Finite Group Databasebeta

← #10592583⋊C3 #13142225

Subgroups — #10592583⋊C3 #13142225

Order 1728.

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

normal subgroups only

This group has 400 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#64106⋊C3C211yesmaximal, characteristic
2C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
4#63647⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592583C311yesmaximal, characteristic, complemented
9C3×#4884—31nomaximal
10C2×C4×(C3⋊D3)⋊C3—21no—
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64106C2×C311yescharacteristic
22C4⋊C4×C3⋊D3C2×C311yescharacteristic
23C3×#897—31no—
24C3×D3×C4⋊C4—31no—
25C32×#107—21no—
26C32×#107D311yescharacteristic, complemented
27C32×#107C2×C311yescharacteristic, complemented
28#63647C2×C311yescharacteristic
29#63643C2×C311yescharacteristic, complemented
30#63637C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C3×(C2×D3)⋊C8—31no—
33C3×#816—31no—
34C3×#810—31no—
35C3×#807—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4×(C3⋊D3)⋊C3—21no—
44C4×(C3⋊D3)⋊C3—21no—
45C4×(C3⋊D3)⋊C3—42no—
46#634⋊C3—21no—
47#634⋊C3—21no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50C3×#484—91no—
51#4884—31no—
52#4884—91no—
53C2×C4×C3⋊D3—21no—
54C2×C4×C3⋊D3C22×C311yescharacteristic
55C2×#634C22×C311yescharacteristic
56C2×C3×C4×D3—31no—
57C2×C3×C4×D3—61no—
58C2×C3×Dic6—31no—
59C2×C32×C8—21no—
60C2×C32×C8C2×D311yescharacteristic
61C2×C32×C8C22×C311yescharacteristic
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C4⋊C4C22×C311yescharacteristic
67C32×C4⋊C4C22×C311yescharacteristic
68#3752—21no—
69C4⋊(C3⋊Dic3)—21no—
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71#3750—21no—
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C3×(C2×D3)⋊C4—61no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C3×Dic3⋊C4—61no—
77C2×C3×C3⋊C8—31no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3Q1612yes—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)—41no—
84C4×H(3,𝔽3)#4111yescharacteristic
85C4×H(3,𝔽3)#4111yescharacteristic
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89C3×C22⋊Q8—91no—
90C2×C3×C4⋊C4—91no—
91#897—31no—
92#897—91no—
93D3×C4⋊C4—31no—
94D3×C4⋊C4—91no—
95C3×#107—31no—
96C3×#107—31no—
97C3×#107—61no—
98C3×#107—91no—
99C3×#107C3×D311yescharacteristic
100C3×#103—91no—
101C3×#103—91no—
102C3×C22⋊C8—91no—
103(C2×D3)⋊C8—31no—
104(C2×D3)⋊C8—91no—
105#816—31no—
106#816—91no—
107#810—31no—
108#810—91no—
109#807—31no—
110#807—91no—
111C22×C3⋊D3C3×D411yescharacteristic
112C22×C3×D3—31no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C23×C311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3C3×D411yescharacteristic
125C4×C3⋊D3—21no—
126C4×C3⋊D3—21no—
127C4×C3⋊D3—42no—
128#634—21no—
129#634—21no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×C4×D3—122no—
136C3×Dic6—61no—
137C3×Dic6—61no—
138C32×C8—22no—
139C32×C8—21no—
140C32×C8—21no—
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—61no—
143#484—91noSylow
144C2×H(3,𝔽3)#12311yescharacteristic
145C2×H(3,𝔽3)C2×Q1611yescharacteristic
146C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
147(C3⋊D3)⋊C3—22no—
148C2×C3×Q8—91no—
149C22×C3×C4—91no—
150C22×C3×C4—181no—
151C2×C4×D3—31no—
152C2×C4×D3—61no—
153C2×C4×D3—91no—
154C2×C4×D3—181no—
155C2×Dic6—31no—
156C2×Dic6—91no—
157C2×C3×C8—31no—
158C2×C3×C8—31no—
159C2×C3×C8—61no—
160C2×C3×C8—91no—
161C2×C3×C8C2×C3×D311yescharacteristic
162C3×C4⋊C4—31no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4—181no—
171C3×C4⋊C4C2×C3×D311yescharacteristic
172C3×C4⋊C4C2×C3×D311yescharacteristic
173C3×C22⋊C4—181no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—31no—
177C4⋊Dic3—61no—
178C4⋊Dic3—91no—
179C4⋊Dic3—181no—
180Dic3⋊C4—61no—
181Dic3⋊C4—181no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—91no—
184C22×C32—21no—
185C22×C32D3×D411yescharacteristic
186C22×C32C2×C3×D411yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3C3×Q1612yes—
190C2×C3×D3—32no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—81no—
