Finite Group Databasebeta

← #10592452⋊C3 #13141699

Subgroups — #10592452⋊C3 #13141699

Order 1728.

549 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#11722627C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic, complemented
3#64117⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×#101C211yesmaximal, characteristic, complemented
5#63652⋊C3C212yesmaximal, complemented
6#10592452C311yesmaximal, characteristic, complemented
7C3×#4753—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9#88722—22no—
10Q8×(C3⋊D3)⋊C3—22no—
11#3828⋊C3—22no—
12#3828⋊C3C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14M16×H(3,𝔽3)C2212yes—
15#3752⋊C3—22no—
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#3745⋊C3—22no—
18#3744⋊C3—22no—
19#64364C2×C311yescharacteristic, complemented
20C3×#1007—31no—
21#64156C2×C311yescharacteristic, complemented
22#64117C2×C312yescomplemented
23C3×#947—31no—
24C3×#908—32no—
25C32×#101—21no—
26C32×#101D311yescharacteristic, complemented
27C32×#101C2×C311yescharacteristic, complemented
28#63652C2×C312yescomplemented
29C3×#825—32no—
30C22×(C3⋊D3)⋊C3—21no—
31Q8×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)D412yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34#638⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—21no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4⋊((C3⋊D3)⋊C3)D412yes—
39C4×(C3⋊D3)⋊C3—24no—
40C4×(C3⋊D3)⋊C3—22no—
41#634⋊C3—22no—
42#634⋊C3D412yes—
43C8×H(3,𝔽3)—22no—
44C3×#454—91no—
45#4753—31no—
46#4753—91no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C32×Q8C22×C311yescharacteristic
50#3832—22no—
51Q8×C3⋊D3—22no—
52#3828—22no—
53#3828C22×C312yes—
54C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
55C3×#239—62no—
56C3×D3×Q8—62no—
57C3×#235—32no—
58C3×#235—62no—
59C2×C3×D12—31no—
60C32×M16—22no—
61C32×M16C2×D312yes—
62C32×M16C22×C312yes—
63#3752—22no—
64C4×C3⋊Dic3C22×C311yescharacteristic
65#3745—22no—
66#3744—22no—
67C3×(C2×D3)⋊C4—62no—
68C3×C4×Dic3—31no—
69C3×D24—62no—
70C3×#204—62no—
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C2×(C3⋊D3)⋊C3—22no—
73C2×(C3⋊D3)⋊C3—42no—
74C2×(C3⋊D3)⋊C3—41no—
75C4×H(3,𝔽3)—22no—
76C4×H(3,𝔽3)C2×D412yes—
77(C3⋊Dic3)⋊C3—22no—
78(C3⋊Dic3)⋊C3—41no—
79C3×2−1+4—91no—
80#1007—31no—
81#1007—91no—
82C3×AGL(1,ℤ/8)—92no—
83C3×#124—91no—
84#947—31no—
85#947—91no—
86#908—32no—
87#908—92no—
88C3×#104—92no—
89C3×#101—31no—
90C3×#101—31no—
91C3×#101—61no—
92C3×#101C3×D311yescharacteristic
93#825—32no—
94#825—92no—
95C22×C3⋊D3—21no—
96C22×C3×D3—61no—
97C32×Q8—22no—
98C32×Q8—22no—
99C32×Q8—42no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4D1212yes—
103C2×C32×C4C3×D412yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C23×C311yescharacteristic
106#638—22no—
107C2×C3⋊Dic3—21no—
108C4⋊(C3⋊D3)—22no—
109C4⋊(C3⋊D3)—22no—
110C4⋊(C3⋊D3)C3×D412yes—
111C4×C3⋊D3—24no—
112C4×C3⋊D3—22no—
113#634—22no—
114#634C3×D412yes—
115C3×#67—62no—
116C2×C3×Dic3—61no—
117C3×D12—32no—
118C3×D12—62no—
119C3×D12—62no—
120C3×C4×D3—64no—
121C3×C4×D3—62no—
122C3×Dic6—32no—
123C3×Dic6—62no—
124C32×C8—22no—
125C32×C8—22no—
126C32×C8—42no—
127#454—91noSylow
128C2×H(3,𝔽3)—21no—
129C2×H(3,𝔽3)#12011yescharacteristic
130(C3⋊D3)⋊C3—42no—
131(C3⋊D3)⋊C3—81no—
132C3×#41—92no—
133C3×#41—182no—
134C3×#41—182no—
135C2×C3×Q8—31no—
136C2×C3×Q8—31no—
137C2×C3×Q8—61no—
138C2×C3×Q8—182no—
139C2×C3×Q8C2×C3×D311yescharacteristic
140C2×C3×D4—91no—
141#239—62no—
142#239—182no—
143D3×Q8—62no—
144D3×Q8—182no—
145#235—32no—
146#235—62no—
147#235—92no—
148#235—182no—
149C2×D12—31no—
150C2×D12—91no—
151C3×SD16—182no—
152C3×D8—182no—
153C3×M16—32no—
154C3×M16—32no—
155C3×M16—62no—
156C3×M16C2×C3×D312yes—
157C3×C22⋊C4—182no—
158C3×C42—91no—
159(C2×D3)⋊C4—62no—
160(C2×D3)⋊C4—182no—
161C4×Dic3—31no—
162C4×Dic3—91no—
163D24—62no—
164D24—182no—
165#204—62no—
166#204—182no—
167C22×C32—21no—
