Finite Group Databasebeta

← #10592444⋊C3 #13141673

Subgroups — #10592444⋊C3 #13141673

Order 1728.

571 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3830⋊C3C211yesmaximal, characteristic
2M16×(C3⋊D3)⋊C3C212yesmaximal, complemented
3H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
4#63660⋊C3C211yesmaximal, characteristic, complemented
5#63651⋊C3C212yesmaximal, complemented
6#10592444C311yesmaximal, characteristic, complemented
7C3×#4745—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3C412yes—
10C22×(C3⋊Dic3)⋊C3C412yes—
11#3830⋊C3—22no—
12#3830⋊C3—22no—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic
15M16×H(3,𝔽3)C2212yes—
16#3748⋊C3C2212yes—
17#3743⋊C3—22no—
18C8×(C3⋊D3)⋊C3—22no—
19C2×#3830C2×C311yescharacteristic
20C2×C3×#237—31no—
21M16×C3⋊D3C2×C312yescomplemented
22C3×D3×M16—32no—
23C32×#100—21no—
24C32×#100D311yescharacteristic, complemented
25C32×#100C2×C311yescharacteristic, complemented
26#63660C2×C311yescharacteristic, complemented
27#63651C2×C312yescomplemented
28C3×#833—31no—
29C3×#824—32no—
30C23×H(3,𝔽3)C2×C412yes—
31C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
32D4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34#638⋊C3—24no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3C2×C412yes—
38C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
39C4×(C3⋊D3)⋊C3D412yes—
40C4×(C3⋊D3)⋊C3D412yes—
41#634⋊C3—22no—
42C8×H(3,𝔽3)—22no—
43(C3⋊(C3⋊C8))⋊C3—22no—
44C3×#413—91no—
45#4745—31no—
46#4745—91no—
47C2×C32×D4—21no—
48C2×C32×D4C2×D311yescharacteristic
49C2×C32×D4C22×C311yescharacteristic
50C2×#638C3×C412yes—
51C22×C3⋊Dic3C3×C412yes—
52#3830—22no—
53#3830—22no—
54C2×C4×C3⋊D3C22×C311yescharacteristic
55C2×#634C22×C311yescharacteristic
56C2×C3×#67—32no—
57C22×C3×Dic3—32no—
58C3×#237—62no—
59C3×#237—62no—
60C2×C3×C4×D3—31no—
61C2×C3×Dic6—31no—
62C32×M16—22no—
63C32×M16C2×D312yes—
64C32×M16C22×C312yes—
65#3748C22×C312yes—
66#3743—22no—
67C8×C3⋊D3—22no—
68C3×#208—32no—
69C3×ΓL(1,𝔽25)—62no—
70C3×D3×C8—62no—
71C22×H(3,𝔽3)—22no—
72C22×H(3,𝔽3)—42no—
73C22×H(3,𝔽3)C22×C411yescharacteristic
74C2×(C3⋊D3)⋊C3—41no—
75C2×(C3⋊D3)⋊C3C22⋊C412yes—
76C4×H(3,𝔽3)C2×D412yes—
77(C3⋊Dic3)⋊C3—22no—
78(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
79(C3⋊Dic3)⋊C3C22⋊C411yescharacteristic
80C2×C3×#41—91no—
81C2×#237—31no—
82C2×#237—91no—
83C2×C3×M16—92no—
84D3×M16—32no—
85D3×M16—92no—
86C3×#101—92no—
87C3×#100—31no—
88C3×#100—31no—
89C3×#100—61no—
90C3×#100—91no—
91C3×#100C3×D311yescharacteristic
92#833—31no—
93#833—91no—
94#824—32no—
95#824—92no—
96C23×C32—22no—
97C23×C32C4×D312yes—
98C23×C32C2×C3×C412yes—
99C22×C3⋊D3C2×C3×C411yescharacteristic
100C22×C3×D3—31no—
101C32×D4—22no—
102C32×D4—22no—
103C32×D4—42no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C23×C311yescharacteristic
107#638—24no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3C2×C3×C412yes—
111C2×C3⋊Dic3C2×C3×C411yescharacteristic
112C4×C3⋊D3C3×D412yes—
113C4×C3⋊D3C3×D412yes—
114#634—22no—
115C3×#67—64no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C3×C4×D3—32no—
121C3×C4×D3—32no—
122C3×Dic6—62no—
123C32×C8—22no—
124C32×C8—22no—
125C32×C8—42no—
126C3⋊(C3⋊C8)—22no—
127C3×C3⋊C8—62no—
128#413—91noSylow
129C2×H(3,𝔽3)—21no—
130C2×H(3,𝔽3)—42no—
131C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
132(C3⋊D3)⋊C3—22no—
133C3×#41—182no—
134C3×#41—182no—
135C2×C3×Q8—91no—
136C2×C3×D4—31no—
137C2×C3×D4—31no—
138C2×C3×D4—61no—
139C2×C3×D4—92no—
140C2×C3×D4C2×C3×D311yescharacteristic
141C22×C3×C4—91no—
142C22×C3×C4—92no—
143C2×#67—32no—
144C2×#67—92no—
145C22×Dic3—32no—
146C22×Dic3—92no—
147#237—62no—
148#237—62no—
149#237—182no—
150#237—182no—
151C2×C4×D3—31no—
152C2×C4×D3—91no—
153C2×Dic6—31no—
154C2×Dic6—91no—
155C3×M16—32no—
156C3×M16—32no—
157C3×M16—62no—
158C3×M16—92no—
159C3×M16—182no—
160C3×M16C2×C3×D312yes—
161C2×C3×C8—182no—
162#208—32no—
163#208—92no—
164ΓL(1,𝔽25)—62no—
165ΓL(1,𝔽25)—182no—
166D3×C8—62no—
167D3×C8—182no—
168C22×C32—22no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—42no—
172C22×C32—42no—
