Finite Group Databasebeta

← #10592841⋊C3 #13143259

Subgroups — #10592841⋊C3 #13143259

Order 1728.

440 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×#131C211yesmaximal, characteristic, complemented
2#64158⋊C3C212yesmaximal, complemented
3C2×#3745⋊C3C211yesmaximal, characteristic, complemented
4#63650⋊C3C212yesmaximal, complemented
5#63646⋊C3C211yesmaximal, characteristic, complemented
6#10592841C311yesmaximal, characteristic, complemented
7C3×#5142—31nomaximal
8#41×H(3,𝔽3)C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10M16×H(3,𝔽3)—21no—
11M16×H(3,𝔽3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#3755⋊C3—22no—
15#3753⋊C3—22no—
16#3748⋊C3C2212yes—
17#3745⋊C3—21no—
18#3745⋊C3—21no—
19C32×#131—21no—
20C32×#131D311yescharacteristic, complemented
21C32×#131C2×C311yescharacteristic, complemented
22#64158C2×C312yescomplemented
23C2×#3745C2×C311yescharacteristic, complemented
24C3×#949—32no—
25C2×C3×D24—31no—
26#63650C2×C312yescomplemented
27#63646C2×C311yescharacteristic, complemented
28C3×#823—32no—
29C3×#819—31no—
30C22×(C3⋊D3)⋊C3—22no—
31Q8×H(3,𝔽3)D411yescharacteristic
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C4⋊((C3⋊D3)⋊C3)—22no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C8×H(3,𝔽3)—21no—
39C8×H(3,𝔽3)D411yescharacteristic
40C8×H(3,𝔽3)D411yescharacteristic
41(C3⋊(C3⋊C8))⋊C3—22no—
42C3×#472—91no—
43#5142—31no—
44#5142—91no—
45C32×#41—21no—
46C32×#41C2×D311yescharacteristic
47C32×#41C22×C311yescharacteristic
48C2×C4⋊(C3⋊D3)C22×C312yes—
49C2×C3×D12—32no—
50C32×M16—22no—
51C32×M16—21no—
52C32×M16—21no—
53C32×M16—21no—
54C32×M16C2×D311yescharacteristic
55C32×M16C22×C311yescharacteristic
56C2×C32×C8—22no—
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C2×C32×C8C2×D311yescharacteristic
61C2×C32×C8C22×C311yescharacteristic
62#3755—22no—
63#3753—22no—
64#3748C22×C312yes—
65#3745—21no—
66#3745—21no—
67C3×#215—62no—
68C3×#213—62no—
69C3×#208—32no—
70C3×D24—61no—
71C3×D24—61no—
72C22×H(3,𝔽3)—21no—
73C22×H(3,𝔽3)#4111yescharacteristic
74C2×(C3⋊D3)⋊C3—42no—
75C2×(C3⋊D3)⋊C3—82no—
76C4×H(3,𝔽3)—21no—
77C4×H(3,𝔽3)C2×D411yescharacteristic
78C4×H(3,𝔽3)C2×D411yescharacteristic
79C3×AGL(1,ℤ/8)—92no—
80C2×C3×D8—91no—
81C3×#131—31no—
82C3×#131—31no—
83C3×#131—61no—
84C3×#131C3×D311yescharacteristic
85#949—32no—
86#949—92no—
87C2×D24—31no—
88C2×D24—91no—
89C3×#108—91no—
90C3×#100—92no—
91#823—32no—
92#823—92no—
93#819—31no—
94#819—91no—
95C22×C3⋊D3—22no—
96C22×C3×D3—62no—
97C32×Q8—21no—
98C32×Q8D1211yescharacteristic
99C32×Q8C3×D411yescharacteristic
100C32×D4—22no—
101C32×D4—21no—
102C32×D4—21no—
103C32×D4—21no—
104C32×D4D1211yescharacteristic
105C32×D4C3×D411yescharacteristic
106C2×C32×C4—22no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C23×C311yescharacteristic
112C4⋊(C3⋊D3)—22no—
113C4⋊(C3⋊D3)—22no—
114C3×D12—62no—
115C3×D12—62no—
116C32×C8—22no—
117C32×C8—21no—
118C32×C8—21no—
119C32×C8—21no—
120C32×C8—21no—
121C32×C8#6711yescharacteristic
122C32×C8#6711yescharacteristic
123C32×C8C3×D411yescharacteristic
124C32×C8C3×D411yescharacteristic
125C3⋊(C3⋊C8)—22no—
126C3×C3⋊C8—62no—
127#472—91noSylow
128C2×H(3,𝔽3)—21no—
129C2×H(3,𝔽3)—41no—
130C2×H(3,𝔽3)#12111yescharacteristic
131(C3⋊D3)⋊C3—82no—
132C3×#41—31no—
133C3×#41—31no—
134C3×#41—61no—
135C3×#41C2×C3×D311yescharacteristic
136C2×C3×D4—92no—
137C2×D12—32no—
138C2×D12—92no—
139C3×SD16—182no—
140C3×D8—181no—
141C3×D8—181no—
142C3×D8—182no—
143C3×M16—21no—
144C3×M16—31no—
145C3×M16—31no—
146C3×M16—61no—
147C3×M16—61no—
148C3×M16—62no—
149C3×M16—61no—
150C3×M16—92no—
151C3×M16C2×C3×D311yescharacteristic
152C2×C3×C8—21no—
153C2×C3×C8—31no—
154C2×C3×C8—31no—
155C2×C3×C8—61no—
156C2×C3×C8—61no—
157C2×C3×C8—62no—
158C2×C3×C8—61no—
159C2×C3×C8C2×C3×D311yescharacteristic
160#215—62no—
161#215—182no—
162#213—62no—
163#213—182no—
164#208—32no—
165#208—92no—
166D24—61no—
167D24—61no—
168D24—181no—
169D24—181no—
170C22×C32—22no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32#23511yescharacteristic