200C32×C4—81no—
201C32×C4#23511yescharacteristic
202C32×C4#23911yescharacteristic
203C32×C4C3×#4111yescharacteristic
204C32×C4C3×#4111yescharacteristic
205C3⋊Dic3—21no—
206C3⋊Dic3—41no—
207C3⋊Dic3—41no—
208C3×Dic3—61no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C22⋊Q8—91no—
212C2×C4⋊C4—91no—
213#107—91no—
214#107(C3⋊D3)⋊C311yescharacteristic, complemented
215#103—91no—
216#103—91no—
217C22⋊C8—91no—
218H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—31no—
221C22×D3—91no—
222C3×Q8—181no—
223C3×Q8—181no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—91no—
234C2×C3×C4—121no—
235C2×C3×C4—121no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4—362no—
241C2×C3×C4C22×C3×D311yescharacteristic
242C2×Dic3—31no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—91no—
246C2×Dic3—181no—
247C2×Dic3—181no—
248C4×D3—61no—
249C4×D3—61no—
250C4×D3—122no—
251C4×D3—181no—
252C4×D3—181no—
253C4×D3—362no—
254Dic6—61no—
255Dic6—61no—
256Dic6—181no—
257Dic6—181no—
258C3×C8—21no—
259C3×C8—61no—
260C3×C8—61no—
261C3×C8—62no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8—181no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32#89411yescharacteristic
269C2×C32#91111yescharacteristic
270C2×C32D3×Q1611yescharacteristic
271C2×C32C3×#12311yescharacteristic
272C2×C32C2×C3×Q1611yescharacteristic
273C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
274C3⋊D3—22no—
275C3×D3—62no—
276C2×Q8—91no—
277C22×C4—91no—
278C22×C4—181no—
279C2×C8—91no—
280C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
281C4⋊C4—91no—
282C4⋊C4—181no—
283C4⋊C4—181no—
284C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
285C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
286C22⋊C4—181no—
287C22×C3—31no—
288C22×C3—31no—
289C22×C3—61no—
290C22×C3—92no—
291C22×C3—181no—
292C22×C3—181no—
293C22×C3C3×D3×D411yescharacteristic
294C2×D3—32no—
295C2×D3—61no—
296C2×D3—61no—
297C2×D3—92no—
298C2×D3—181no—
299C2×D3—181no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—61no—
307C3×C4—61no—
308C3×C4—121no—
309C3×C4—121no—
310C3×C4—121no—
311C3×C4—121no—
312C3×C4—181no—
313C3×C4—241no—
314C3×C4—241no—
315C3×C4—361no—
316C3×C4—361no—
317C3×C4C3×#23511yescharacteristic
318C3×C4C3×#23911yescharacteristic
319Dic3—61no—
320Dic3—121no—
321Dic3—121no—
322Dic3—181no—
323Dic3—361no—
324Dic3—361no—
325C32—21no—
326C32#488411yescharacteristic, complemented
327C32C3×#48411yescharacteristic, complemented
328C23—91no—
329Q8—181no—
330Q8—181no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—91no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—362no—
339C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
340C8—21no—
341C8—181no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—61no—
349C2×C3—61no—
350C2×C3—61no—
351C2×C3—182no—
352C2×C3C3×#89411yescharacteristic
353C2×C3C3×#91111yescharacteristic
354C2×C3C3×D3×Q1611yescharacteristic
355D3—62no—
356D3—182no—
357C22—92no—
358C22—181no—
359C22—181no—
360C22D4×(C3⋊D3)⋊C311yescharacteristic
361C4—41no—
362C4—41no—
363C4—181no—
364C4—361no—
365C4—361no—
366C4#3828⋊C311yescharacteristic
367C4#8872211yescharacteristic
368C3—31no—
369C3—31no—
370C3—61no—
371C3C3×#488411yescharacteristic
372C2—182no—
373C2#64103⋊C311yescharacteristic
374C2#64120⋊C311yescharacteristic
375C2Q16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#484
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#107C2semidirect
5. #63643⋊C3C2semidirect
6. #63637⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592583C3semidirect
26. C32×#107D3semidirect
27. C32×#107C2×C3semidirect
29. #63643C2×C3semidirect
30. #63637C2×C3semidirect
31. #63634C2×C3semidirect
214. #107(C3⋊D3)⋊C3semidirect
218. H(3,𝔽3)#484semidirect
326. C32#4884semidirect
327. C32C3×#484semidirect

Maximal subgroups

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

subgroupindex
#64106⋊C32
C4⋊C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1072
#63647⋊C32
#63643⋊C32
#63637⋊C32
#63634⋊C32
#105925833
C3×#48843