Finite Group Databasebeta

← #10592588⋊C3 #13142230

Subgroups — #10592588⋊C3 #13142230

Order 1728.

456 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C2×#3744⋊C3C211yesmaximal, characteristic
2#64102⋊C3C211yesmaximal, characteristic
3#64096⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
5#63642⋊C3C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592588C311yesmaximal, characteristic, complemented
9C3×#4889—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
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—
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3744⋊C3C414yescomplemented
22C2×#3744C2×C311yescharacteristic
23#64102C2×C311yescharacteristic
24#64096C2×C311yescharacteristic, complemented
25C2×C3×#204—31no—
26C3×#893—31no—
27C3×#887—31no—
28C32×#107—21no—
29C32×#107D311yescharacteristic, complemented
30C32×#107C2×C311yescharacteristic, complemented
31#63642C2×C311yescharacteristic, complemented
32#63637C2×C311yescharacteristic, complemented
33#63636C2×C311yescharacteristic
34C3×#815—31no—
35C3×#810—31no—
36C3×#809—31no—
37C22×(C3⋊D3)⋊C3—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)⋊C3D411yescharacteristic
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)C2×C412yes—
46C4×(C3⋊D3)⋊C3—42no—
47#634⋊C3—21no—
48#634⋊C3C2×C412yescomplemented
49C8×H(3,𝔽3)C2×C412yescomplemented
50(C3⋊(C3⋊C8))⋊C3—21no—
51C3×#440—91no—
52#4889—31no—
53#4889—91no—
54C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×#634C22×C311yescharacteristic
57C2×C3×D12—31no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—31no—
60C2×C32×C8—21no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C22×C311yescharacteristic
63C32×C4⋊C4—21no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C4⋊C4C2×D311yescharacteristic
67C32×C4⋊C4C22×C311yescharacteristic
68C32×C4⋊C4C22×C311yescharacteristic
69#3752—21no—
70#3750—21no—
71C4×C3⋊Dic3—21no—
72C4×C3⋊Dic3C22×C311yescharacteristic
73C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
74#3744C3×C414yescomplemented
75C3×(C2×D3)⋊C4—61no—
76C3×Dic3⋊C4—61no—
77C3×C4×Dic3—31no—
78C3×C4×Dic3—61no—
79C2×C3×C3⋊C8—31no—
80C3×#204—34no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)—41no—
87C4×H(3,𝔽3)C22×C411yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—21no—
92C2×C3×SD16—91no—
93C3×C4×Q8—91no—
94C3×C4×D4—91no—
95C2×#204—31no—
96C2×#204—91no—
97#893—31no—
98#893—91no—
99#887—31no—
100#887—91no—
101C3×#107—31no—
102C3×#107—31no—
103C3×#107—61no—
104C3×#107C3×D311yescharacteristic
105C3×#103—91no—
106C3×#102—91no—
107C3×#97—91no—
108#815—31no—
109#815—91no—
110#810—31no—
111#810—91no—
112#809—31no—
113#809—91no—
114C22×C3⋊D3—21no—
115C22×C3×D3—61no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4—41no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3C3×D411yescharacteristic
127C2×C3⋊Dic3C3×D411yescharacteristic
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)C2×C3×C412yes—
130C4×C3⋊D3—42no—
131#634—21no—
132#634C2×C3×C412yescomplemented
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—61no—
136C3×D12—32no—
137C3×D12—61no—
138C3×C4×D3—122no—
139C3×Dic6—32no—
140C3×Dic6—61no—
141C32×C8—22no—
142C32×C8C4×D312yescomplemented
143C32×C8C2×C3×C412yescomplemented
144C3⋊(C3⋊C8)—21no—
145C3×C3⋊C8—61no—
146#440—91noSylow
147C2×H(3,𝔽3)C4×D411yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149C2×H(3,𝔽3)#13711yescharacteristic
150(C3⋊D3)⋊C3—42no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C22×C3×C4—181no—
154C2×D12—31no—
155C2×D12—91no—
156C2×C4×D3—61no—
157C2×C4×D3—181no—
158C2×Dic6—31no—
159C2×Dic6—91no—
160C3×SD16—94no—
161C2×C3×C8—31no—
162C2×C3×C8—31no—
163C2×C3×C8—61no—
164C2×C3×C8—91no—
165C2×C3×C8C2×C3×D311yescharacteristic
166C3×C4⋊C4—31no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—61no—
172C3×C4⋊C4—181no—
173C3×C4⋊C4C2×C3×D311yescharacteristic
174C3×C4⋊C4C2×C3×D311yescharacteristic
175C3×C22⋊C4—181no—
176C3×C42—91no—
177C3×C42—181no—
178(C2×D3)⋊C4—61no—
179(C2×D3)⋊C4—181no—
180Dic3⋊C4—61no—
181Dic3⋊C4—181no—
182C4×Dic3—31no—
183C4×Dic3—61no—
184C4×Dic3—91no—
185C4×Dic3—181no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—91no—
188#204—34no—
189#204—94no—
190C22×C32—21no—
191C22×C32D3×D411yescharacteristic
192C22×C32C2×C3×D411yescharacteristic
193C2×C3⋊D3—22no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3×D3—62no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4—41no—