175C22×C32C3×#4111yescharacteristic
176C2×C3⋊D3—42no—
177C2×C3⋊D3—82no—
178C2×C3×D3—122no—
179C2×C3×D3—242no—
180C32×C4—22no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—21no—
185C32×C4C2×D1211yescharacteristic
186C32×C4C2×#6711yescharacteristic
187C32×C4C2×C3×D411yescharacteristic
188C32×C4C2×C3×D411yescharacteristic
189AGL(1,ℤ/8)—92no—
190C2×D8—91no—
191#131(C3⋊D3)⋊C311yescharacteristic, complemented
192#108—91no—
193#100—92no—
194H(3,𝔽3)#47211yescharacteristic, complemented, Sylow
195C23×C3—182no—
196C22×D3—62no—
197C22×D3—182no—
198C3×Q8—31no—
199C3×Q8—31no—
200C3×Q8—61no—
201C3×Q8C3×D1211yescharacteristic
202C3×D4—21no—
203C3×D4—31no—
204C3×D4—31no—
205C3×D4—61no—
206C3×D4—61no—
207C3×D4—62no—
208C3×D4—61no—
209C3×D4—182no—
210C3×D4—182no—
211C3×D4C3×D1211yescharacteristic
212C2×C3×C4—21no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—62no—
218C2×C3×C4—61no—
219C2×C3×C4C22×C3×D311yescharacteristic
220D12—62no—
221D12—62no—
222D12—182no—
223D12—182no—
224C3×C8—21no—
225C3×C8—31no—
226C3×C8—31no—
227C3×C8—31no—
228C3×C8—31no—
229C3×C8—61no—
230C3×C8—61no—
231C3×C8—62no—
232C3×C8—61no—
233C3×C8—61no—
234C3×C8—182no—
235C3×C8C3×#6711yescharacteristic
236C3×C8C3×#6711yescharacteristic
237C3⋊C8—62no—
238C3⋊C8—182no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—41no—
243C2×C32—42no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32#93011yescharacteristic
247C2×C32C3×#12111yescharacteristic
248C3⋊D3—82no—
249C3×D3—242no—
250#41C2×(C3⋊D3)⋊C311yescharacteristic
251C2×D4—92no—
252SD16—182no—
253D8—182no—
254D8—181no—
255D8—181no—
256M16—21no—
257M16—92no—
258M16C2×(C3⋊D3)⋊C311yescharacteristic
259C2×C8—21no—
260C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
261C22×C3—21no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—61no—
265C22×C3—61no—
266C22×C3—62no—
267C22×C3—61no—
268C22×C3—362no—
269C22×C3—722no—
270C22×C3C3×#23511yescharacteristic
271C2×D3—122no—
272C2×D3—242no—
273C2×D3—362no—
274C2×D3—722no—
275C3×C4—21no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—61no—
281C3×C4—61no—
282C3×C4—62no—
283C3×C4—61no—
284C3×C4—61no—
285C3×C4C2×C3×D1211yescharacteristic
286C3×C4C2×C3×#6711yescharacteristic
287C32—21no—
288C32#514211yescharacteristic, complemented
289C32C3×#47211yescharacteristic, complemented
290C23—182no—
291Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
292D4—21no—
293D4—182no—
294D4—182no—
295D4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
296C2×C4—21no—
297C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
298C8—21no—
299C8—182no—
300C8#638⋊C311yescharacteristic
301C8#638⋊C311yescharacteristic
302C2×C3—21no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—41no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—121no—
310C2×C3—121no—
311C2×C3—121no—
312C2×C3—122no—
313C2×C3—722no—
314C2×C3C3×#93011yescharacteristic
315D3—242no—
316D3—722no—
317C22—21no—
318C22—362no—
319C22—722no—
320C22#3828⋊C311yescharacteristic
321C4—21no—
322C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
323C4C2×#638⋊C311yescharacteristic
324C3—31no—
325C3—31no—
326C3—61no—
327C3C3×#514211yescharacteristic
328C2—21no—
329C2—41no—
330C2—722no—
331C2#64139⋊C311yescharacteristic
subgroupG/H
centerC2#64139⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#131C2
socleC2×C3C3×#930
hypercenter#131(C3⋊D3)⋊C3
hypocenterC32C3×#472
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#131C2semidirect
2. #64158⋊C3C2semidirect
3. C2×#3745⋊C3C2semidirect
4. #63650⋊C3C2semidirect
5. #63646⋊C3C2semidirect
6. #10592841C3semidirect
20. C32×#131D3semidirect
21. C32×#131C2×C3semidirect
22. #64158C2×C3semidirect
23. C2×#3745C2×C3semidirect
26. #63650C2×C3semidirect
27. #63646C2×C3semidirect
191. #131(C3⋊D3)⋊C3semidirect
194. H(3,𝔽3)#472semidirect
288. C32#5142semidirect
289. C32C3×#472semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1312
#64158⋊C32
C2×#3745⋊C32
#63650⋊C32
#63646⋊C32
#105928413
C3×#51423