Finite Group Databasebeta

← #13139943

Subgroups — #13139943

Order 1728.

493 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 85 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#11721830C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3#63657⋊C3C211yesmaximal, characteristic, complemented
4#11720324C212yesmaximal, complemented
5#11720307C212yesmaximal, complemented
6#10591092—31nomaximal
7#10591091—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235—21no—
10C2×C4⋊#272C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3—21no—
12#88542—21no—
13C4⋊#1084C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
18C3×#930—31no—
19C3×#930—31no—
20C3×#930—61no—
21C2×C3×C4⋊Dic3—31no—
22#63657D311yescharacteristic, complemented
23C3×#830—31no—
24#63574—32no—
25#63573—32no—
26#63563—32no—
27#63562—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C22×#272—21no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33#6235—42no—
34C2×#1084—21no—
35C4⋊#272—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—42no—
38(C3⋊(C3⋊C8))⋊C3—22no—
39#5034—91no—
40#4968—91no—
41#4735—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3—21no—
46C2×C3×#67—62no—
47C2×C3×#67—61no—
48C2×C3×#67—61no—
49C22×C3×Dic3—61no—
50C2×C3×D12—31no—
51C2×C3×D12—31no—
52C2×C3×D12—61no—
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)C2×D312yes—
55C2×C3⋊(C3⋊C8)C2×D312yes—
56C3×(C2×D3)⋊C4—62no—
57C3×(C2×D3)⋊C4—61no—
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—61no—
63C3×C4⋊Dic3—61no—
64C2×C3×C3⋊C8—32no—
65C22×H(3,𝔽3)—21no—
66C22×H(3,𝔽3)—21no—
67C22×H(3,𝔽3)SD1612yes—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C2×#272—41no—
70C2×#272—41no—
71C2×#272—41no—
72C4×H(3,𝔽3)—21no—
73C4×H(3,𝔽3)#4112yes—
74#1084—41no—
75(C3⋊Dic3)⋊C3—42no—
76C3×#121—91no—
77C2×C3×C4⋊C4—91no—
78#930—31no—
79#930—31no—
80#930—61no—
81C2×C4⋊Dic3—31no—
82C2×C4⋊Dic3—91no—
83C3×C22⋊C8—91no—
84#832—92no—
85#830—31no—
86#830—91no—
87#821—92no—
88#817—92no—
89#809—92no—
90#808—92no—
91#808—92no—
92C23×C32—21no—
93C23×C32D1211yescharacteristic
94C23×C32#6711yescharacteristic
95C22×C3×D3—62no—
96C22×C3×D3—61no—
97C22×C3×D3—61no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—42no—
103C2×C32×C4D1211yescharacteristic
104C2×C32×C4#6711yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—42no—
109C3×#67—124no—
110C3×#67—122no—
111C3×#67—122no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—122no—
117C3×D12—62no—
118C3×D12—62no—
119C3×D12—122no—
120C3⋊(C3⋊C8)—22no—
121C3×C3⋊C8—62no—
122#481—271noSylow
123C2×H(3,𝔽3)—22no—
124C2×H(3,𝔽3)#12311yescharacteristic
125C2×H(3,𝔽3)C2×SD1611yescharacteristic
126C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
127#272—81no—
128C2×C3×D4—91no—
129C2×C3×D4—181no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—61no—
133C22×C3×C4—181no—
134C22×C3×C4D3211yescharacteristic
135C2×#67—61no—
136C2×#67—61no—
137C2×#67—62no—
138C22×Dic3—61no—
139C22×Dic3—181no—
140C2×D12—31no—
141C2×D12—31no—
142C2×D12—61no—
143C2×C3×C8—92no—
144C3×C4⋊C4—92no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—181no—
147C3×C22⋊C4—181no—
148(C2×D3)⋊C4—61no—
149(C2×D3)⋊C4—61no—
150(C2×D3)⋊C4—62no—
151C4⋊Dic3—31no—
152C4⋊Dic3—31no—
153C4⋊Dic3—32no—
154C4⋊Dic3—61no—
155C4⋊Dic3—61no—
156C4⋊Dic3—92no—
157C4⋊Dic3—181no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—92no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32—41no—
167C22×C32—41no—
168C22×C32#20412yes—
169C22×C32#21512yes—
170C22×C32C2×D1211yescharacteristic
171C22×C32C2×#6711yescharacteristic
172C2×C3×D3—122no—
173C2×C3×D3—122no—
174C2×C3×D3—122no—
175C2×C3×D3—121no—
176C2×C3×D3—121no—
177C2×C3×D3—121no—
178C2×C3×D3—121no—
179C2×C3×D3—121no—
180C2×C3×D3—121no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—41no—