168C22×C32C2×D1211yescharacteristic
169C22×C32C2×C3×D411yescharacteristic
170C2×C3⋊D3—22no—
171C2×C3⋊D3—42no—
172C2×C3⋊D3—41no—
173C2×C3×D3—62no—
174C2×C3×D3—122no—
175C2×C3×D3—121no—
176C32×C4—22no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—42no—
180C32×C4D3×D412yes—
181C32×C4C2×C3×D412yes—
182C3⋊Dic3—22no—
183C3⋊Dic3—41no—
184C3×Dic3—62no—
185C3×Dic3—121no—
1862−1+4—91no—
187AGL(1,ℤ/8)—92no—
188#124—91no—
189#104—92no—
190#101(C3⋊D3)⋊C311yescharacteristic, complemented
191H(3,𝔽3)#45411yescharacteristic, complemented, Sylow
192C23×C3—181no—
193C22×D3—61no—
194C22×D3—181no—
195C3×Q8—22no—
196C3×Q8—62no—
197C3×Q8—62no—
198C3×Q8—92no—
199C3×Q8—122no—
200C3×Q8—182no—
201C3×D4—92no—
202C3×D4—182no—
203C3×D4—182no—
204C3×D4—182no—
205C2×C3×C4—32no—
206C2×C3×C4—32no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—62no—
210C2×C3×C4—61no—
211C2×C3×C4—182no—
212C2×C3×C4—184no—
213C2×C3×C4—181no—
214C2×C3×C4C3×D1212yes—
215C2×C3×C4C22×C3×D311yescharacteristic
216#67—62no—
217#67—182no—
218C2×Dic3—61no—
219C2×Dic3—181no—
220D12—32no—
221D12—62no—
222D12—62no—
223D12—92no—
224D12—182no—
225D12—182no—
226C4×D3—64no—
227C4×D3—62no—
228C4×D3—182no—
229C4×D3—184no—
230Dic6—32no—
231Dic6—62no—
232Dic6—92no—
233Dic6—182no—
234C3×C8—22no—
235C3×C8—62no—
236C3×C8—62no—
237C3×C8—122no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—41no—
242C2×C32#88211yescharacteristic
243C2×C32C3×#12011yescharacteristic
244C3⋊D3—42no—
245C3⋊D3—81no—
246C3×D3—122no—
247C3×D3—241no—
248#41—92no—
249#41—182no—
250#41—182no—
251C2×Q8—182no—
252C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
253C2×D4—91no—
254SD16—182no—
255D8—182no—
256M16C2×(C3⋊D3)⋊C312yes—
257C22⋊C4—182no—
258C42—91no—
259C22×C3—31no—
260C22×C3—31no—
261C22×C3—61no—
262C22×C3—182no—
263C22×C3—361no—
264C22×C3—362no—
265C22×C3C2×C3×D1211yescharacteristic
266C2×D3—62no—
267C2×D3—122no—
268C2×D3—121no—
269C2×D3—182no—
270C2×D3—361no—
271C2×D3—362no—
272C3×C4—22no—
273C3×C4—32no—
274C3×C4—32no—
275C3×C4—62no—
276C3×C4—62no—
277C3×C4—62no—
278C3×C4—122no—
279C3×C4—182no—
280C3×C4—361no—
281C3×C4C3×D3×D412yes—
282Dic3—62no—
283Dic3—121no—
284Dic3—182no—
285Dic3—361no—
286C32—21no—
287C32#475311yescharacteristic, complemented
288C32C3×#45411yescharacteristic, complemented
289C23—181no—
290Q8—22no—
291Q8—92no—
292Q8—182no—
293D4—92no—
294D4—182no—
295D4—182no—
296D4—182no—
297C2×C4—184no—
298C2×C4—182no—
299C2×C4—181no—
300C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
301C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
302C8—22no—
303C2×C3—21no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—121no—
310C2×C3—362no—
311C2×C3—721no—
312C2×C3C3×#88211yescharacteristic
313D3—122no—
314D3—241no—
315D3—362no—
316D3—721no—
317C22—182no—
318C22—362no—
319C22—361no—
320C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
321C4—22no—
322C4—182no—
323C4—361no—
324C4D4×(C3⋊D3)⋊C312yes—
325C3—31no—
326C3—31no—
327C3—61no—
328C3C3×#475311yescharacteristic
329C2—21no—
330C2—362no—
331C2—721no—
332C2#64091⋊C311yescharacteristic
subgroupG/H
centerC2#64091⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#101C2
socleC2×C3C3×#882
hypercenter#101(C3⋊D3)⋊C3
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #11722627C2semidirect
2. #64156⋊C3C2semidirect
3. #64117⋊C3C2semidirect
4. H(3,𝔽3)×#101C2semidirect
5. #63652⋊C3C2semidirect
6. #10592452C3semidirect
19. #64364C2×C3semidirect
21. #64156C2×C3semidirect
22. #64117C2×C3semidirect
26. C32×#101D3semidirect
27. C32×#101C2×C3semidirect
28. #63652C2×C3semidirect
190. #101(C3⋊D3)⋊C3semidirect
191. H(3,𝔽3)#454semidirect
287. C32#4753semidirect
288. C32C3×#454semidirect

Maximal subgroups

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

subgroupindex
#117226272
#64156⋊C32
#64117⋊C32
H(3,𝔽3)×#1012
#63652⋊C32
#105924523
C3×#47533