Finite Group Databasebeta

← #10595997⋊C3 #13152520

Subgroups — #10595997⋊C3 #13152520

Order 1728.

1021 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
4#64106⋊C3C214yesmaximal, complemented
5#64095⋊C3C214yesmaximal, complemented
6#64084⋊C3C212yesmaximal, complemented
7#64078⋊C3C212yesmaximal, complemented
8#10595997C311yesmaximal, characteristic, complemented
9C3×#5526—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13#3828⋊C3—24no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×#634⋊C3—24no—
17C2×#634⋊C3C2212yes—
18C2×#634⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2212yes—
21C22⋊C4×H(3,𝔽3)C2212yes—
22C42×H(3,𝔽3)C2212yes—
23#3752⋊C3C2218yes—
24C4⋊((C3⋊Dic3)⋊C3)C2218yes—
25C2×#3828C2×C311yescharacteristic
26C22×#634C2×C311yescharacteristic, complemented
27C2×C3×#235—31no—
28C22×C3×Dic6—31no—
29C32×#117—21no—
30C32×#117D311yescharacteristic, complemented
31C32×#117C2×C311yescharacteristic, complemented
32#64106C2×C314yescomplemented
33#64095C2×C314yescomplemented
34#64084C2×C312yescomplemented
35#64078C2×C312yescomplemented
36C3×#897—34no—
37C3×#886—34no—
38C3×#875—32no—
39C3×#869—32no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×(C3⋊D3)⋊C3C2312yes—
42C2×C4×H(3,𝔽3)D414yes—
43C2×C4×H(3,𝔽3)C2314yes—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46#638⋊C3—24no—
47C2×(C3⋊Dic3)⋊C3—24no—
48C2×(C3⋊Dic3)⋊C3C2314yes—
49C2×(C3⋊Dic3)⋊C3C2312yes—
50C4⋊((C3⋊D3)⋊C3)—22no—
51C4×(C3⋊D3)⋊C3—24no—
52#634⋊C3—24no—
53#634⋊C3—24no—
54#634⋊C3—22no—
55C3×#536—91no—
56#5526—31no—
57#5526—91no—
58C22×C32×C4—21no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C22×C311yescharacteristic
61C2×#638C22×C312yes—
62C22×C3⋊Dic3C22×C312yes—
63#3828—24no—
64C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
65C2×C4×C3⋊D3C22×C312yes—
66C2×#634—24no—
67C2×#634C22×C311yescharacteristic
68C2×#634C22×C312yes—
69C2×#634C22×C312yes—
70C2×C3×#67—32no—
71C22×C3×Dic3—32no—
72C3×#235—64no—
73C2×C3×D12—31no—
74C2×C3×C4×D3—32no—
75C2×C3×Dic6—32no—
76C2×C3×Dic6—32no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—64no—
79C32×C4⋊C4—22no—
80C32×C4⋊C4C2×D312yes—
81C32×C4⋊C4C22×C312yes—
82C32×C22⋊C4—22no—
83C32×C22⋊C4C2×D312yes—
84C32×C22⋊C4C22×C312yes—
85C32×C42—22no—
86C32×C42C2×D312yes—
87C32×C42C22×C312yes—
88#3752C22×C318yes—
89C4⋊(C3⋊Dic3)C22×C318yes—
90C3×(C2×D3)⋊C4—38no—
91C3×C4⋊Dic3—38no—
92C22×H(3,𝔽3)—22no—
93C22×H(3,𝔽3)C2×D412yes—
94C22×H(3,𝔽3)C2411yescharacteristic
95C2×(C3⋊D3)⋊C3—22no—
96C2×(C3⋊D3)⋊C3—24no—
97C4×H(3,𝔽3)—24no—
98C4×H(3,𝔽3)C2×D412yes—
99C4×H(3,𝔽3)C2×D412yes—
100(C3⋊Dic3)⋊C3—24no—
101(C3⋊Dic3)⋊C3—22no—
102C2×C3×#41—91no—
103C22×C3×Q8—91no—
104C2×#235—31no—
105C2×#235—91no—
106C22×Dic6—31no—
107C22×Dic6—91no—
108C3×C4⋊Q8—92no—
109C3×#124—92no—
110C3×#123—94no—
111C3×C22⋊Q8—94no—
112C3×#117—31no—
113C3×#117—31no—
114C3×#117—61no—
115C3×#117C3×D311yescharacteristic
116#897—34no—
117#897—94no—
118#886—34no—
119#886—94no—
120#875—32no—
121#875—92no—
122#869—32no—
123#869—92no—
124C23×C32—21no—
125C23×C32C22×D311yescharacteristic
126C23×C32C23×C311yescharacteristic
127C22×C3⋊D3C23×C312yes—
128C22×C3×D3—32no—
