Finite Group Databasebeta

← #10592491⋊C3 #13141902

Subgroups — #10592491⋊C3 #13141902

Order 1728.

385 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 94 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#64153⋊C3C211yesmaximal, characteristic
2#64097⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#103C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic
6C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
7#63637⋊C3C211yesmaximal, characteristic
8#10592491C311yesmaximal, characteristic, complemented
9C3×#4792—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3—21no—
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17#3750⋊C3—21no—
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64153C2×C311yescharacteristic
22#64097C2×C311yescharacteristic
23C3×#944—31no—
24C3×#888—31no—
25C32×#103—21no—
26C32×#103D311yescharacteristic
27C32×#103C2×C311yescharacteristic
28#63662C2×C311yescharacteristic
29#63643C2×C311yescharacteristic
30C8×C3⋊Dic3C2×C311yescharacteristic
31#63637C2×C311yescharacteristic
32C3×#835—31no—
33C3×#816—31no—
34C3×C8×Dic3—31no—
35C3×#810—31no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#634⋊C3—21no—
46#634⋊C3—21no—
47#634⋊C3—42no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50C3×#487—91no—
51#4792—31no—
52#4792—91no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C22×C311yescharacteristic
56C2×#634—21no—
57C2×#634C22×C311yescharacteristic
58C2×C3×Dic6—31no—
59C2×C3×Dic6—61no—
60C2×C32×C8—21no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C22×C311yescharacteristic
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67#3750—21no—
68#3750—21no—
69#3750—21no—
70C4×C3⋊Dic3C22×C311yescharacteristic
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×C4⋊Dic3—31no—
73C3×Dic3⋊C4—61no—
74C3×Dic3⋊C4—61no—
75C3×Dic3⋊C4—61no—
76C3×C4×Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)—41no—
81C4×H(3,𝔽3)#4111yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83(C3⋊Dic3)⋊C3—41no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3SD1612yes—
86(C3⋊Dic3)⋊C3Q1612yes—
87C3×C4⋊Q8—91no—
88C3×C4⋊Q8—91no—
89#944—31no—
90#944—91no—
91#888—31no—
92#888—91no—
93C3×#103—31no—
94C3×#103—31no—
95C3×#103—61no—
96C3×#103—91no—
97C3×#103—91no—
98C3×#103—91no—
99C3×#103C3×D311yescharacteristic
100C3×C4×C8—91no—
101#835—31no—
102#835—91no—
103#816—31no—
104#816—91no—
105C8×Dic3—31no—
106C8×Dic3—91no—
107#810—31no—
108#810—91no—
109C32×Q8—22no—
110C32×Q8—21no—
111C32×Q8—21no—
112C32×Q8—21no—
113C32×Q8—21no—
114C32×Q8—41no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C23×C311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3C3×D411yescharacteristic
127C2×C3⋊Dic3C3×D411yescharacteristic
128#634—21no—
129#634—21no—
130#634—42no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C3×Dic6—61no—
136C3×Dic6—61no—
137C3×Dic6—122no—
138C32×C8—21no—
139C32×C8—21no—
140C32×C8—41no—
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—61no—
143#487—91noSylow
144C2×H(3,𝔽3)#12411yescharacteristic
145C2×H(3,𝔽3)C2×SD1611yescharacteristic
146C2×H(3,𝔽3)C2×Q1611yescharacteristic
147C2×C3×Q8—31no—
148C2×C3×Q8—31no—
149C2×C3×Q8—61no—
150C2×C3×Q8—91no—
151C2×C3×Q8—181no—
152C2×C3×Q8C2×C3×D311yescharacteristic
153C2×Dic6—31no—
154C2×Dic6—61no—
155C2×Dic6—91no—
156C2×Dic6—181no—
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—61no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—181no—
167C3×C4⋊C4—181no—
168C3×C4⋊C4—181no—
169C3×C4⋊C4C2×C3×D311yescharacteristic
170C3×C42—91no—
171C4⋊Dic3—31no—
172C4⋊Dic3—91no—
173Dic3⋊C4—61no—
174Dic3⋊C4—61no—
175Dic3⋊C4—61no—
176Dic3⋊C4—181no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C4×Dic3—31no—
