Finite Group Databasebeta

← #10596050⋊C3 #13152720

Subgroups — #10596050⋊C3 #13152720

Order 1728.

868 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64155⋊C3C212yesmaximal, complemented
3#64105⋊C3C214yesmaximal, complemented
4C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
5C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
6C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal
7#64079⋊C3C212yesmaximal, complemented
8#64078⋊C3C211yesmaximal, characteristic, complemented
9#10596050C311yesmaximal, characteristic, complemented
10C3×#5579—31nomaximal
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×C4×(C3⋊D3)⋊C3C2214yes—
15C2×#634⋊C3C2212yes—
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2212yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19C42×H(3,𝔽3)C2212yes—
20#3752⋊C3C2214yes—
21#3752⋊C3C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2214yes—
26#3750⋊C3C2214yes—
27C32×C4×Q8—21no—
28C32×C4×Q8D311yescharacteristic, complemented
29C32×C4×Q8C2×C311yescharacteristic, complemented
30#64155C2×C312yescomplemented
31#64105C2×C314yescomplemented
32C4⋊C4×C3⋊D3C2×C312yescomplemented
33C4×C4⋊(C3⋊D3)C2×C312yes—
34C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
35#64079C2×C312yescomplemented
36#64078C2×C311yescharacteristic, complemented
37C3×#946—32no—
38C3×#896—34no—
39C3×D3×C4⋊C4—32no—
40C3×C4×D12—32no—
41C3×C4×D12—31no—
42C3×#870—32no—
43C3×#869—31no—
44C22×(C3⋊D3)⋊C3C2312yes—
45Q8×H(3,𝔽3)—22no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2312yes—
48C2×C4×H(3,𝔽3)C2312yes—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)Q814yescomplemented
54C4×(C3⋊D3)⋊C3—24no—
55C4×(C3⋊D3)⋊C3—44no—
56#634⋊C3—42no—
57C3×#554—91no—
58#5579—31no—
59#5579—91no—
60C2×C32×Q8—21no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×C32×Q8C22×C311yescharacteristic
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×C4×C3⋊D3C22×C312yes—
65C2×C4×C3⋊D3C22×C314yes—
66C2×#634C22×C312yes—
67C2×C3×D12—31no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—34no—
70C2×C3×Dic6—32no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—22no—
73C32×C4⋊C4C2×D311yescharacteristic
74C32×C4⋊C4C2×D312yes—
75C32×C4⋊C4C22×C311yescharacteristic
76C32×C4⋊C4C22×C312yes—
77C32×C42—21no—
78C32×C42—22no—
79C32×C42C2×D311yescharacteristic
80C32×C42C2×D312yes—
81C32×C42C22×C311yescharacteristic
82C32×C42C22×C312yes—
83#3752C22×C312yes—
84#3752C22×C314yes—
85C4⋊(C3⋊Dic3)C22×C312yes—
86C4⋊(C3⋊Dic3)C22×C312yes—
87C4⋊(C3⋊Dic3)C22×C311yescharacteristic
88#3750C22×C314yes—
89#3750C22×C314yes—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—34no—
92C3×C4⋊Dic3—32no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—31no—
95C3×Dic3⋊C4—34no—
96C3×Dic3⋊C4—34no—
97C22×H(3,𝔽3)C2411yescharacteristic
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—22no—
100C2×(C3⋊D3)⋊C3C2×Q814yes—
101C4×H(3,𝔽3)—22no—
102C4×H(3,𝔽3)—21no—
103C4×H(3,𝔽3)—22no—
104C4×H(3,𝔽3)C2×Q812yes—
105C4×H(3,𝔽3)#4112yes—
106(C3⋊Dic3)⋊C3—22no—
107(C3⋊Dic3)⋊C3—24no—
108C3×C4⋊Q8—91no—
109C3×#125—92no—
110C3×C22⋊Q8—94no—
111C3×C22⋊Q8—92no—
112C3×C4×Q8—31no—
113C3×C4×Q8—31no—
114C3×C4×Q8—61no—
115C3×C4×Q8C3×D311yescharacteristic
116C3×C4×D4—92no—
117C3×C4×D4—91no—
118C2×C3×C4⋊C4—92no—
119#946—32no—
120#946—92no—
121#896—34no—
122#896—94no—
123D3×C4⋊C4—32no—
124D3×C4⋊C4—92no—
125C4×D12—32no—
126C4×D12—31no—