129C2×C32×C4—24no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—24no—
133C2×C32×C4D1214yes—
134C2×C32×C4C3×D414yes—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C23×C314yes—
139C2×C32×C4C23×C311yescharacteristic
140C2×C32×C4C23×C311yescharacteristic
141#638—24no—
142C2×C3⋊Dic3—24no—
143C2×C3⋊Dic3C23×C312yes—
144C2×C3⋊Dic3C23×C314yes—
145C4⋊(C3⋊D3)—22no—
146C4×C3⋊D3—24no—
147#634—24no—
148#634—24no—
149#634—22no—
150C3×#67—64no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—64no—
154C3×D12—62no—
155C3×C4×D3—64no—
156C3×Dic6—64no—
157C3×Dic6—64no—
158C3×Dic6—62no—
159#536—91noSylow
160C2×H(3,𝔽3)—22no—
161C2×H(3,𝔽3)C22×D411yescharacteristic
162C2×H(3,𝔽3)2−1+412yes—
163(C3⋊D3)⋊C3—42no—
164C3×#41—184no—
165C2×C3×Q8—92no—
166C2×C3×Q8—92no—
167C2×C3×Q8—91no—
168C2×C3×Q8—184no—
169C2×C3×D4—91no—
170C2×C3×D4—92no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—61no—
174C22×C3×C4—92no—
175C22×C3×C4—92no—
176C22×C3×C4C2×C3×D311yescharacteristic
177C2×#67—32no—
178C2×#67—92no—
179C22×Dic3—32no—
180C22×Dic3—92no—
181#235—64no—
182#235—184no—
183C2×D12—31no—
184C2×D12—91no—
185C2×C4×D3—32no—
186C2×C4×D3—92no—
187C2×Dic6—31no—
188C2×Dic6—32no—
189C2×Dic6—32no—
190C2×Dic6—64no—
191C2×Dic6—91no—
192C2×Dic6—92no—
193C2×Dic6—92no—
194C2×Dic6—184no—
195C3×C4⋊C4—32no—
196C3×C4⋊C4—32no—
197C3×C4⋊C4—62no—
198C3×C4⋊C4—98no—
199C3×C4⋊C4C2×C3×D312yes—
200C3×C22⋊C4—32no—
201C3×C22⋊C4—32no—
202C3×C22⋊C4—62no—
203C3×C22⋊C4—98no—
204C3×C22⋊C4C2×C3×D312yes—
205C3×C42—32no—
206C3×C42—32no—
207C3×C42—62no—
208C3×C42C2×C3×D312yes—
209(C2×D3)⋊C4—38no—
210(C2×D3)⋊C4—98no—
211C4⋊Dic3—38no—
212C4⋊Dic3—98no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—42no—
218C22×C32C2×D1212yes—
219C22×C32C2×C3×D412yes—
220C22×C32C23×D311yescharacteristic
221C22×C32C24×C311yescharacteristic
222C2×C3⋊D3—22no—
223C2×C3⋊D3—24no—
224C2×C3×D3—62no—
225C2×C3×D3—64no—
226C32×C4—24no—
227C32×C4—24no—
228C32×C4—22no—
229C32×C4—22no—
230C32×C4—44no—
231C32×C4C2×D1212yes—
232C32×C4C2×D1212yes—
233C32×C4C2×C3×D412yes—
234C32×C4C2×C3×D412yes—
235C3⋊Dic3—24no—
236C3⋊Dic3—22no—
237C3×Dic3—64no—
238C3×Dic3—62no—
239C2×#41—91no—
240C22×Q8—91no—
241C4⋊Q8—92no—
242#124—92no—
243#123—94no—
244C22⋊Q8—94no—
245#117(C3⋊D3)⋊C311yescharacteristic, complemented
246H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
247C23×C3—31no—
248C23×C3—31no—
249C23×C3—61no—
250C23×C3—92no—
251C23×C3C22×C3×D311yescharacteristic
252C22×D3—32no—
253C22×D3—92no—
254C3×Q8—184no—
255C3×Q8—184no—
256C3×Q8—182no—
257C3×D4—182no—
258C3×D4—184no—
259C2×C3×C4—34no—
260C2×C3×C4—34no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—34no—
266C2×C3×C4—34no—
267C2×C3×C4—64no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—64no—
271C2×C3×C4—94no—
272C2×C3×C4—92no—
273C2×C3×C4—184no—
274C2×C3×C4—184no—
275C2×C3×C4C3×D1214yes—
276C2×C3×C4C22×C3×D314yes—
277C2×C3×C4C22×C3×D311yescharacteristic
278C2×C3×C4C22×C3×D311yescharacteristic
279#67—64no—
280#67—184no—
281C2×Dic3—32no—
282C2×Dic3—34no—
283C2×Dic3—64no—
284C2×Dic3—92no—
285C2×Dic3—94no—
286C2×Dic3—184no—
287D12—62no—
288D12—182no—
289C4×D3—64no—
