Finite Group Databasebeta

← #13140044

Subgroups — #13140044

Order 1728.

568 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#634⋊C3C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3C2×#88433C212yesmaximal, complemented
4#11720573C211yesmaximal, characteristic
5#11720313C212yesmaximal, complemented
6#10591190—31nomaximal
7#10591186—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3—21no—
10C2×#634⋊C3—21no—
11C2×#634⋊C3—24no—
12C2×#634⋊C3C2211yescharacteristic
13C2×#634⋊C3C2212yes—
14C2×#6180C2212yes—
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18#88433—24no—
19C22×#634D311yescharacteristic
20C22×C3×Dic6—31no—
21#64133D311yescharacteristic
22C3×#924—31no—
23C2×#3719—32no—
24C2×#3719—32no—
25C3×#830—31no—
26C3×#830—31no—
27C3×#830—61no—
28#63567—32no—
29#63566—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊Dic3)⋊C3—24no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C2×(C3⋊Dic3)⋊C3—21no—
38#634⋊C3—24no—
39#634⋊C3—22no—
40#634⋊C3—22no—
41#634⋊C3D414yes—
42#6180—22no—
43#5228—91no—
44#5050—91no—
45#4739—91no—
46C22×C32×C4—21no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C3⋊Dic3—21no—
50C2×#634—21no—
51C2×#634—24no—
52C2×#634C2×D311yescharacteristic
53C2×#634C2×D312yes—
54C22×C3×Dic3—61no—
55C2×C3×Dic6—32no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—64no—
58C2×C3×Dic6—61no—
59#3757—21no—
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61#3750—21no—
62C3×#217—61no—
63C3×C4⋊Dic3—31no—
64C3×Dic3⋊C4—61no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—62no—
68#3719—64no—
69#3719—64no—
70C22×H(3,𝔽3)—21no—
71C22×H(3,𝔽3)—21no—
72C22×H(3,𝔽3)Q1612yescomplemented
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)—21no—
75C4×H(3,𝔽3)C2×D412yes—
76(C3⋊Dic3)⋊C3—24no—
77(C3⋊Dic3)⋊C3—41no—
78(C3⋊Dic3)⋊C3—41no—
79C22×C3×Q8—91no—
80C22×Dic6—31no—
81C22×Dic6—91no—
82C3×C22⋊Q8—91no—
83C2×#216—92no—
84C2×#216—92no—
85#924—31no—
86#924—91no—
87C2×Dic12—92no—
88C3×C22⋊C8—91no—
89#835—92no—
90#830—31no—
91#830—31no—
92#830—61no—
93#816—92no—
94#810—92no—
95C23×C32—21no—
96C23×C32#6711yescharacteristic
97C23×C32#6711yescharacteristic
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—42no—
103C2×C32×C4#6711yescharacteristic
104C2×C32×C4#6711yescharacteristic
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—24no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—21no—
111#634—24no—
112#634—22no—
113#634—22no—
114#634#6714yes—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C3×Dic6—34no—
120C3×Dic6—64no—
121C3×Dic6—62no—
122C3×Dic6—62no—
123C3×C3⋊C8—62no—
124C3×C3⋊C8—62no—
125C3×C3⋊C8—122no—
126#451—271noSylow
127C2×H(3,𝔽3)—22no—
128C2×H(3,𝔽3)#12011yescharacteristic
129C2×H(3,𝔽3)C2×Q1611yescharacteristic
130C2×H(3,𝔽3)#13711yescharacteristic
131C2×C3×Q8—92no—
132C2×C3×Q8—91no—
133C2×C3×Q8—181no—
134C2×C3×Q8—184no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—61no—
138C22×C3×C4—181no—
139C22×C3×C4D3211yescharacteristic
140C22×Dic3—61no—
141C22×Dic3—181no—
142C2×Dic6—32no—
143C2×Dic6—31no—
144C2×Dic6—64no—
145C2×Dic6—61no—
146C2×Dic6—92no—
147C2×Dic6—91no—
148C2×Dic6—184no—
149C2×Dic6—181no—
150C2×C3×C8—92no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C22⋊C4—181no—
154#217—61no—
155#217—181no—
156#216—184no—
157#216—184no—
158C4⋊Dic3—31no—
159C4⋊Dic3—91no—
160Dic3⋊C4—61no—
161Dic3⋊C4—181no—
162C2×C3⋊C8—32no—
163C2×C3⋊C8—32no—
164C2×C3⋊C8—62no—
165Dic12—184no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—21no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32#21612yescomplemented
