Finite Group Databasebeta

← #13159166

Subgroups — #13159166

Order 1728.

828 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C32×D3×C4⋊C4C211yesmaximal, characteristic, complemented
2#11723515C211yesmaximal, characteristic
3#11722178C211yesmaximal, characteristic, complemented
4C3×#63636C211yesmaximal, characteristic, complemented
5#11722054C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592667—31nomaximal
9#10590500—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×C4×D3—21no—
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6290—21no—
14C33×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20#88803—21no—
21C2×C32×C4⋊C4—31no—
22#64139—31no—
23C3×D3×C4⋊C4D314yescomplemented
24#63908—34no—
25#63657—31no—
26#63636—31no—
27#63636D311yescharacteristic, complemented
28#63635—31no—
29#63635—31no—
30#63635—34no—
31#63635—64no—
32C3×#809—34no—
33#63573—34no—
34#63562—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3×C3⋊D3—21no—
37C22×C32×D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C32×C4×D3—21no—
46C32×C4×D3—21no—
47C32×C4×D3—42no—
48#6290—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C3×C3⋊(C3⋊C8)—21no—
51#4968—94no—
52#4777—91no—
53C22×C32×C4—31no—
54C22×C32×C4—61no—
55C2×#638—61no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C3×D12—34no—
58C2×C3×C4×D3—24no—
59C2×C3×C4×D3C2×D314yes—
60C2×#625—64no—
61C32×C4⋊C4—24no—
62C32×C4⋊C4—31no—
63C32×C4⋊C4—61no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—61no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—64no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—64no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×C4⋊Dic3C2×D314yes—
75C3×Dic3⋊C4—24no—
76C2×C3×C3⋊C8—34no—
77(C2×D3)⋊Dic3—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3—41no—
82C2×C32×D3—21no—
83C2×C32×D3—21no—
84C2×C32×D3SD1612yes—
85C33×C4—41no—
86C33×C4#4111yescharacteristic
87C33×C4#4111yescharacteristic
88C3⋊(C3⋊Dic3)—41no—
89C32×Dic3—21no—
90C32×Dic3—41no—
91C2×C3×C4⋊C4—34no—
92#938—91no—
93#930—94no—
94D3×C4⋊C4C3⋊D311yescharacteristic, complemented
95C3×#102—91no—
96#832—91no—
97#830—94no—
98(C2×D3)⋊C8—91no—
99#817—91no—
100#809—34no—
101#809—94no—
102#808—94no—
103#808—94no—
104#808—91no—
105#808—184no—
106C23×C32—31no—
107C22×C3⋊D3—21no—
108C22×C3×D3—64no—
109C22×C3×D3#6714yes—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—44no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117C2×C32×C4—61no—
118C2×C32×C4—122no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121#638—122no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—124no—
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)—21no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3#6714yes—
129C3×D12—64no—
130C3×D12—64no—
131C3×C4×D3—24no—
132C3×C4×D3—24no—
133C3×C4×D3—48no—
134#625—128no—
135C3⋊(C3⋊C8)—21no—
136C3⋊(C3⋊C8)—68no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—61no—
139C3×C3⋊C8—64no—
140#481—271noSylow
141C2×C33#12311yescharacteristic
142C2×C33C2×SD1611yescharacteristic
143C2×C33AGL(1,ℤ/8)11yescharacteristic
144C3×C3⋊D3—81no—
145C32×D3—22no—
146C2×C3×D4—91no—
147C22×C3×C4—34no—
148C22×C3×C4—64no—
149C2×#67—184no—
150C2×#67—181no—
151C2×D12—34no—
152C2×C4×D3—21no—
153C2×C4×D3C2×C3⋊D311yescharacteristic
154C2×C3×C8—91no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—34no—
157C3×C4⋊C4—64no—
158C3×C4⋊C4D3214yes—
159C3×C4⋊C4C2×C3⋊D311yescharacteristic
160#217—181no—
161(C2×D3)⋊C4—184no—
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—184no—
165C4⋊Dic3C2×C3⋊D311yescharacteristic
166Dic3⋊C4—21no—
167C2×C3⋊C8—34no—
168C2×C3⋊C8—94no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—184no—
171C22×C32—24no—
172C22×C32—32no—
173C22×C32—61no—
174C22×C32—61no—
175C22×C32D3×D411yescharacteristic
176C22×C32C2×#6714yes—
177C2×C3⋊D3—41no—
178C2×C3⋊D3—41no—
179C2×C3⋊D3—41no—
180C2×C3×D3—24no—
181C2×C3×D3—24no—
182C2×C3×D3—124no—
183C2×C3×D3—124no—
184C2×C3×D3—124no—
