Finite Group Databasebeta

← #10592431⋊C3 #13141616

Subgroups — #10592431⋊C3 #13141616

Order 1728.

743 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3C2×#3745⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
5#63648⋊C3C212yesmaximal, complemented
6#10592431C311yesmaximal, characteristic, complemented
7C3×#4732—31nomaximal
8C23×(C3⋊D3)⋊C3—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3—21no—
11C2×C4⋊((C3⋊D3)⋊C3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)—24no—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
15C2×C8×H(3,𝔽3)C2212yescomplemented
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
18#3745⋊C3—24no—
19C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
20C22×C3×D12—31no—
21#64139C2×C311yescharacteristic, complemented
22C2×#3745C2×C312yescomplemented
23C3×#930—31no—
24C2×C3×D24—32no—
25C32×C22⋊C8—21no—
26C32×C22⋊C8D311yescharacteristic, complemented
27C32×C22⋊C8C2×C311yescharacteristic, complemented
28#63648C2×C312yescomplemented
29C3×#821—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×(C3⋊D3)⋊C3—24no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—21no—
34C22×(C3⋊D3)⋊C3—42no—
35C22×(C3⋊D3)⋊C3—42no—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#638⋊C3—42no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C4⋊((C3⋊D3)⋊C3)—24no—
42C4⋊((C3⋊D3)⋊C3)—22no—
43C4⋊((C3⋊D3)⋊C3)—22no—
44C4⋊((C3⋊D3)⋊C3)D414yescomplemented
45C8×H(3,𝔽3)—22no—
46C3×#447—91no—
47#4732—31no—
48#4732—91no—
49C23×C3⋊D3—21no—
50C23×C3×D3—61no—
51C22×C32×C4—21no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C22×C311yescharacteristic
54C2×#638—21no—
55C2×C4⋊(C3⋊D3)—21no—
56C2×C4⋊(C3⋊D3)—24no—
57C2×C4⋊(C3⋊D3)C22×C312yes—
58C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
59C2×C3×#67—61no—
60C2×C3×D12—32no—
61C2×C3×D12—31no—
62C2×C3×D12—64no—
63C2×C3×D12—61no—
64C2×C32×C8—22no—
65C2×C32×C8C2×D312yescomplemented
66C2×C32×C8C22×C312yescomplemented
67#3752—21no—
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
69#3745—24no—
70C3×(C2×D3)⋊C4—61no—
71C3×C4⋊Dic3—31no—
72C3×D24—64no—
73C22×H(3,𝔽3)—21no—
74C22×H(3,𝔽3)—21no—
75C22×H(3,𝔽3)D812yescomplemented
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C2×(C3⋊D3)⋊C3—24no—
78C2×(C3⋊D3)⋊C3—48no—
79C2×(C3⋊D3)⋊C3—42no—
80C2×(C3⋊D3)⋊C3—42no—
81C2×(C3⋊D3)⋊C3—41no—
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C4×H(3,𝔽3)—21no—
85C4×H(3,𝔽3)C2×D412yes—
86(C3⋊Dic3)⋊C3—41no—
87C22×C3×D4—91no—
88C22×D12—31no—
89C22×D12—91no—
90C2×C3×D8—92no—
91C3×#121—91no—
92#930—31no—
93#930—91no—
94C2×D24—32no—
95C2×D24—92no—
96C3×#102—92no—
97C3×C22⋊C8—31no—
98C3×C22⋊C8—31no—
99C3×C22⋊C8—61no—
100C3×C22⋊C8C3×D311yescharacteristic
101#821—32no—
102#821—92no—
103C23×C32—21no—
104C23×C32D1211yescharacteristic
105C23×C32C3×D411yescharacteristic
106C22×C3⋊D3—24no—
107C22×C3⋊D3—22no—
108C22×C3⋊D3—21no—
109C22×C3⋊D3—42no—
110C22×C3⋊D3—42no—
111C22×C3×D3—64no—
112C22×C3×D3—62no—
113C22×C3×D3—61no—
114C22×C3×D3—122no—
115C22×C3×D3—122no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—42no—
121C2×C32×C4D1211yescharacteristic
122C2×C32×C4C3×D411yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125#638—42no—
126C2×C3⋊Dic3—21no—
127C4⋊(C3⋊D3)—24no—
128C4⋊(C3⋊D3)—22no—
129C4⋊(C3⋊D3)—22no—