175C22×C32#21612yescomplemented
176C22×C32C2×#6711yescharacteristic
177C22×C32C2×#6711yescharacteristic
178C32×C4—21no—
179C32×C4—21no—
180C32×C4—22no—
181C32×C4—41no—
182C32×C4D3×D412yes—
183C32×C4C2×#6712yes—
184C3⋊Dic3—24no—
185C3⋊Dic3—41no—
186C3⋊Dic3—41no—
187C3×Dic3—64no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C22×Q8—91no—
191C2×Q16—272no—
192C22⋊Q8—91no—
193#103—272no—
194C22⋊C8—91no—
195H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
196C23×C3—31no—
197C23×C3—31no—
198C23×C3—61no—
199C23×C3#62511yescharacteristic
200C3×Q8—94no—
201C3×Q8—182no—
202C3×Q8—182no—
203C3×Q8—184no—
204C2×C3×C4—22no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4—122no—
214C2×C3×C4—182no—
215C2×C3×C4—184no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4#62511yescharacteristic
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—64no—
221C2×Dic3—62no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—182no—
225C2×Dic3—184no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228Dic6—34no—
229Dic6—64no—
230Dic6—62no—
231Dic6—62no—
232Dic6—94no—
233Dic6—182no—
234Dic6—184no—
235Dic6—182no—
236C3×C8—182no—
237C3⋊C8—62no—
238C3⋊C8—62no—
239C3⋊C8—122no—
240C2×C32—22no—
241C2×C32—22no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—42no—
246C2×C32#93711yescharacteristic
247C2×C32#94211yescharacteristic
248C2×C32C2×#21611yescharacteristic
249C2×C32C2×#21611yescharacteristic
250C2×C32#95111yescharacteristic
251C2×C32#95311yescharacteristic
252C2×Q8—92no—
253C2×Q8—91no—
254C2×Q8—184no—
255C2×Q8—181no—
256C22×C4—181no—
257C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
258Q16—544no—
259C2×C8—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#371812yes—
278C22×C3C2×#62511yescharacteristic
279C3×C4—21no—
280C3×C4—32no—
281C3×C4—32no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—62no—
285C3×C4—121no—
286C3×C4—184no—
287C3×C4—361no—
288C3×C4—361no—
289C3×C4D3×#6712yes—
290Dic3—64no—
291Dic3—121no—
292Dic3—121no—
293Dic3—184no—
294Dic3—361no—
295Dic3—361no—
296C32—21no—
297C32#505011yescharacteristic, complemented
298C32#522811yescharacteristic, complemented
299C23#619811yescharacteristic
300Q8—94no—
301Q8—184no—
302Q8—182no—
303Q8—182no—
304C2×C4—22no—
305C2×C4—184no—
306C2×C4—182no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4#619811yescharacteristic
310C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C8—182no—
312C2×C3—22no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—62no—
320C2×C3—62no—
321C2×C3—61no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—122no—
325C2×C3#6382711yescharacteristic
326C2×C3C2×#371811yescharacteristic
327C2×C3#6397611yescharacteristic
328C22—21no—
329C22—21no—
330C22#8843312yescomplemented
331C22C2×#619811yescharacteristic
332C4—21no—
333C4—184no—
334C4—361no—
335C4—361no—
336C4#8867612yes—
337C3—31no—
338C3—31no—
339C3—61no—
340C3#1059118511yescharacteristic
341C2—22no—
342C2#1172105811yescharacteristic
343C2C2×#8843311yescharacteristic
344C2#1172131611yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#451
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
3. C2×#88433C2semidirect
5. #11720313C2semidirect
72. C22×H(3,𝔽3)Q16semidirect
174. C22×C32#216semidirect
175. C22×C32#216semidirect
195. H(3,𝔽3)#451semidirect
297. C32#5050semidirect
298. C32#5228semidirect
330. C22#88433semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
#64133⋊C32
C2×#884332
#117205732
#117203132
#105911903
#105911863