Finite Group Databasebeta

← #13140925

Subgroups — #13140925

Order 1728.

549 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 101 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×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721053C212yesmaximal
3#11720584C211yesmaximal, characteristic, complemented
4#63663⋊C3C211yesmaximal, characteristic, complemented
5#11720321C212yesmaximal, complemented
6#10591639—31nomaximal
7D3×#836—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9Q8×(C3⋊D3)⋊C3—22no—
10Q8×(C3⋊D3)⋊C3—22no—
11C2×C4×(C3⋊D3)⋊C3C412yes—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#634⋊C3C412yes—
14C2×#634⋊C3C2211yescharacteristic
15#88538C2212yes—
16#3748⋊C3C2212yes—
17#88430—22no—
18((C3⋊D3)⋊C3)⋊C8—22no—
19C2×Q8×C3⋊D3D311yescharacteristic
20C2×C3×D3×Q8—31no—
21#63818—32no—
22D3×#208—32no—
23#63663D311yescharacteristic, complemented
24C3×#836—31no—
25C3×#836—31no—
26C3×#836—31no—
27C3×#836—61no—
28#63561—32no—
29#63560—32no—
30C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
31Q8×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)C2×C412yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊Dic3)⋊C3C2×C412yes—
35C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
36C4×(C3⋊D3)⋊C3—22no—
37C4×(C3⋊D3)⋊C3—22no—
38C4×(C3⋊D3)⋊C3D412yes—
39C4×(C3⋊D3)⋊C3D412yes—
40#634⋊C3—22no—
41#634⋊C3—24no—
42#6180—22no—
43(C3⋊(C3⋊C8))⋊C3—22no—
44C2×#836—91no—
45D3×#101—91no—
46D3×#101—91no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C32×Q8C2×D311yescharacteristic
50Q8×C3⋊D3—22no—
51Q8×C3⋊D3—22no—
52C2×C4×C3⋊D3Dic312yes—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634Dic312yes—
55C2×#634C2×D311yescharacteristic
56C3×D3×Q8—62no—
57C3×D3×Q8—62no—
58C2×C3×C4×D3—31no—
59C2×C3×C4×D3—32no—
60C2×C3×Dic6—32no—
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)C22×C411yescharacteristic
72C2×(C3⋊D3)⋊C3—22no—
73C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
74C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
75C4×H(3,𝔽3)—22no—
76C4×H(3,𝔽3)C2×D412yes—
77(C3⋊Dic3)⋊C3—22no—
78(C3⋊Dic3)⋊C3C22⋊C412yes—
79C22×C3×Q8—91no—
80C2×D3×Q8—31no—
81C2×D3×Q8—91no—
82C2×#208—92no—
83D3×M16—92no—
84D3×M16—92no—
85C3×#101—91no—
86C3×#101—91no—
87#836—31no—
88#836—31no—
89#836—31no—
90#836—61no—
91#836—91no—
92#836—92no—
93#824—92no—
94#824—92no—
95C22×C3⋊D3C2×Dic311yescharacteristic
96C22×C3×D3—31no—
97C32×Q8—22no—
98C32×Q8—22no—
99C32×Q8—42no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4C4×D312yes—
103C2×C32×C4C2×Dic312yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3C2×Dic312yes—
107C2×C3⋊Dic3C2×Dic311yescharacteristic
108C4×C3⋊D3—22no—
109C4×C3⋊D3—22no—
110C4×C3⋊D3#6712yes—
111C4×C3⋊D3#6712yes—
112#634—22no—
113#634—24no—
114C2×C3×Dic3—31no—
115C2×C3×Dic3—32no—
116C3×C4×D3—32no—
117C3×C4×D3—32no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120C3×Dic6—64no—
121C3×Dic6—62no—
122C3⋊(C3⋊C8)—22no—
123C3×C3⋊C8—62no—
124C3×C3⋊C8—62no—
125C3×C3⋊C8—62no—
126C3×C3⋊C8—122no—
127C2×#101—271noSylow
128C2×H(3,𝔽3)—21no—
129C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
130(C3⋊D3)⋊C3—21no—
131(C3⋊D3)⋊C3#10112yescomplemented
132C2×C3×Q8—31no—
133C2×C3×Q8—31no—
134C2×C3×Q8—61no—
135C2×C3×Q8—91no—
136C2×C3×Q8—92no—
137C2×C3×Q8—182no—
138C2×C3×Q8—182no—
139C2×C3×Q8D3211yescharacteristic
140C22×C3×C4—91no—
141C22×C3×C4—92no—
142D3×Q8—62no—
143D3×Q8—62no—
144D3×Q8—182no—
145D3×Q8—182no—
146C2×C4×D3—31no—
147C2×C4×D3—32no—
148C2×C4×D3—91no—
149C2×C4×D3—92no—
150C2×Dic6—32no—
151C2×Dic6—31no—
152C2×Dic6—91no—
153C2×Dic6—92no—
154C3×M16—92no—
155C3×M16—92no—
156#208—32no—
157#208—32no—
158#208—32no—
159#208—62no—
160#208—92no—
161#208—182no—
162C2×C3⋊C8—182no—