130C4⋊(C3⋊D3)C3×D414yescomplemented
131C3×#67—122no—
132C2×C3×Dic3—61no—
133C3×D12—34no—
134C3×D12—64no—
135C3×D12—62no—
136C3×D12—62no—
137C32×C8—22no—
138C32×C8—22no—
139C32×C8—42no—
140#447—91noSylow
141C2×H(3,𝔽3)—22no—
142C2×H(3,𝔽3)#12011yescharacteristic
143C2×H(3,𝔽3)C2×D811yescharacteristic
144C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
145(C3⋊D3)⋊C3—44no—
146(C3⋊D3)⋊C3—81no—
147C24×C3—181no—
148C23×D3—61no—
149C23×D3—181no—
150C2×C3×D4—92no—
151C2×C3×D4—91no—
152C2×C3×D4—181no—
153C2×C3×D4—184no—
154C2×C3×D4—181no—
155C22×C3×C4—31no—
156C22×C3×C4—31no—
157C22×C3×C4—61no—
158C22×C3×C4C2×C3×D311yescharacteristic
159C2×#67—61no—
160C2×#67—181no—
161C2×D12—32no—
162C2×D12—31no—
163C2×D12—64no—
164C2×D12—61no—
165C2×D12—92no—
166C2×D12—91no—
167C2×D12—184no—
168C2×D12—181no—
169C3×D8—184no—
170C2×C3×C8—32no—
171C2×C3×C8—32no—
172C2×C3×C8—62no—
173C2×C3×C8C2×C3×D312yes—
174C3×C4⋊C4—91no—
175C3×C22⋊C4—181no—
176(C2×D3)⋊C4—61no—
177(C2×D3)⋊C4—181no—
178C4⋊Dic3—31no—
179C4⋊Dic3—91no—
180D24—64no—
181D24—184no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—22no—
187C22×C32—21no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32D2412yescomplemented
191C22×C32C3×D812yescomplemented
192C22×C32C2×D1211yescharacteristic
193C22×C32C2×C3×D411yescharacteristic
194C2×C3⋊D3—24no—
195C2×C3⋊D3—48no—
196C2×C3⋊D3—42no—
197C2×C3⋊D3—42no—
198C2×C3⋊D3—41no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3×D3—64no—
202C2×C3×D3—128no—
203C2×C3×D3—122no—
204C2×C3×D3—122no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—22no—
211C32×C4—41no—
212C32×C4D3×D412yes—
213C32×C4C2×C3×D412yes—
214C3⋊Dic3—41no—
215C3×Dic3—121no—
216C22×D4—91no—
217C2×D8—92no—
218#121—91no—
219#102—92no—
220C22⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
221H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
222C23×C3—31no—
223C23×C3—31no—
224C23×C3—61no—
225C23×C3—182no—
226C23×C3—181no—
227C23×C3—184no—
228C23×C3—362no—
229C23×C3—362no—
230C23×C3C3×D1211yescharacteristic
231C22×D3—64no—
232C22×D3—62no—
233C22×D3—61no—
234C22×D3—122no—
235C22×D3—122no—
236C22×D3—182no—
237C22×D3—184no—
238C22×D3—181no—
239C22×D3—362no—
240C22×D3—362no—
241C3×D4—94no—
242C3×D4—182no—
243C3×D4—182no—
244C3×D4—184no—
245C3×D4—362no—
246C2×C3×C4—22no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—122no—
256C2×C3×C4—181no—
257C2×C3×C4C3×D1211yescharacteristic
258C2×C3×C4C22×C3×D311yescharacteristic
259#67—122no—
260#67—362no—
261C2×Dic3—61no—
262C2×Dic3—181no—
263D12—34no—
264D12—64no—
265D12—62no—
266D12—62no—
267D12—94no—
268D12—182no—
269D12—182no—
270D12—184no—
271C3×C8—22no—
272C3×C8—62no—
273C3×C8—62no—
274C3×C8—122no—
275C2×C32—22no—
276C2×C32—22no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—42no—
281C2×C32#88211yescharacteristic
282C2×C32C2×D2411yescharacteristic
283C2×C32#90811yescharacteristic
284C2×C32C3×#12011yescharacteristic
285C2×C32C2×C3×D811yescharacteristic
286C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
287C3⋊D3—44no—
288C3⋊D3—81no—
289C3×D3—124no—
290C3×D3—241no—
291C24—181no—
292C2×D4—92no—
293C2×D4—91no—
294C2×D4—184no—
295C2×D4—181no—
296C2×D4—181no—
297C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
298D8—184no—
299C2×C8C2×(C3⋊D3)⋊C312yescomplemented
300C4⋊C4—91no—