290C4×D3—184no—
291Dic6—64no—
292Dic6—64no—
293Dic6—62no—
294Dic6—184no—
295Dic6—184no—
296Dic6—182no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—21no—
300C2×C32—22no—
301C2×C32—42no—
302C2×C32C22×D1211yescharacteristic
303C2×C32#101012yes—
304C2×C32C22×C3×D411yescharacteristic
305C2×C32C3×2−1+412yes—
306C3⋊D3—42no—
307C3×D3—122no—
308#41—184no—
309C2×Q8—92no—
310C2×Q8—92no—
311C2×Q8—91no—
312C2×Q8—184no—
313C2×D4—92no—
314C2×D4—91no—
315C22×C4—92no—
316C22×C4—92no—
317C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
318C4⋊C4—98no—
319C4⋊C4C2×(C3⋊D3)⋊C312yes—
320C22⋊C4—98no—
321C22⋊C4C2×(C3⋊D3)⋊C312yes—
322C42C2×(C3⋊D3)⋊C312yes—
323C22×C3—22no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—32no—
327C22×C3—32no—
328C22×C3—62no—
329C22×C3—62no—
330C22×C3—61no—
331C22×C3—62no—
332C22×C3—122no—
333C22×C3—182no—
334C22×C3—184no—
335C22×C3C2×C3×D1212yes—
336C22×C3C23×C3×D311yescharacteristic
337C2×D3—62no—
338C2×D3—64no—
339C2×D3—182no—
340C2×D3—184no—
341C3×C4—24no—
342C3×C4—32no—
343C3×C4—32no—
344C3×C4—32no—
345C3×C4—32no—
346C3×C4—64no—
347C3×C4—64no—
348C3×C4—62no—
349C3×C4—62no—
350C3×C4—124no—
351C3×C4—182no—
352C3×C4—184no—
353C3×C4C2×C3×D1212yes—
354C3×C4C2×C3×D1212yes—
355Dic3—64no—
356Dic3—62no—
357Dic3—184no—
358Dic3—182no—
359C32—21no—
360C32#552611yescharacteristic, complemented
361C32C3×#53611yescharacteristic, complemented
362C23—92no—
363C23C22×(C3⋊D3)⋊C311yescharacteristic
364Q8—184no—
365Q8—184no—
366Q8—182no—
367D4—184no—
368D4—182no—
369C2×C4—94no—
370C2×C4—92no—
371C2×C4—184no—
372C2×C4—184no—
373C2×C4C4⋊((C3⋊D3)⋊C3)14yes—
374C2×C4C22×(C3⋊D3)⋊C314yes—
375C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
376C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
377C2×C3—22no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—32no—
381C2×C3—32no—
382C2×C3—62no—
383C2×C3—62no—
384C2×C3—61no—
385C2×C3—62no—
386C2×C3—122no—
387C2×C3—362no—
388C2×C3C22×C3×D1211yescharacteristic
389C2×C3C3×#101012yes—
390D3—122no—
391D3—362no—
392C22—22no—
393C22—182no—
394C22—184no—
395C22C2×C4⋊((C3⋊D3)⋊C3)12yes—
396C22C23×(C3⋊D3)⋊C311yescharacteristic
397C4—24no—
398C4—182no—
399C4—184no—
400C4C2×C4⋊((C3⋊D3)⋊C3)12yes—
401C4C2×C4⋊((C3⋊D3)⋊C3)12yes—
402C3—31no—
403C3—31no—
404C3—61no—
405C3C3×#552611yescharacteristic
406C2—22no—
407C2—362no—
408C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
409C2#1172263912yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
prime radicalC2×#3828⋊C3C2
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
3. H(3,𝔽3)×#117C2semidirect
4. #64106⋊C3C2semidirect
5. #64095⋊C3C2semidirect
6. #64084⋊C3C2semidirect
7. #64078⋊C3C2semidirect
8. #10595997C3semidirect
26. C22×#634C2×C3semidirect
30. C32×#117D3semidirect
31. C32×#117C2×C3semidirect
32. #64106C2×C3semidirect
33. #64095C2×C3semidirect
34. #64084C2×C3semidirect
35. #64078C2×C3semidirect
245. #117(C3⋊D3)⋊C3semidirect
246. H(3,𝔽3)#536semidirect
360. C32#5526semidirect
361. C32C3×#536semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C22×#634⋊C32
H(3,𝔽3)×#1172
#64106⋊C32
#64095⋊C32
#64084⋊C32
#64078⋊C32
#105959973
C3×#55263