173C22×C32—42no—
174C22×C32—82no—
175C22×C32C2×C4×D311yescharacteristic
176C22×C32C22×C3×C411yescharacteristic
177C2×C3⋊D3—41no—
178C2×C3⋊D3C3×C22⋊C412yes—
179C2×C3×D3—32no—
180C2×C3×D3—121no—
181C32×C4—22no—
182C32×C4D3×D412yes—
183C32×C4C2×C3×D412yes—
184C3⋊Dic3—22no—
185C3⋊Dic3C3×C22⋊C411yescharacteristic
186C3⋊Dic3C3×C22⋊C411yescharacteristic
187C3×Dic3—31no—
188C3×Dic3—31no—
189C3×Dic3—62no—
190C2×#41—91no—
191C2×M16—92no—
192#101—92no—
193#100—91no—
194#100(C3⋊D3)⋊C311yescharacteristic, complemented
195H(3,𝔽3)#41311yescharacteristic, complemented, Sylow
196C23×C3—32no—
197C23×C3—32no—
198C23×C3—62no—
199C23×C3—91no—
200C23×C3C3×C4×D312yes—
201C22×D3—31no—
202C22×D3—91no—
203C3×Q8—182no—
204C3×D4—22no—
205C3×D4—62no—
206C3×D4—62no—
207C3×D4—122no—
208C3×D4—184no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—61no—
212C2×C3×C4—92no—
213C2×C3×C4—92no—
214C2×C3×C4—91no—
215C2×C3×C4—92no—
216C2×C3×C4—182no—
217C2×C3×C4—182no—
218C2×C3×C4C22×C3×D311yescharacteristic
219#67—64no—
220#67—184no—
221C2×Dic3—31no—
222C2×Dic3—32no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—91no—
226C2×Dic3—92no—
227C2×Dic3—182no—
228C2×Dic3—182no—
229C4×D3—32no—
230C4×D3—32no—
231C4×D3—92no—
232C4×D3—92no—
233Dic6—62no—
234Dic6—182no—
235C3×C8—22no—
236C3×C8—62no—
237C3×C8—62no—
238C3×C8—122no—
239C3×C8—182no—
240C3⋊C8—62no—
241C3⋊C8—182no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—41no—
246C2×C32—42no—
247C2×C32—42no—
248C2×C32—82no—
249C2×C32D3×C22⋊C411yescharacteristic
250C2×C32C2×C3×C22⋊C411yescharacteristic
251C3⋊D3—22no—
252C3×D3—62no—
253#41—182no—
254#41—182no—
255C2×Q8—91no—
256C2×D4—92no—
257C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
258C22×C4—91no—
259C22×C4—92no—
260M16—92no—
261M16—182no—
262M16C2×(C3⋊D3)⋊C312yes—
263C2×C8—182no—
264C22×C3—22no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—42no—
268C22×C3—62no—
269C22×C3—62no—
270C22×C3—61no—
271C22×C3—92no—
272C22×C3—122no—
273C22×C3—122no—
274C22×C3—122no—
275C22×C3—242no—
276C22×C3—361no—
277C22×C3C2×C3×C4×D311yescharacteristic
278C2×D3—32no—
279C2×D3—92no—
280C2×D3—121no—
281C2×D3—361no—
282C3×C4—32no—
283C3×C4—32no—
284C3×C4—62no—
285C3×C4—91no—
286C3×C4—91no—
287C3×C4—182no—
288C3×C4C3×D3×D412yes—
289Dic3—31no—
290Dic3—31no—
291Dic3—62no—
292Dic3—91no—
293Dic3—91no—
294Dic3—182no—
295C32—21no—
296C32#474511yescharacteristic, complemented
297C32C3×#41311yescharacteristic, complemented
298C23—91no—
299C23C4×(C3⋊D3)⋊C312yes—
300Q8—182no—
301D4—22no—
302D4—184no—
303C2×C4—91no—
304C2×C4—92no—
305C2×C4—92no—
306C2×C4—92no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
310C8—22no—
311C8—182no—
312C2×C3—21no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—42no—
316C2×C3—61no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3—121no—
320C2×C3—122no—
321C2×C3—122no—
322C2×C3—182no—
323C2×C3—242no—
324C2×C3C3×D3×C22⋊C411yescharacteristic
325D3—62no—
326D3—182no—
327C22—22no—
328C22—42no—
329C22—92no—
330C22—361no—
331C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
332C4—91no—
333C4—91no—
334C4—182no—
335C4D4×(C3⋊D3)⋊C312yes—
336C3—31no—
337C3—31no—
338C3—61no—
339C3C3×#474511yescharacteristic
340C2—21no—
341C2—42no—
342C2—182no—
343C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2C22⋊C4×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#100C2
socleC2×C3C3×D3×C22⋊C4
prime radicalC2×#3830⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#100(C3⋊D3)⋊C3
hypocenterC32C3×#413
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. M16×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×#100C2semidirect
4. #63660⋊C3C2semidirect
5. #63651⋊C3C2semidirect
6. #10592444C3semidirect
21. M16×C3⋊D3C2×C3semidirect
24. C32×#100D3semidirect
25. C32×#100C2×C3semidirect
26. #63660C2×C3semidirect
27. #63651C2×C3semidirect
194. #100(C3⋊D3)⋊C3semidirect
195. H(3,𝔽3)#413semidirect
296. C32#4745semidirect
297. C32C3×#413semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
M16×(C3⋊D3)⋊C32
H(3,𝔽3)×#1002
#63660⋊C32
#63651⋊C32
#105924443
C3×#47453