Finite Group Databasebeta

← #13154294

Subgroups — #13154294

Order 1728.

1485 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1485 conjugacy classes of subgroups — too many to draw. Only its 213 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×#88748C211yesmaximal, characteristic
2C2×Q8×#272C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
4#11721864C211yesmaximal, characteristic, complemented
5#11721861C211yesmaximal, characteristic, complemented
6#11721764C214yesmaximal, complemented
7#11721762C214yesmaximal, complemented
8#11721721C211yesmaximal, characteristic, complemented
9#11721718C211yesmaximal, characteristic, complemented
10C3×#5724—34nomaximal
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13#88748—24no—
14Q8×#272—24no—
15C2×C4⋊#272C2212yes—
16C2×C4⋊#272C2211yescharacteristic
17C2×C4×#272C2212yes—
18C2×C4×#272C2212yes—
19C2×C4×#272C2211yescharacteristic
20C2×#6229C2212yes—
21C2×#6229C2211yescharacteristic
22C4⋊C4×H(3,𝔽3)C2214yes—
23C42×H(3,𝔽3)C2211yescharacteristic
24#88542C2212yes—
25#88542C2214yes—
26#88542C2214yes—
27#88540C2214yes—
28#88540C2212yes—
29C4×#1084C2211yescharacteristic
30C2×C3×#239—34no—
31C2×C3×D3×Q8—34no—
32C32×C4⋊Q8D314yescomplemented
33C3×#947—34no—
34C3×#944—34no—
35C3×#896—316no—
36C3×#894—316no—
37C3×#875—34no—
38C3×#872—34no—
39C22×#272C2312yes—
40C22×#272C2311yescharacteristic
41Q8×H(3,𝔽3)—22no—
42Q8×H(3,𝔽3)—22no—
43C2×C4×H(3,𝔽3)C2312yes—
44C2×C4×H(3,𝔽3)C2312yes—
45C2×C4×H(3,𝔽3)C2312yes—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×#1084C2312yes—
48C2×#1084C2312yes—
49C2×#1084C2311yescharacteristic
50C4⋊#272—24no—
51C4⋊#272—22no—
52C4×#272—24no—
53C4×#272—24no—
54C4×#272D414yes—
55#6229—22no—
56#6229—24no—
57C3×#536—91no—
58#5724—34no—
59C2×C32×Q8C2×D314yes—
60C2×C32×Q8C2×D314yes—
61C3×#239—616no—
62C3×D3×Q8—616no—
63C2×C3×D12—38no—
64C2×C3×D12—34no—
65C2×C3×C4×D3—34no—
66C2×C3×C4×D3—38no—
67C2×C3×C4×D3—38no—
68C2×C3×Dic6—38no—
69C2×C3×Dic6—34no—
70C32×C4⋊C4C2×D3116yes—
71C32×C42C2×D314yes—
72C3×(C2×D3)⋊C4—38no—
73C3×(C2×D3)⋊C4—316no—
74C3×(C2×D3)⋊C4—316no—
75C3×Dic3⋊C4—316no—
76C3×Dic3⋊C4—38no—
77C3×C4×Dic3—34no—
78C22×H(3,𝔽3)C2411yescharacteristic
79C2×#272—22no—
80C2×#272—22no—
81C2×#272—24no—
82C2×#272C2×D412yes—
83C4×H(3,𝔽3)—22no—
84C4×H(3,𝔽3)—22no—
85C4×H(3,𝔽3)—22no—
86C4×H(3,𝔽3)C2×D412yes—
87#1084—22no—
88#1084—22no—
89#1084C2×D412yes—
90C2×C3×#41—91no—
91C22×C3×Q8—91no—
92C2×#239—34no—
93C2×D3×Q8—34no—
94C3×C4⋊Q8—34no—
95C3×C4⋊Q8—91no—
96C3×C4⋊Q8C3⋊D311yescharacteristic
97C3×#124—91no—
98C3×#124—91no—
99C3×#123—94no—
100C3×C22⋊Q8—94no—
101C3×#117—91no—
102#947—34no—
103#944—34no—
104#896—316no—
105#894—316no—
106#875—34no—
107#872—34no—
108C22×C3×D3—34no—
109C22×C3×D3—38no—
110C32×Q8—28no—
111C32×Q8—28no—
112C2×C32×C4C22×D318yes—
113C2×C32×C4C22×D318yes—
114C2×C32×C4C22×D318yes—
115C2×C32×C4C22×D314yes—
116C2×C3×Dic3—34no—
117C2×C3×Dic3—38no—
118C2×C3×Dic3—38no—
119C3×D12—616no—
120C3×D12—68no—
121C3×C4×D3—316no—
122C3×C4×D3—616no—
123C3×C4×D3—616no—
124C3×Dic6—616no—
125C3×Dic6—68no—
126#536—91noSylow
127C2×H(3,𝔽3)C22×D411yescharacteristic
128C2×H(3,𝔽3)2−1+412yes—
129#272—22no—
130#272—42no—
131C3×#41—184no—
132C2×C3×Q8—34no—
133C2×C3×Q8—34no—
134C2×C3×Q8—92no—
135C2×C3×Q8—91no—
136C2×C3×Q8—184no—
137C2×C3×Q8C2×C3⋊D311yescharacteristic
138C2×C3×Q8C2×C3⋊D311yescharacteristic
139C2×C3×D4—92no—
140C2×C3×D4—91no—
141C22×C3×C4—91no—
142C22×C3×C4—92no—
143C22×C3×C4—92no—
144#239—616no—
145D3×Q8—616no—
146C2×D12—34no—
147C2×D12—38no—
148C2×C4×D3—34no—
149C2×C4×D3—38no—
150C2×C4×D3—38no—
151C2×Dic6—34no—
152C2×Dic6—38no—
153C3×C4⋊C4—316no—
154C3×C4⋊C4—92no—
