Finite Group Databasebeta

← #10596201⋊C3 #13153326

Subgroups — #10596201⋊C3 #13153326

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64155⋊C3C214yesmaximal, complemented
3#64106⋊C3C212yesmaximal, complemented
4#64102⋊C3C212yesmaximal
5C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
6#64099⋊C3C212yesmaximal, complemented
7C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
8C4×#634⋊C3C211yesmaximal, characteristic, complemented
9#10596201C311yesmaximal, characteristic, complemented
10C3×#5730—31nomaximal
11C2×Q8×H(3,𝔽3)C2212yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×C4×(C3⋊D3)⋊C3C2214yes—
15C2×#634⋊C3C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2212yes—
17C4⋊C4×H(3,𝔽3)C2212yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19#3752⋊C3C2212yes—
20#3752⋊C3C2214yes—
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2214yes—
24#3750⋊C3C2212yes—
25#3750⋊C3C2214yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27C32×C4⋊Q8—21no—
28C32×C4⋊Q8D311yescharacteristic, complemented
29C32×C4⋊Q8C2×C311yescharacteristic, complemented
30#64155C2×C314yescomplemented
31#64106C2×C312yescomplemented
32#64102C2×C312yes—
33C4⋊C4×C3⋊D3C2×C312yescomplemented
34#64099C2×C312yescomplemented
35C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
36C4×#634C2×C311yescharacteristic, complemented
37C3×#946—34no—
38C3×#897—32no—
39C3×#893—32no—
40C3×D3×C4⋊C4—32no—
41C3×#890—32no—
42C3×C4×D12—31no—
43C3×C4×Dic6—31no—
44C22×(C3⋊D3)⋊C3C2312yes—
45Q8×H(3,𝔽3)—42no—
46C2×C4×H(3,𝔽3)C2314yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×(C3⋊Dic3)⋊C3C2312yes—
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)Q814yescomplemented
54C4×(C3⋊D3)⋊C3—24no—
55C4×(C3⋊D3)⋊C3—44no—
56#634⋊C3—22no—
57C3×#554—91no—
58#5730—31no—
59#5730—91no—
60C2×C32×Q8—22no—
61C2×C32×Q8C2×D312yes—
62C2×C32×Q8C22×C312yes—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×C4×C3⋊D3C22×C312yes—
65C2×C4×C3⋊D3C22×C314yes—
66C2×#634C22×C311yescharacteristic
67C2×C3×D12—31no—
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—34no—
70C2×C3×Dic6—31no—
71C32×C4⋊C4—22no—
72C32×C4⋊C4—22no—
73C32×C4⋊C4C2×D312yes—
74C32×C4⋊C4C2×D312yes—
75C32×C4⋊C4C22×C312yes—
76C32×C4⋊C4C22×C312yes—
77C32×C42—21no—
78C32×C42C2×D311yescharacteristic
79C32×C42C22×C311yescharacteristic
80#3752C22×C312yes—
81#3752C22×C314yes—
82C4⋊(C3⋊Dic3)C22×C311yescharacteristic
83C4⋊(C3⋊Dic3)C22×C311yescharacteristic
84C4⋊(C3⋊Dic3)C22×C314yes—
85#3750C22×C312yes—
86#3750C22×C314yes—
87C4×C3⋊Dic3C22×C312yes—
88C3×(C2×D3)⋊C4—34no—
89C3×(C2×D3)⋊C4—32no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—34no—
92C3×C4⋊Dic3—31no—
93C3×Dic3⋊C4—34no—
94C3×Dic3⋊C4—32no—