127C4×D12—92no—
128C4×D12—91no—
129#870—32no—
130#870—92no—
131#869—31no—
132#869—91no—
133C22×C3⋊D3C23×C312yes—
134C22×C3×D3—32no—
135C32×Q8—24no—
136C32×Q8—22no—
137C32×Q8—22no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4—21no—
143C2×C32×C4C22×D312yes—
144C2×C32×C4C22×D312yes—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C23×C312yes—
149C2×C32×C4C23×C312yes—
150C2×C32×C4C23×C311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152C2×C32×C4C23×C311yescharacteristic
153C2×C3⋊Dic3C23×C312yes—
154C2×C3⋊Dic3C23×C314yes—
155C4⋊(C3⋊D3)C3×Q814yescomplemented
156C4×C3⋊D3—24no—
157C4×C3⋊D3—44no—
158#634—42no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—34no—
161C3×D12—34no—
162C3×C4×D3—64no—
163C3×C4×D3—124no—
164C3×Dic6—122no—
165#554—91noSylow
166C2×H(3,𝔽3)C22×Q811yescharacteristic
167C2×H(3,𝔽3)C2×#4111yescharacteristic
168C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
169(C3⋊D3)⋊C3—24no—
170C2×C3×Q8—31no—
171C2×C3×Q8—31no—
172C2×C3×Q8—61no—
173C2×C3×Q8—92no—
174C2×C3×Q8C2×C3×D311yescharacteristic
175C2×C3×D4—91no—
176C22×C3×C4—92no—
177C22×C3×C4—94no—
178C2×D12—31no—
179C2×D12—91no—
180C2×C4×D3—32no—
181C2×C4×D3—34no—
182C2×C4×D3—92no—
183C2×C4×D3—94no—
184C2×Dic6—32no—
185C2×Dic6—92no—
186C3×C4⋊C4—31no—
187C3×C4⋊C4—32no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—32no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—62no—
192C3×C4⋊C4—94no—
193C3×C4⋊C4—94no—
194C3×C4⋊C4—92no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4—92no—
197C3×C4⋊C4C2×C3×D311yescharacteristic
198C3×C4⋊C4C2×C3×D312yes—
199C3×C22⋊C4—92no—
200C3×C22⋊C4—94no—
201C3×C42—31no—
202C3×C42—32no—
203C3×C42—32no—
204C3×C42—31no—
205C3×C42—62no—
206C3×C42—61no—
207C3×C42C2×C3×D311yescharacteristic
208C3×C42C2×C3×D312yes—
209(C2×D3)⋊C4—32no—
210(C2×D3)⋊C4—34no—
211(C2×D3)⋊C4—92no—
212(C2×D3)⋊C4—94no—
213C4⋊Dic3—32no—
214C4⋊Dic3—32no—
215C4⋊Dic3—31no—
216C4⋊Dic3—92no—
217C4⋊Dic3—92no—
218C4⋊Dic3—91no—
219Dic3⋊C4—34no—
220Dic3⋊C4—34no—
221Dic3⋊C4—94no—
222Dic3⋊C4—94no—
223C22×C32—21no—
224C22×C32C23×D311yescharacteristic
225C22×C32C24×C311yescharacteristic
226C2×C3⋊D3—22no—
227C2×C3⋊D3—22no—
228C2×C3⋊D3C2×C3×Q814yes—
229C2×C3×D3—34no—
230C2×C3×D3—62no—
231C2×C3×D3—62no—
232C32×C4—24no—
233C32×C4—22no—
234C32×C4—21no—
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—21no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—22no—
241C32×C4—41no—
242C32×C4—42no—
243C32×C4#23512yes—
244C32×C4D3×Q812yes—
245C32×C4C2×C3×Q812yes—
246C32×C4C3×#4112yes—
247C3⋊Dic3—22no—
248C3⋊Dic3—24no—
249C3×Dic3—62no—
250C3×Dic3—64no—
251C4⋊Q8—91no—
252#125—92no—
253C22⋊Q8—92no—
254C22⋊Q8—94no—
255C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
256C4×D4—92no—
257C4×D4—91no—
258C2×C4⋊C4—92no—
259H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
260C23×C3—92no—
261C22×D3—32no—
262C22×D3—92no—
263C3×Q8—22no—
264C3×Q8—62no—
265C3×Q8—62no—
266C3×Q8—64no—
267C3×Q8—362no—
268C3×D4—94no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—32no—
275C2×C3×C4—32no—
276C2×C3×C4—32no—
277C2×C3×C4—32no—
278C2×C3×C4—31no—
279C2×C3×C4—61no—
280C2×C3×C4—62no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—62no—
284C2×C3×C4—92no—
285C2×C3×C4—94no—
286C2×C3×C4—184no—
287C2×C3×C4—364no—
288C2×C3×C4C22×C3×D312yes—
289C2×C3×C4C22×C3×D312yes—
290C2×C3×C4C22×C3×D311yescharacteristic