163ΓL(1,𝔽25)—182no—
164ΓL(1,𝔽25)—182no—
165D3×C8—182no—
166D3×C8—182no—
167C22×C32—21no—
168C22×C32C2×C4×D311yescharacteristic
169C22×C32C22×Dic311yescharacteristic
170C2×C3⋊D3—22no—
171C2×C3⋊D3#21711yescharacteristic
172C2×C3⋊D3#21711yescharacteristic
173C2×C3×D3—31no—
174C2×C3×D3—31no—
175C2×C3×D3—62no—
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—42no—
180C32×C4D3×D412yes—
181C32×C4C2×#6712yes—
182C3⋊Dic3—22no—
183C3⋊Dic3#21712yes—
184C3×Dic3—32no—
185C3×Dic3—62no—
186C22×Q8—91no—
187C2×M16—272no—
188#101—91no—
189#101—91no—
190#101—272no—
191H(3,𝔽3)C2×#10111yescharacteristic, complemented, Sylow
192C23×C3—91no—
193C22×D3—31no—
194C22×D3—91no—
195C3×Q8—22no—
196C3×Q8—62no—
197C3×Q8—62no—
198C3×Q8—122no—
199C3×Q8—182no—
200C3×Q8—184no—
201C2×C3×C4—32no—
202C2×C3×C4—32no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—62no—
206C2×C3×C4—61no—
207C2×C3×C4—92no—
208C2×C3×C4—92no—
209C2×C3×C4—91no—
210C2×C3×C4—92no—
211C2×C3×C4—182no—
212C2×C3×C4—182no—
213C2×C3×C4D3×Dic312yes—
214C2×C3×C4C2×D3211yescharacteristic
215C2×Dic3—31no—
216C2×Dic3—32no—
217C2×Dic3—91no—
218C2×Dic3—92no—
219C4×D3—32no—
220C4×D3—32no—
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—92no—
224C4×D3—92no—
225C4×D3—182no—
226C4×D3—182no—
227Dic6—64no—
228Dic6—62no—
229Dic6—184no—
230Dic6—182no—
231C3×C8—182no—
232C3×C8—182no—
233C3⋊C8—62no—
234C3⋊C8—62no—
235C3⋊C8—62no—
236C3⋊C8—122no—
237C3⋊C8—182no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—41no—
242C2×C32D3×C22⋊C411yescharacteristic
243C2×C32C2×#21711yescharacteristic
244C3⋊D3—21no—
245C3⋊D3#83612yescomplemented
246C3×D3—32no—
247C3×D3—61no—
248C2×Q8—92no—
249C2×Q8—91no—
250C2×Q8—182no—
251C2×Q8—182no—
252C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
253C22×C4—91no—
254C22×C4—92no—
255M16—92no—
256M16—92no—
257M16—542no—
258C2×C8—542no—
259C22×C3—31no—
260C22×C3—31no—
261C22×C3—61no—
262C22×C3—91no—
263C22×C3—91no—
264C22×C3—182no—
265C22×C3C2×D3×Dic311yescharacteristic
266C2×D3—31no—
267C2×D3—31no—
268C2×D3—62no—
269C2×D3—91no—
270C2×D3—91no—
271C2×D3—182no—
272C3×C4—22no—
273C3×C4—32no—
274C3×C4—32no—
275C3×C4—62no—
276C3×C4—62no—
277C3×C4—62no—
278C3×C4—92no—
279C3×C4—122no—
280C3×C4—182no—
281C3×C4D3×#6712yes—
282Dic3—32no—
283Dic3—62no—
284Dic3—92no—
285Dic3—182no—
286C32—21no—
287C32D3×#10111yescharacteristic, complemented
288C32C2×#83611yescharacteristic, complemented
289C23—91no—
290Q8—22no—
291Q8—184no—
292Q8—182no—
293C2×C4—91no—
294C2×C4—92no—
295C2×C4—92no—
296C2×C4—92no—
297C2×C4—182no—
298C2×C4—182no—
299C2×C4((C3⋊D3)⋊C3)⋊C412yes—
300C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
301C8—182no—
302C8—182no—
303C2×C3—21no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—92no—
310C2×C3—121no—
311C2×C3—181no—
312C2×C3D3×#21711yescharacteristic
313D3—32no—
314D3—61no—
315D3—92no—
316D3—181no—
317C22—91no—
318C22—91no—
319C22—182no—
320C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
321C4—22no—
322C4—92no—
323C4—182no—
324C4#8867612yes—
325C3—31no—
326C3—31no—
327C3—61no—
328C3D3×#83611yescharacteristic
329C2—21no—
330C2—92no—
331C2—181no—
332C2#1172131011yescharacteristic
subgroupG/H
centerC2#11721310
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC2×C3D3×#217
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#101
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. #11720584C2semidirect
4. #63663⋊C3C2semidirect
5. #11720321C2semidirect
23. #63663D3semidirect
131. (C3⋊D3)⋊C3#101semidirect
191. H(3,𝔽3)C2×#101semidirect
245. C3⋊D3#836semidirect
287. C32D3×#101semidirect
288. C32C2×#836semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
#117210532
#117205842
#63663⋊C32
#117203212
#105916393
D3×#8363