185C32×C4#23512yes—
186C32×C4#23712yes—
187C3⋊Dic3—42no—
188C3×Dic3—122no—
189C3×Dic3—121no—
190C3×Dic3—122no—
191C3×Dic3—121no—
192#121—91no—
193C2×C4⋊C4—91no—
194#106—272no—
195#102—272no—
196C22⋊C8—91no—
197H(3,𝔽3)#48111yescharacteristic, complemented, Sylow
198C23×C3—31no—
199C23×C3—31no—
200C23×C3—61no—
201C23×C3—181no—
202C23×C3#62611yescharacteristic
203C22×D3—61no—
204C22×D3—61no—
205C22×D3—62no—
206C3×D4—182no—
207C3×D4—362no—
208C2×C3×C4—22no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—62no—
214C2×C3×C4—62no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—122no—
218C2×C3×C4—182no—
219C2×C3×C4—181no—
220C2×C3×C4—362no—
221C2×C3×C4#62611yescharacteristic
222C2×C3×C4C2×D3211yescharacteristic
223#67—122no—
224#67—122no—
225#67—124no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—122no—
231C2×Dic3—182no—
232C2×Dic3—362no—
233D12—62no—
234D12—62no—
235D12—122no—
236C3×C8—182no—
237C3⋊C8—62no—
238C3⋊C8—182no—
239C2×C32—22no—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—42no—
245C2×C32#88611yescharacteristic
246C2×C32C2×#20411yescharacteristic
247C2×C32#90811yescharacteristic
248C2×C32#92911yescharacteristic
249C2×C32#93211yescharacteristic
250C2×C32C2×#21511yescharacteristic
251C3×D3—242no—
252C3×D3—241no—
253C3×D3—241no—
254C2×D4—91no—
255C2×D4—181no—
256C22×C4—181no—
257C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C2×C8—92no—
259C4⋊C4—92no—
260C4⋊C4—91no—
261C4⋊C4—181no—
262C22⋊C4—181no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—32no—
268C22×C3—32no—
269C22×C3—61no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—61no—
273C22×C3—61no—
274C22×C3—62no—
275C22×C3—121no—
276C22×C3—121no—
277C22×C3—361no—
278C22×C3—361no—
279C22×C3—361no—
280C22×C3#371712yes—
281C22×C3C2×#62611yescharacteristic
282C2×D3—121no—
283C2×D3—121no—
284C2×D3—121no—
285C2×D3—121no—
286C2×D3—121no—
287C2×D3—121no—
288C2×D3—122no—
289C2×D3—122no—
290C2×D3—122no—
291C3×C4—21no—
292C3×C4—32no—
293C3×C4—32no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—62no—
297C3×C4—121no—
298C3×C4—362no—
299C3×C4—361no—
300C3×C4#380412yes—
301Dic3—121no—
302Dic3—121no—
303Dic3—122no—
304Dic3—122no—
305Dic3—362no—
306C32—21no—
307C32#473511yescharacteristic, complemented
308C32#496811yescharacteristic, complemented
309C23—181no—
310C23#620011yescharacteristic
311D4—182no—
312D4—362no—
313C2×C4—22no—
314C2×C4—181no—
315C2×C4—182no—
316C2×C4—362no—
317C2×C4#620011yescharacteristic
318C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
319C8—182no—
320C2×C3—22no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—62no—
328C2×C3—62no—
329C2×C3—61no—
330C2×C3—61no—
331C2×C3—61no—
332C2×C3—122no—
333C2×C3—721no—
334C2×C3#6383011yescharacteristic
335C2×C3C2×#371711yescharacteristic
336C2×C3#6396611yescharacteristic
337D3—241no—
338D3—241no—
339D3—242no—
340C22—21no—
341C22—21no—
342C22—361no—
343C22—361no—
344C22—361no—
345C22#8843712yes—
346C22C2×#620011yescharacteristic
347C4—21no—
348C4—361no—
349C4—362no—
350C4#8867112yes—
351C3—31no—
352C3—31no—
353C3—61no—
354C3#1059112511yescharacteristic
355C2—22no—
356C2—721no—
357C2C2×#8843711yescharacteristic
358C2#1172107111yescharacteristic
359C2#1172130111yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#11721830C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#481
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
3. #63657⋊C3C2semidirect
4. #11720324C2semidirect
5. #11720307C2semidirect
17. C2×C4⋊(C3⋊Dic3)D3semidirect
22. #63657D3semidirect
197. H(3,𝔽3)#481semidirect
307. C32#4735semidirect
308. C32#4968semidirect

Maximal subgroups

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

subgroupindex
#117218302
C2×C4⋊((C3⋊Dic3)⋊C3)2
#63657⋊C32
#117203242
#117203072
#105910923
#105910913