301C22⋊C4—181no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—31no—
305C22×C3—31no—
306C22×C3—32no—
307C22×C3—32no—
308C22×C3—61no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—62no—
314C22×C3—121no—
315C22×C3—121no—
316C22×C3—184no—
317C22×C3—362no—
318C22×C3—362no—
319C22×C3—361no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3—368no—
323C22×C3C3×D2412yes—
324C22×C3C2×C3×D1211yescharacteristic
325C2×D3—64no—
326C2×D3—128no—
327C2×D3—122no—
328C2×D3—122no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—184no—
333C2×D3—362no—
334C2×D3—362no—
335C2×D3—368no—
336C2×D3—361no—
337C2×D3—361no—
338C2×D3—361no—
339C3×C4—21no—
340C3×C4—32no—
341C3×C4—32no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—62no—
345C3×C4—121no—
346C3×C4—361no—
347C3×C4C3×D3×D412yes—
348Dic3—121no—
349Dic3—361no—
350C32—21no—
351C32#473211yescharacteristic, complemented
352C32C3×#44711yescharacteristic, complemented
353C23—184no—
354C23—182no—
355C23—181no—
356C23—362no—
357C23—362no—
358C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
359D4—94no—
360D4—184no—
361D4—182no—
362D4—182no—
363D4—362no—
364C2×C4—22no—
365C2×C4—181no—
366C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
367C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
368C8—22no—
369C2×C3—22no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—62no—
377C2×C3—62no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—122no—
382C2×C3—364no—
383C2×C3—721no—
384C2×C3C3×#88211yescharacteristic
385C2×C3C2×C3×D2411yescharacteristic
386C2×C3C3×#90811yescharacteristic
387D3—124no—
388D3—241no—
389D3—364no—
390D3—721no—
391C22—21no—
392C22—21no—
393C22—184no—
394C22—368no—
395C22—362no—
396C22—362no—
397C22—361no—
398C22—361no—
399C22—361no—
400C22#3745⋊C312yescomplemented
401C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
402C4—21no—
403C4—361no—
404C4D4×(C3⋊D3)⋊C312yes—
405C3—31no—
406C3—31no—
407C3—61no—
408C3C3×#473211yescharacteristic
409C2—22no—
410C2—364no—
411C2—721no—
412C2#64091⋊C311yescharacteristic
413C2C2×#3745⋊C311yescharacteristic
414C2#64117⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C3×D12
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#447
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
2. #64139⋊C3C2semidirect
3. C2×#3745⋊C3C2semidirect
4. H(3,𝔽3)×C22⋊C8C2semidirect
5. #63648⋊C3C2semidirect
6. #10592431C3semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. C2×C8×H(3,𝔽3)C22semidirect
17. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
19. C22×C4⋊(C3⋊D3)C2×C3semidirect
21. #64139C2×C3semidirect
22. C2×#3745C2×C3semidirect
26. C32×C22⋊C8D3semidirect
27. C32×C22⋊C8C2×C3semidirect
28. #63648C2×C3semidirect
44. C4⋊((C3⋊D3)⋊C3)D4semidirect
58. C2×C4⋊(C3⋊D3)C22×C3semidirect
65. C2×C32×C8C2×D3semidirect
66. C2×C32×C8C22×C3semidirect
68. C4⋊(C3⋊Dic3)C22×C3semidirect
75. C22×H(3,𝔽3)D8semidirect
130. C4⋊(C3⋊D3)C3×D4semidirect
190. C22×C32D24semidirect
191. C22×C32C3×D8semidirect
220. C22⋊C8(C3⋊D3)⋊C3semidirect
221. H(3,𝔽3)#447semidirect
299. C2×C8C2×(C3⋊D3)⋊C3semidirect
351. C32#4732semidirect
352. C32C3×#447semidirect
400. C22#3745⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
#64139⋊C32
C2×#3745⋊C32
H(3,𝔽3)×C22⋊C82
#63648⋊C32
#105924313
C3×#47323