180C4×Dic3—91no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32D3×D411yescharacteristic
185C22×C32C2×C3×D411yescharacteristic
186C32×C4—21no—
187C32×C4—21no—
188C32×C4—42no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4—81no—
194C32×C4#23511yescharacteristic
195C32×C4#23711yescharacteristic
196C32×C4C3×#4111yescharacteristic
197C32×C4C3×#4111yescharacteristic
198C3⋊Dic3—41no—
199C3⋊Dic3—41no—
200C3⋊Dic3C3×SD1612yes—
201C3⋊Dic3C3×Q1612yes—
202C3×Dic3—32no—
203C3×Dic3—32no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C4⋊Q8—91no—
207C4⋊Q8—91no—
208#103—91no—
209#103—91no—
210#103—91no—
211#103(C3⋊D3)⋊C311yescharacteristic
212C4×C8—91no—
213H(3,𝔽3)#48711yescharacteristic, complemented, Sylow
214C3×Q8—21no—
215C3×Q8—21no—
216C3×Q8—61no—
217C3×Q8—61no—
218C3×Q8—61no—
219C3×Q8—61no—
220C3×Q8—62no—
221C3×Q8—121no—
222C3×Q8—181no—
223C3×Q8—181no—
224C3×Q8—362no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—91no—
235C2×C3×C4—91no—
236C2×C3×C4—121no—
237C2×C3×C4—121no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4C22×C3×D311yescharacteristic
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—91no—
246C2×Dic3—91no—
247C2×Dic3—181no—
248C2×Dic3—181no—
249Dic6—61no—
250Dic6—61no—
251Dic6—122no—
252Dic6—181no—
253Dic6—181no—
254Dic6—362no—
255C3×C8—21no—
256C3×C8—61no—
257C3×C8—61no—
258C3×C8—121no—
259C3×C8—181no—
260C3⋊C8—61no—
261C3⋊C8—181no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32#88511yescharacteristic
266C2×C32D3×SD1611yescharacteristic
267C2×C32D3×Q1611yescharacteristic
268C2×C32C3×#12411yescharacteristic
269C2×C32C2×C3×SD1611yescharacteristic
270C2×C32C2×C3×Q1611yescharacteristic
271C2×Q8—91no—
272C2×Q8—181no—
273C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
274C2×C8—91no—
275C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
276C4⋊C4—91no—
277C4⋊C4—181no—
278C4⋊C4—181no—
279C4⋊C4—181no—
280C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
281C42—91no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—61no—
285C22×C3C3×D3×D411yescharacteristic
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—61no—
293C3×C4—61no—
294C3×C4—92no—
295C3×C4—92no—
296C3×C4—121no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—121no—
300C3×C4—122no—
301C3×C4—241no—
302C3×C4—361no—
303C3×C4—361no—
304C3×C4C3×#23511yescharacteristic
305C3×C4C3×#23711yescharacteristic
306Dic3—32no—
307Dic3—32no—
308Dic3—92no—
309Dic3—92no—
310Dic3—121no—
311Dic3—121no—
312Dic3—361no—
313Dic3—361no—
314C32—21no—
315C32#479211yescharacteristic, complemented
316C32C3×#48711yescharacteristic, complemented
317Q8—21no—
318Q8—21no—
319Q8—181no—
320Q8—181no—
321Q8—362no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
329C8—21no—
330C8—181no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—61no—
338C2×C3—61no—
339C2×C3—61no—
340C2×C3C3×#88511yescharacteristic
341C2×C3C3×D3×SD1611yescharacteristic
342C2×C3C3×D3×Q1611yescharacteristic
343C22D4×(C3⋊D3)⋊C311yescharacteristic
344C4—41no—
345C4—41no—
346C4—92no—
347C4—92no—
348C4—361no—
349C4—361no—
350C4#3828⋊C311yescharacteristic
351C4#3830⋊C311yescharacteristic
352C3—31no—
353C3—31no—
354C3—61no—
355C3C3×#479211yescharacteristic
356C2#64094⋊C311yescharacteristic
357C2SD16×(C3⋊D3)⋊C311yescharacteristic
358C2Q16×(C3⋊D3)⋊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 radicalC22×H(3,𝔽3)C2×D4
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#487
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
8. #10592491C3semidirect
213. H(3,𝔽3)#487semidirect
315. C32#4792semidirect
316. C32C3×#487semidirect

Maximal subgroups

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

subgroupindex
#64153⋊C32
#64097⋊C32
H(3,𝔽3)×#1032
#63662⋊C32
#63643⋊C32
C8×(C3⋊Dic3)⋊C32
#63637⋊C32
#105924913
C3×#47923