185C2×C3×D3#21518yes—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—41no—
189C32×C4—44no—
190C32×C4—61no—
191C32×C4—84no—
192C32×C4—121no—
193C32×C4#23514yes—
194C32×C4#23514yes—
195C32×C4#23511yescharacteristic
196C32×C4#23711yescharacteristic
197C3⋊Dic3—121no—
198C3⋊Dic3—124no—
199C3⋊Dic3—244no—
200C3×Dic3—24no—
201C3×Dic3—44no—
202#121—271no—
203C2×C4⋊C4—31no—
204#106—271no—
205#106—271no—
206#102—91no—
207#102—271no—
208C22⋊C8—271no—
209C33#48111yescharacteristic, complemented, Sylow
210C23×C3—34no—
211C23×C3—181no—
212C22×D3—64no—
213C22×D3#63811yescharacteristic
214C3×D4—181no—
215C3×D4—181no—
216C2×C3×C4—24no—
217C2×C3×C4—21no—
218C2×C3×C4—24no—
219C2×C3×C4—34no—
220C2×C3×C4—44no—
221C2×C3×C4—64no—
222C2×C3×C4—64no—
223C2×C3×C4—64no—
224C2×C3×C4—128no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227#67—368no—
228#67—362no—
229C2×Dic3—21no—
230C2×Dic3—184no—
231C2×Dic3—181no—
232C2×Dic3—364no—
233C2×Dic3#63811yescharacteristic
234D12—64no—
235D12—64no—
236C4×D3—21no—
237C4×D3—21no—
238C4×D3—42no—
239C3×C8—181no—
240C3⋊C8—64no—
241C3⋊C8—188no—
242C3⋊C8—184no—
243C3⋊C8—181no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—62no—
248C2×C32#88311yescharacteristic
249C2×C32#91111yescharacteristic
250C2×C32D3×SD1611yescharacteristic
251C2×C32#92914yes—
252C2×C32#93214yes—
253C2×C32C2×#21514yes—
254C3⋊D3—81no—
255C3×D3—28no—
256C3×D3—244no—
257C2×D4—91no—
258C2×D4—541no—
259C22×C4—31no—
260C22×C4—61no—
261C2×C8—91no—
262C2×C8—271no—
263C4⋊C4—31no—
264C4⋊C4—61no—
265C4⋊C4—271no—
266C4⋊C4D3×C3⋊D311yescharacteristic
267C22⋊C4—541no—
268C22×C3—24no—
269C22×C3—38no—
270C22×C3—64no—
271C22×C3—64no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3—361no—
275C22×C3D3×#6714yes—
276C22×C3C2×#63811yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—124no—
280C2×D3—124no—
281C2×D3—124no—
282C2×D3#375512yes—
283C3×C4—24no—
284C3×C4—24no—
285C3×C4—41no—
286C3×C4—44no—
287C3×C4—64no—
288C3×C4—84no—
289C3×C4—124no—
290C3×C4#379514yes—
291C3×C4#379814yes—
292C3×C4#382811yescharacteristic
293C3×C4#382811yescharacteristic
294Dic3—21no—
295Dic3—41no—
296Dic3—361no—
297Dic3—364no—
298Dic3—724no—
299C32—24no—
300C32#477711yescharacteristic, complemented
301C32#496814yescomplemented
302C23—31no—
303C23—181no—
304D4—181no—
305D4—181no—
306D4—1082no—
307C2×C4—21no—
308C2×C4—31no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—122no—
313C2×C4—541no—
314C2×C4C2×D3×C3⋊D311yescharacteristic
315C8—181no—
316C8—541no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—68no—
321C2×C3—721no—
322C2×C3#6390514yes—
323C2×C3#6393214yes—
324C2×C3D3×#21514yes—
325C2×C3#6413811yescharacteristic
326C2×C3#6414111yescharacteristic
327C2×C3C2×#375511yescharacteristic
328D3—22no—
329D3—244no—
330C22—32no—
331C22—61no—
332C22—61no—
333C22—361no—
334C22—361no—
335C22—361no—
336C22D3×#63811yescharacteristic
337C4—41no—
338C4—61no—
339C4—121no—
340C4—1081no—
341C4#8902011yescharacteristic
342C4#8902111yescharacteristic
343C3—24no—
344C3#1059050014yescomplemented
345C3#1059266711yescharacteristic, complemented
346C2—62no—
347C2—721no—
348C2#1172351311yescharacteristic
349C2#1172355511yescharacteristic
350C2D3×#375511yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723515C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#481
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×D3×C4⋊C4C2semidirect
3. #11722178C2semidirect
4. C3×#63636C2semidirect
5. #11722054C2semidirect
6. #11722033C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
23. C3×D3×C4⋊C4D3semidirect
27. #63636D3semidirect
94. D3×C4⋊C4C3⋊D3semidirect
209. C33#481semidirect
300. C32#4777semidirect
301. C32#4968semidirect
344. C3#10590500semidirect
345. C3#10592667semidirect

Maximal subgroups

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

subgroupindex
C32×D3×C4⋊C42
#117235152
#117221782
C3×#636362
#117220542
#117220332
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105926673
#105905003