Finite Group Databasebeta

← #13140566

Subgroups — #13140566

Order 1728.

571 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 97 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
2#11721053C212yesmaximal, complemented
3#11720579C211yesmaximal, characteristic, complemented
4#63660⋊C3C211yesmaximal, characteristic, complemented
5#11720321C212yesmaximal, complemented
6#10591504—31nomaximal
7#10591501—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
15#88538C2212yes—
16#3748⋊C3C2212yes—
17#88430—22no—
18((C3⋊D3)⋊C3)⋊C8—22no—
19C2×#3830D311yescharacteristic
20C2×C3×#237—31no—
21#63818—32no—
22D3×#208—32no—
23#63660D311yescharacteristic, complemented
24C3×#833—31no—
25C3×#833—31no—
26C3×#833—31no—
27C3×#833—61no—
28#63561—32no—
29#63560—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—
42#6180—22no—
43(C3⋊(C3⋊C8))⋊C3—22no—
44#5237—91no—
45#4745—91no—
46#4745—91no—
47C2×C32×D4—21no—
48C2×C32×D4C2×D311yescharacteristic
49C2×C32×D4C2×D311yescharacteristic
50C2×#638Dic312yes—
51C22×C3⋊Dic3Dic312yes—
52#3830—22no—
53#3830—22no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×#634C2×D311yescharacteristic
56C2×C3×#67—32no—
57C22×C3×Dic3—32no—
58C3×#237—62no—
59C3×#237—62no—
60C2×C3×C4×D3—31no—
61C2×C3×Dic6—31no—
62#3748C2×D312yes—
63C3×#208—32no—
64C3×#208—32no—
65C3×#208—32no—
66C3×#208—62no—
67#3712—62no—
68#3711—62no—
69#3710—62no—
70D3×C3⋊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×#208—92no—
84D3×M16—92no—
85D3×M16—92no—
86C3×#100—91no—
87C3×#100—91no—
88#836—92no—
89#833—31no—
90#833—31no—
91#833—31no—
92#833—61no—
93#833—91no—
94#824—92no—
95#824—92no—
96C23×C32—22no—
97C23×C32C4×D312yes—
98C23×C32C2×Dic312yes—
99C22×C3⋊D3C2×Dic311yescharacteristic
100C22×C3×D3—31no—
101C32×D4—22no—
102C32×D4—22no—
103C32×D4—42no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107#638—24no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3C2×Dic312yes—
111C2×C3⋊Dic3C2×Dic311yescharacteristic
112C4×C3⋊D3#6712yes—
113C4×C3⋊D3#6712yes—
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—
123C3⋊(C3⋊C8)—22no—
124C3×C3⋊C8—62no—
125C3×C3⋊C8—62no—
126C3×C3⋊C8—62no—
127C3×C3⋊C8—122no—
128#413—271noSylow
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×D4D3211yescharacteristic
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—92no—
156C3×M16—92no—
157#208—32no—
158#208—32no—
159#208—32no—
160#208—62no—
161#208—92no—
162#208—182no—
163C2×C3⋊C8—182no—
164ΓL(1,𝔽25)—182no—
165ΓL(1,𝔽25)—182no—
166D3×C8—182no—
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×Dic311yescharacteristic
177C2×C3⋊D3—41no—
178C2×C3⋊D3#21712yes—
179C2×C3×D3—32no—
180C2×C3×D3—121no—
181C32×C4—22no—
182C32×C4D3×D412yes—
183C32×C4C2×#6712yes—
184C3⋊Dic3—22no—
185C3⋊Dic3#21711yescharacteristic
186C3⋊Dic3#21711yescharacteristic
187C3×Dic3—31no—
188C3×Dic3—31no—
189C3×Dic3—62no—
190C2×#41—91no—
191C2×M16—272no—
192#101—272no—
193#100—91no—
194#100—91no—
195H(3,𝔽3)#41311yescharacteristic, complemented, Sylow
196C23×C3—32no—
197C23×C3—32no—
198C23×C3—62no—
199C23×C3—91no—
200C23×C3D3×Dic312yes—
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×C4C2×D3211yescharacteristic
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—182no—
236C3×C8—182no—
237C3⋊C8—62no—
238C3⋊C8—62no—
239C3⋊C8—62no—
240C3⋊C8—122no—
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×#21711yescharacteristic
251C3⋊D3—22no—
252C3×D3—62no—
253#41—182no—
254#41—182no—
255C2×Q8—91no—
256C2×D4—92no—
257C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C22×C4—91no—
259C22×C4—92no—
260M16—92no—
261M16—92no—
262M16—542no—
263C2×C8—542no—
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×D3×Dic311yescharacteristic
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×C4D3×#6712yes—
289Dic3—31no—
290Dic3—31no—
291Dic3—62no—
292Dic3—91no—
293Dic3—91no—
294Dic3—182no—
295C32—21no—
296C32#474511yescharacteristic, complemented
297C32#523711yescharacteristic, complemented
298C23—91no—
299C23((C3⋊D3)⋊C3)⋊C412yes—
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×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
310C8—182no—
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×C3D3×#21711yescharacteristic
325D3—62no—
326D3—182no—
327C22—22no—
328C22—42no—
329C22—92no—
330C22—361no—
331C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
332C4—91no—
333C4—91no—
334C4—182no—
335C4#8867612yes—
336C3—31no—
337C3—31no—
338C3—61no—
339C3#1059150111yescharacteristic
340C2—21no—
341C2—42no—
342C2—182no—
343C2#1172131011yescharacteristic
subgroupG/H
centerC2#11721310
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC2×C3D3×#217
prime radicalC2×#3830⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#413
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721053C2semidirect
3. #11720579C2semidirect
4. #63660⋊C3C2semidirect
5. #11720321C2semidirect
23. #63660D3semidirect
195. H(3,𝔽3)#413semidirect
296. C32#4745semidirect
297. C32#5237semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
#117210532
#117205792
#63660⋊C32
#117203212
#105915043
#105915013