Finite Group Databasebeta

← #10592497⋊C3 #13141910

Subgroups — #10592497⋊C3 #13141910

Order 1728.

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

normal subgroups only

This group has 434 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#64156⋊C3C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3C2×#3744⋊C3C211yesmaximal, characteristic
4#64096⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic
8#10592497C311yesmaximal, characteristic, complemented
9C3×#4798—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#3756⋊C3—22no—
16#3752⋊C3—21no—
17#3752⋊C3—21no—
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3744⋊C3—22no—
23#64156C2×C311yescharacteristic
24C2×#3756C2×C311yescharacteristic, complemented
25C2×#3744C2×C311yescharacteristic
26#64096C2×C311yescharacteristic, complemented
27C3×#947—31no—
28C2×C3×#216—31no—
29C2×C3×#204—31no—
30C3×#887—31no—
31C32×#103—21no—
32C32×#103D311yescharacteristic, complemented
33C32×#103C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic
36C3×Dic3⋊C8—31no—
37C3×#809—31no—
38C22×(C3⋊D3)⋊C3—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51C8×H(3,𝔽3)—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53C3×#464—91no—
54#4798—31no—
55#4798—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C22×C311yescharacteristic
59C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
60C2×#634C22×C311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×Dic6—31no—
63C2×C32×C8—21no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C22×C311yescharacteristic
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C22×C311yescharacteristic
69#3756—22no—
70#3752—21no—
71#3752—21no—
72#3750—21no—
73C4×C3⋊Dic3—21no—
74C4×C3⋊Dic3C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76#3744—22no—
77C3×#216—62no—
78C3×(C2×D3)⋊C4—61no—
79C3×(C2×D3)⋊C4—61no—
80C3×Dic3⋊C4—61no—
81C3×C4×Dic3—31no—
82C3×C4×Dic3—61no—
83C2×C3×C3⋊C8—31no—
84C3×#204—62no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×(C3⋊D3)⋊C3—41no—
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)—41no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93(C3⋊Dic3)⋊C3—22no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—21no—
96C2×C3×Q16—91no—
97C2×C3×SD16—91no—
98C3×#124—91no—
99C3×C4×Q8—91no—
100#947—31no—
101#947—91no—
102C2×#216—31no—
103C2×#216—91no—
104C2×#204—31no—
105C2×#204—91no—
106#887—31no—
107#887—91no—
108C3×C4⋊C8—91no—
109C3×#103—31no—
110C3×#103—31no—
111C3×#103—61no—
112C3×#103C3×D311yescharacteristic
113C3×#102—91no—
114Dic3⋊C8—31no—
115Dic3⋊C8—91no—
116#809—31no—
117#809—91no—
118C22×C3⋊D3—21no—
119C22×C3×D3—61no—
120C32×Q8—22no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—41no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C23×C311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3C3×D411yescharacteristic
137C2×C3⋊Dic3C3×D411yescharacteristic
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140#634—21no—
141#634C3×D412yes—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×Dic6—32no—
148C3×Dic6—61no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8—41no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—61no—
154#464—91noSylow
155C2×H(3,𝔽3)#12111yescharacteristic
156C2×H(3,𝔽3)#13511yescharacteristic
157C2×H(3,𝔽3)#13711yescharacteristic
158(C3⋊D3)⋊C3—81no—
159C2×C3×Q8—31no—
160C2×C3×Q8—31no—
161C2×C3×Q8—61no—
162C2×C3×Q8—91no—
163C2×C3×Q8C2×C3×D311yescharacteristic
164C2×C3×D4—91no—
165C2×D12—31no—
166C2×D12—91no—
167C2×Dic6—31no—
168C2×Dic6—91no—
169C3×Q16—182no—
170C3×SD16—182no—
171C2×C3×C8—31no—
172C2×C3×C8—31no—
173C2×C3×C8—61no—
174C2×C3×C8—91no—
175C2×C3×C8C2×C3×D311yescharacteristic
176C3×C4⋊C4—31no—
177C3×C4⋊C4—31no—
178C3×C4⋊C4—61no—
179C3×C4⋊C4—181no—
180C3×C4⋊C4C2×C3×D311yescharacteristic
181C3×C22⋊C4—181no—
182C3×C22⋊C4—181no—
183C3×C42—91no—
184C3×C42—181no—
185#216—62no—
186#216—182no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—61no—
189(C2×D3)⋊C4—181no—
190(C2×D3)⋊C4—181no—
191Dic3⋊C4—61no—
192Dic3⋊C4—181no—
193C4×Dic3—31no—
194C4×Dic3—61no—
195C4×Dic3—91no—
196C4×Dic3—181no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—91no—
199#204—62no—
