
212#102—271no—
213C33#46411yescharacteristic, complemented, Sylow
214C23×C3—64no—
215C22×D3—21no—
216C3×Q8—38no—
217C3×Q8—64no—
218C3×Q8—181no—
219C3×Q8—181no—
220C3×D4—64no—
221C3×D4—64no—
222C2×C3×C4—24no—
223C2×C3×C4—64no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4C2×D3214yes—
227C2×C3×C4C22×C3⋊D311yescharacteristic
228C2×Dic3—21no—
229C2×Dic3—64no—
230C2×Dic3—64no—
231C2×Dic3—94no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—184no—
236C2×Dic3—184no—
237D12—21no—
238D12—21no—
239Dic6—21no—
240Dic6—64no—
241Dic6—64no—
242Dic6#63812yes—
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—241no—
253C2×C32#88411yescharacteristic
254C2×C32#91514yes—
255C2×C32#91614yes—
256C2×C32#91611yescharacteristic
257C2×C32#91811yescharacteristic
258C2×C32#93914yes—
259C3×D3—84no—
260C2×Q8—31no—
261C2×Q8—91no—
262C2×D4—31no—
263Q16—542no—
264SD16—62no—
265C2×C8—271no—
266C2×C8D3×C3⋊D311yescharacteristic
267C4⋊C4—91no—
268C4⋊C4—541no—
269C22⋊C4—541no—
270C22⋊C4—541no—
271C42—271no—
272C42—541no—
273C22×C3—24no—
274C22×C3—124no—
275C22×C3—124no—
276C22×C3—124no—
277C22×C3#380214yes—
278C22×C3D4×C3⋊D311yescharacteristic
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C3×C4—24no—
283C3×C4—24no—
284C3×C4—68no—
285C3×C4—361no—
286C3×C4—361no—
287C3×C4#379514yes—
288C3×C4D3×#6714yes—
289C3×C4#383011yescharacteristic
290C3×C4C2×#63811yescharacteristic
291Dic3—22no—
292Dic3—124no—
293Dic3—124no—
294Dic3—188no—
295Dic3—184no—
296Dic3—184no—
297Dic3—181no—
298Dic3—181no—
299Dic3—364no—
300C32—24no—
301C32#479811yescharacteristic, complemented
302C32#516614yescomplemented
303C23—61no—
304Q8—32no—
305Q8—61no—
306Q8—181no—
307Q8—181no—
308D4—61no—
309D4—61no—
310C2×C4—61no—
311C2×C4—181no—
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—244no—
322C2×C3#6380914yes—
323C2×C3#6381114yes—
324C2×C3#6388514yes—
325C2×C3#6412411yescharacteristic
326C2×C3#6412511yescharacteristic
327C2×C3#6414811yescharacteristic
328D3—81no—
329C22—121no—
330C22—121no—
331C22—121no—
332C22#8902611yescharacteristic
333C4—62no—
334C4—361no—
335C4—361no—
336C4—541no—
337C4—541no—
338C4#8901411yescharacteristic
339C4D3×#63811yescharacteristic
340C3—24no—
341C3#1059079614yescomplemented
342C3#1059286511yescharacteristic, complemented
343C2—241no—
344C2#1172334211yescharacteristic
345C2#1172334411yescharacteristic
346C2#1172347411yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
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. #11723397C2semidirect
4. C2×#88798C2semidirect
5. #11722184C2semidirect
6. C3×#63643C2semidirect
32. #63643D3semidirect
213. C33#464semidirect
301. C32#4798semidirect
302. C32#5166semidirect
341. C3#10590796semidirect
342. C3#10592865semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2042
#117234342
#117233972
C2×#887982
#117221842
C3×#636432
#117220312
#105928653
#105907963