155C3×C4⋊C4—94no—
156C3×C4⋊C4C2×C3⋊D314yes—
157C3×C22⋊C4—92no—
158C3×C22⋊C4—94no—
159C3×C22⋊C4—94no—
160C3×C42—34no—
161C3×C42—91no—
162C3×C42C2×C3⋊D311yescharacteristic
163(C2×D3)⋊C4—38no—
164(C2×D3)⋊C4—316no—
165(C2×D3)⋊C4—316no—
166Dic3⋊C4—38no—
167Dic3⋊C4—316no—
168C4×Dic3—34no—
169C22×C32C23×D314yes—
170C2×C3×D3—38no—
171C2×C3×D3—68no—
172C2×C3×D3—68no—
173C2×C3×D3—616no—
174C32×C4—28no—
175C32×C4—28no—
176C32×C4—28no—
177C32×C4D3×D418yes—
178C3×Dic3—38no—
179C3×Dic3—68no—
180C3×Dic3—68no—
181C2×#41—91no—
182C22×Q8—91no—
183C4⋊Q8—91no—
184C4⋊Q8#27211yescharacteristic, complemented
185#124—91no—
186#124—91no—
187#123—94no—
188C22⋊Q8—94no—
189#117—91no—
190H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
191C23×C3—91no—
192C23×C3—92no—
193C22×D3—34no—
194C22×D3—38no—
195C3×Q8—22no—
196C3×Q8—22no—
197C3×Q8—68no—
198C3×Q8—68no—
199C3×Q8—184no—
200C3×Q8—182no—
201C3×D4—184no—
202C3×D4—182no—
203C2×C3×C4—34no—
204C2×C3×C4—38no—
205C2×C3×C4—38no—
206C2×C3×C4—38no—
207C2×C3×C4—91no—
208C2×C3×C4—94no—
209C2×C3×C4—92no—
210C2×C3×C4—92no—
211C2×C3×C4—184no—
212C2×C3×C4—184no—
213C2×C3×C4C22×C3⋊D312yes—
214C2×C3×C4C22×C3⋊D312yes—
215C2×C3×C4C22×C3⋊D312yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—34no—
218C2×Dic3—38no—
219C2×Dic3—38no—
220D12—68no—
221D12—616no—
222C4×D3—316no—
223C4×D3—616no—
224C4×D3—616no—
225Dic6—616no—
226Dic6—68no—
227C2×C32C2×D3×D414yes—
228C2×C32#100718yes—
229C3×D3—68no—
230C3×D3—128no—
231#41—184no—
232C2×Q8—92no—
233C2×Q8—91no—
234C2×Q8—184no—
235C2×Q8C2×#27211yescharacteristic
236C2×Q8C2×#27211yescharacteristic
237C2×D4—92no—
238C2×D4—91no—
239C22×C4—91no—
240C22×C4—92no—
241C22×C4—92no—
242C4⋊C4—92no—
243C4⋊C4—94no—
244C4⋊C4C2×#27214yes—
245C22⋊C4—92no—
246C22⋊C4—94no—
247C22⋊C4—94no—
248C42—91no—
249C42C2×#27211yescharacteristic
250C22×C3—34no—
251C22×C3—92no—
252C22×C3—182no—
253C22×C3—182no—
254C22×C3—184no—
255C22×C3C23×C3⋊D311yescharacteristic
256C2×D3—38no—
257C2×D3—68no—
258C2×D3—68no—
259C2×D3—616no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—38no—
264C3×C4—68no—
265C3×C4—68no—
266C3×C4—68no—
267C3×C4—92no—
268C3×C4—182no—
269C3×C4—182no—
270C3×C4D4×C3⋊D312yes—
271Dic3—38no—
272Dic3—68no—
273Dic3—68no—
274C32#572414yescomplemented
275C23—91no—
276C23—92no—
277Q8—22no—
278Q8—22no—
279Q8—184no—
280Q8—182no—
281D4—184no—
282D4—182no—
283C2×C4—91no—
284C2×C4—94no—
285C2×C4—92no—
286C2×C4—92no—
287C2×C4—184no—
288C2×C4—184no—
289C2×C4C22×#27212yes—
290C2×C4C22×#27212yes—
291C2×C4C22×#27212yes—
292C2×C4C22×#27211yescharacteristic
293C2×C3—34no—
294C2×C3—38no—
295C2×C3—182no—
296C2×C3—362no—
297C2×C3C2×D4×C3⋊D311yescharacteristic
298C2×C3#6436412yes—
299D3—68no—
300D3—128no—
301C22—92no—
302C22—182no—
303C22—182no—
304C22—184no—
305C22C23×#27211yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—92no—
310C4—182no—
311C4—182no—
312C4D4×#27212yes—
313C3—34no—
314C3#1059619511yescharacteristic
315C2—182no—
316C2—362no—
317C2C2×D4×#27211yescharacteristic
318C2#1172267012yes—
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC2×#88748C2
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#536
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#272C2semidirect
3. H(3,𝔽3)×C4⋊Q8C2semidirect
4. #11721864C2semidirect
5. #11721861C2semidirect
6. #11721764C2semidirect
7. #11721762C2semidirect
8. #11721721C2semidirect
9. #11721718C2semidirect
32. C32×C4⋊Q8D3semidirect
184. C4⋊Q8#272semidirect
190. H(3,𝔽3)#536semidirect
274. C32#5724semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
C2×Q8×#2722
H(3,𝔽3)×C4⋊Q82
#117218642
#117218612
#117217642
#117217622
#117217212
#117217182
C3×#57243