200#204—182no—
201C22×C32—21no—
202C22×C32D3×D411yescharacteristic
203C22×C32C2×C3×D411yescharacteristic
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—42no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4—81no—
218C32×C4#23511yescharacteristic
219C32×C4D3×D411yescharacteristic
220C32×C4C2×C3×D411yescharacteristic
221C32×C4C3×#4111yescharacteristic
222C3⋊Dic3—22no—
223C3⋊Dic3—21no—
224C3⋊Dic3—21no—
225C3×Dic3—62no—
226C3×Dic3—61no—
227C3×Dic3—61no—
228C2×Q16—91no—
229C2×SD16—91no—
230#124—91no—
231C4×Q8—91no—
232C4⋊C8—91no—
233#103(C3⋊D3)⋊C311yescharacteristic, complemented
234#102—91no—
235H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
236C23×C3—181no—
237C22×D3—61no—
238C22×D3—181no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—61no—
242C3×Q8—61no—
243C3×Q8—61no—
244C3×Q8—61no—
245C3×Q8—62no—
246C3×Q8—92no—
247C3×Q8—121no—
248C3×Q8—181no—
249C3×D4—181no—
250C3×D4—181no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—121no—
263C2×C3×C4—121no—
264C2×C3×C4—181no—
265C2×C3×C4C22×C3×D311yescharacteristic
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—61no—
269C2×Dic3—91no—
270C2×Dic3—91no—
271C2×Dic3—181no—
272D12—61no—
273D12—61no—
274D12—181no—
275D12—181no—
276Dic6—32no—
277Dic6—61no—
278Dic6—92no—
279Dic6—181no—
280C3×C8—21no—
281C3×C8—61no—
282C3×C8—61no—
283C3×C8—121no—
284C3×C8—181no—
285C3⋊C8—61no—
286C3⋊C8—181no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32#88411yescharacteristic
291C2×C32#91611yescharacteristic
292C2×C32#91811yescharacteristic
293C2×C32C3×#12111yescharacteristic
294C2×C32C3×#13511yescharacteristic
295C2×C32C3×#13711yescharacteristic
296C3⋊D3—81no—
297C3×D3—241no—
298C2×Q8—91no—
299C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
300C2×D4—91no—
301Q16—182no—
302SD16—182no—
303C2×C8—91no—
304C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
305C4⋊C4—181no—
306C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
307C22⋊C4—181no—
308C22⋊C4—181no—
309C42—91no—
310C42—181no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—61no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3C3×D3×D411yescharacteristic
318C2×D3—121no—
319C2×D3—121no—
320C2×D3—121no—
321C2×D3—361no—
322C2×D3—361no—
323C2×D3—361no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—121no—
333C3×C4—121no—
334C3×C4—121no—
335C3×C4—121no—
336C3×C4—122no—
337C3×C4—181no—
338C3×C4—181no—
339C3×C4—182no—
340C3×C4—241no—
341C3×C4C3×#23511yescharacteristic
342C3×C4C3×D3×D411yescharacteristic
343Dic3—62no—
344Dic3—61no—
345Dic3—61no—
346Dic3—182no—
347Dic3—181no—
348Dic3—181no—
349C32—21no—
350C32#479811yescharacteristic, complemented
351C32C3×#46411yescharacteristic, complemented
352C23—181no—
353Q8—21no—
354Q8—21no—
355Q8—92no—
356Q8—181no—
357D4—181no—
358D4—181no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—91no—
362C2×C4—91no—
363C2×C4—181no—
364C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
365C8—21no—
366C8—181no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—61no—
374C2×C3—61no—
375C2×C3—61no—
376C2×C3—721no—
377C2×C3C3×#88411yescharacteristic
378C2×C3C3×#91611yescharacteristic
379C2×C3C3×#91811yescharacteristic
380D3—241no—
381D3—721no—
382C22—361no—
383C22—361no—
384C22—361no—
385C22D4×(C3⋊D3)⋊C311yescharacteristic
386C4—41no—
387C4—41no—
388C4—181no—
389C4—181no—
390C4—182no—
391C4#3828⋊C311yescharacteristic
392C4D4×(C3⋊D3)⋊C311yescharacteristic
393C3—31no—
394C3—31no—
395C3—61no—
396C3C3×#479811yescharacteristic
397C2—721no—
398C2#64093⋊C311yescharacteristic
399C2#64125⋊C311yescharacteristic
400C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
4. #64096⋊C3C2semidirect
5. H(3,𝔽3)×#103C2semidirect
6. #63641⋊C3C2semidirect
8. #10592497C3semidirect
24. C2×#3756C2×C3semidirect
26. #64096C2×C3semidirect
32. C32×#103D3semidirect
33. C32×#103C2×C3semidirect
34. #63641C2×C3semidirect
233. #103(C3⋊D3)⋊C3semidirect
235. H(3,𝔽3)#464semidirect
350. C32#4798semidirect
351. C32C3×#464semidirect

Maximal subgroups

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

subgroupindex
#64156⋊C32
C2×#3756⋊C32
C2×#3744⋊C32
#64096⋊C32
H(3,𝔽3)×#1032
#63641⋊C32
#63636⋊C32
#105924973
C3×#47983