204C32×C4—41no—
205C32×C4—81no—
206C32×C4—81no—
207C32×C4C2×C4×D311yescharacteristic
208C32×C4#23911yescharacteristic
209C32×C4C22×C3×C411yescharacteristic
210C32×C4C3×#4111yescharacteristic
211C3⋊Dic3—22no—
212C3⋊Dic3—21no—
213C3⋊Dic3—21no—
214C3×Dic3—62no—
215C3×Dic3—61no—
216C3×Dic3—61no—
217C2×SD16—91no—
218C4×Q8—91no—
219C4×D4—91no—
220#107(C3⋊D3)⋊C311yescharacteristic, complemented
221#103—91no—
222#102—91no—
223#97—91no—
224H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
225C23×C3—181no—
226C22×D3—61no—
227C22×D3—181no—
228C3×Q8—92no—
229C3×Q8—181no—
230C3×D4—92no—
231C3×D4—181no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—91no—
242C2×C3×C4—91no—
243C2×C3×C4—121no—
244C2×C3×C4—121no—
245C2×C3×C4—181no—
246C2×C3×C4—362no—
247C2×C3×C4C22×C3×D311yescharacteristic
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—61no—
251C2×Dic3—91no—
252C2×Dic3—91no—
253C2×Dic3—181no—
254D12—32no—
255D12—61no—
256D12—92no—
257D12—181no—
258C4×D3—122no—
259C4×D3—362no—
260Dic6—32no—
261Dic6—61no—
262Dic6—92no—
263Dic6—181no—
264C3×C8—32no—
265C3×C8—32no—
266C3×C8—62no—
267C3×C8—181no—
268C3×C8C3×C4×D312yes—
269C3⋊C8—61no—
270C3⋊C8—181no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32#89311yescharacteristic
275C2×C32#91111yescharacteristic
276C2×C32#91811yescharacteristic
277C2×C32C3×C4×D411yescharacteristic
278C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
279C2×C32C3×#13711yescharacteristic
280C3⋊D3—42no—
281C3×D3—122no—
282C2×Q8—91no—
283C2×D4—91no—
284C22×C4—181no—
285SD16—94no—
286C2×C8—91no—
287C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
288C4⋊C4—181no—
289C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
290C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
291C22⋊C4—181no—
292C42—91no—
293C42—181no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—61no—
297C22×C3—182no—
298C22×C3—361no—
299C22×C3—361no—
300C22×C3C3×D3×D411yescharacteristic
301C2×D3—62no—
302C2×D3—121no—
303C2×D3—121no—
304C2×D3—182no—
305C2×D3—361no—
306C2×D3—361no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—121no—
316C3×C4—121no—
317C3×C4—121no—
318C3×C4—121no—
319C3×C4—181no—
320C3×C4—181no—
321C3×C4—182no—
322C3×C4—241no—
323C3×C4—241no—
324C3×C4C2×C3×C4×D311yescharacteristic
325C3×C4C3×#23911yescharacteristic
326Dic3—62no—
327Dic3—61no—
328Dic3—61no—
329Dic3—182no—
330Dic3—181no—
331Dic3—181no—
332C32—21no—
333C32#488911yescharacteristic, complemented
334C32C3×#44011yescharacteristic, complemented
335C23—181no—
336Q8—92no—
337Q8—181no—
338D4—92no—
339D4—181no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—91no—
343C2×C4—91no—
344C2×C4—181no—
345C2×C4—362no—
346C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
347C8—181no—
348C8C4×(C3⋊D3)⋊C312yescomplemented
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—61no—
356C2×C3—61no—
357C2×C3—61no—
358C2×C3—362no—
359C2×C3C3×#89311yescharacteristic
360C2×C3C3×#91111yescharacteristic
361C2×C3C3×#91811yescharacteristic
362D3—122no—
363D3—362no—
364C22—182no—
365C22—361no—
366C22—361no—
367C22D4×(C3⋊D3)⋊C311yescharacteristic
368C4—41no—
369C4—41no—
370C4—181no—
371C4—181no—
372C4—182no—
373C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
374C4#8872211yescharacteristic
375C3—31no—
376C3—31no—
377C3—61no—
378C3C3×#488911yescharacteristic
379C2—362no—
380C2#64102⋊C311yescharacteristic
381C2#64120⋊C311yescharacteristic
382C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C32×C4C23×C3
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#440
LCSG ▷ 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. #64096⋊C3C2semidirect
4. H(3,𝔽3)×#107C2semidirect
5. #63642⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592588C3semidirect
21. #3744⋊C3C4semidirect
24. #64096C2×C3semidirect
29. C32×#107D3semidirect
30. C32×#107C2×C3semidirect
31. #63642C2×C3semidirect
32. #63637C2×C3semidirect
48. #634⋊C3C2×C4semidirect
49. C8×H(3,𝔽3)C2×C4semidirect
74. #3744C3×C4semidirect
132. #634C2×C3×C4semidirect
142. C32×C8C4×D3semidirect
143. C32×C8C2×C3×C4semidirect
220. #107(C3⋊D3)⋊C3semidirect
224. H(3,𝔽3)#440semidirect
333. C32#4889semidirect
334. C32C3×#440semidirect
348. C8C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
#64102⋊C32
#64096⋊C32
H(3,𝔽3)×#1072
#63642⋊C32
#63636⋊C32
#63637⋊C32
#105925883
C3×#48893