291C2×C3×C4C22×C3×D311yescharacteristic
292C2×C3×C4C22×C3×D311yescharacteristic
293C2×Dic3—32no—
294C2×Dic3—34no—
295C2×Dic3—92no—
296C2×Dic3—94no—
297D12—34no—
298D12—94no—
299C4×D3—64no—
300C4×D3—124no—
301C4×D3—184no—
302C4×D3—364no—
303Dic6—122no—
304Dic6—362no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32C2×#23511yescharacteristic
309C2×C32C2×D3×Q811yescharacteristic
310C2×C32#100911yescharacteristic
311C2×C32C22×C3×Q811yescharacteristic
312C2×C32C2×C3×#4111yescharacteristic
313C2×C32C3×H(5,𝔽2)11yescharacteristic
314C3⋊D3—24no—
315C3×D3—64no—
316C2×Q8—92no—
317C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
318C2×D4—91no—
319C22×C4—92no—
320C22×C4—94no—
321C4⋊C4—94no—
322C4⋊C4—92no—
323C4⋊C4—91no—
324C4⋊C4—92no—
325C4⋊C4—94no—
326C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
327C4⋊C4C2×(C3⋊D3)⋊C312yes—
328C22⋊C4—92no—
329C22⋊C4—94no—
330C42C2×(C3⋊D3)⋊C312yes—
331C42C2×(C3⋊D3)⋊C311yescharacteristic
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—61no—
335C22×C3—94no—
336C22×C3—182no—
337C22×C3—182no—
338C22×C3C23×C3×D311yescharacteristic
339C2×D3—34no—
340C2×D3—62no—
341C2×D3—62no—
342C2×D3—94no—
343C2×D3—182no—
344C2×D3—182no—
345C3×C4—22no—
346C3×C4—21no—
347C3×C4—22no—
348C3×C4—32no—
349C3×C4—32no—
350C3×C4—32no—
351C3×C4—32no—
352C3×C4—61no—
353C3×C4—62no—
354C3×C4—62no—
355C3×C4—62no—
356C3×C4—62no—
357C3×C4—61no—
358C3×C4—64no—
359C3×C4—62no—
360C3×C4—62no—
361C3×C4—122no—
362C3×C4—121no—
363C3×C4—182no—
364C3×C4—184no—
365C3×C4C3×#23512yes—
366C3×C4C3×D3×Q812yes—
367Dic3—62no—
368Dic3—64no—
369Dic3—184no—
370Dic3—182no—
371C32—21no—
372C32#557911yescharacteristic, complemented
373C32C3×#55411yescharacteristic, complemented
374C23—92no—
375Q8—22no—
376Q8—362no—
377D4—94no—
378C2×C4—92no—
379C2×C4—94no—
380C2×C4—184no—
381C2×C4—364no—
382C2×C4C22×(C3⋊D3)⋊C312yes—
383C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
384C2×C4C22×(C3⋊D3)⋊C312yes—
385C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
386C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—184no—
397C2×C3C2×C3×#23511yescharacteristic
398C2×C3C2×C3×D3×Q811yescharacteristic
399C2×C3C3×#100911yescharacteristic
400D3—64no—
401D3—184no—
402C22—94no—
403C22—182no—
404C22—182no—
405C22C23×(C3⋊D3)⋊C311yescharacteristic
406C4—22no—
407C4—22no—
408C4—21no—
409C4—182no—
410C4—184no—
411C4#3828⋊C312yes—
412C4Q8×(C3⋊D3)⋊C312yes—
413C3—31no—
414C3—31no—
415C3—61no—
416C3C3×#557911yescharacteristic
417C2—184no—
418C2C2×#3828⋊C311yescharacteristic
419C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
420C2#1172263811yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×H(3,𝔽3)C2semidirect
2. #64155⋊C3C2semidirect
3. #64105⋊C3C2semidirect
4. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
7. #64079⋊C3C2semidirect
8. #64078⋊C3C2semidirect
9. #10596050C3semidirect
28. C32×C4×Q8D3semidirect
29. C32×C4×Q8C2×C3semidirect
30. #64155C2×C3semidirect
31. #64105C2×C3semidirect
32. C4⋊C4×C3⋊D3C2×C3semidirect
35. #64079C2×C3semidirect
36. #64078C2×C3semidirect
53. C4⋊((C3⋊D3)⋊C3)Q8semidirect
155. C4⋊(C3⋊D3)C3×Q8semidirect
255. C4×Q8(C3⋊D3)⋊C3semidirect
259. H(3,𝔽3)#554semidirect
372. C32#5579semidirect
373. C32C3×#554semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
#64155⋊C32
#64105⋊C32
C4⋊C4×(C3⋊D3)⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#64079⋊C32
#64078⋊C32
#105960503
C3×#55793