95C3×C4×Dic3—32no—
96C22×H(3,𝔽3)C2411yescharacteristic
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3C2×Q814yes—
100C4×H(3,𝔽3)—21no—
101C4×H(3,𝔽3)—24no—
102C4×H(3,𝔽3)C2×Q812yes—
103C4×H(3,𝔽3)#4112yes—
104(C3⋊Dic3)⋊C3—22no—
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—22no—
107C3×C4⋊Q8—31no—
108C3×C4⋊Q8—31no—
109C3×C4⋊Q8—61no—
110C3×C4⋊Q8C3×D311yescharacteristic
111C3×#125—92no—
112C3×C22⋊Q8—92no—
113C3×C22⋊Q8—94no—
114C3×C4×Q8—91no—
115C3×C4×D4—91no—
116C3×C4×D4—92no—
117C2×C3×C4⋊C4—92no—
118#946—34no—
119#946—94no—
120#897—32no—
121#897—92no—
122#893—32no—
123#893—92no—
124D3×C4⋊C4—32no—
125D3×C4⋊C4—92no—
126#890—32no—
127#890—92no—
128C4×D12—31no—
129C4×D12—91no—
130C4×Dic6—31no—
131C4×Dic6—91no—
132C22×C3⋊D3C23×C312yes—
133C22×C3×D3—32no—
134C32×Q8—42no—
135C32×Q8—42no—
136C32×Q8—44no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—24no—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C23×C311yescharacteristic
146C2×C32×C4C23×C314yes—
147C2×C32×C4C23×C311yescharacteristic
148C2×C32×C4C23×C311yescharacteristic
149C2×C3⋊Dic3C23×C312yes—
150C2×C3⋊Dic3C23×C312yes—
151C2×C3⋊Dic3C23×C312yes—
152C4⋊(C3⋊D3)C3×Q814yescomplemented
153C4×C3⋊D3—24no—
154C4×C3⋊D3—44no—
155#634—22no—
156C2×C3×Dic3—32no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—32no—
159C3×D12—34no—
160C3×C4×D3—64no—
161C3×C4×D3—124no—
162C3×Dic6—62no—
163#554—91noSylow
164C2×H(3,𝔽3)C22×Q811yescharacteristic
165C2×H(3,𝔽3)C2×#4111yescharacteristic
166C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
167(C3⋊D3)⋊C3—24no—
168C2×C3×Q8—32no—
169C2×C3×Q8—32no—
170C2×C3×Q8—62no—
171C2×C3×Q8—91no—
172C2×C3×Q8C2×C3×D312yes—
173C2×C3×D4—91no—
174C22×C3×C4—92no—
175C22×C3×C4—94no—
176C2×D12—31no—
177C2×D12—91no—
178C2×C4×D3—32no—
179C2×C4×D3—34no—
180C2×C4×D3—92no—
181C2×C4×D3—94no—
182C2×Dic6—31no—
183C2×Dic6—91no—
184C3×C4⋊C4—32no—
185C3×C4⋊C4—32no—
186C3×C4⋊C4—32no—
187C3×C4⋊C4—32no—
188C3×C4⋊C4—62no—
189C3×C4⋊C4—62no—
190C3×C4⋊C4—94no—
191C3×C4⋊C4—92no—
192C3×C4⋊C4—91no—
193C3×C4⋊C4—91no—
194C3×C4⋊C4—94no—
195C3×C4⋊C4C2×C3×D312yes—
196C3×C4⋊C4C2×C3×D312yes—
197C3×C22⋊C4—92no—
198C3×C22⋊C4—94no—
199C3×C42—31no—
200C3×C42—31no—
201C3×C42—61no—
202C3×C42—92no—
203C3×C42C2×C3×D311yescharacteristic
204(C2×D3)⋊C4—32no—
205(C2×D3)⋊C4—34no—
206(C2×D3)⋊C4—92no—
207(C2×D3)⋊C4—94no—
208C4⋊Dic3—31no—
209C4⋊Dic3—31no—
210C4⋊Dic3—34no—
211C4⋊Dic3—91no—
212C4⋊Dic3—91no—
213C4⋊Dic3—94no—
214Dic3⋊C4—32no—
215Dic3⋊C4—34no—
216Dic3⋊C4—92no—
217Dic3⋊C4—94no—
218C4×Dic3—32no—
219C4×Dic3—92no—
220C22×C32—21no—
221C22×C32C23×D311yescharacteristic
222C22×C32C24×C311yescharacteristic
223C2×C3⋊D3—22no—
224C2×C3⋊D3—22no—
225C2×C3⋊D3C2×C3×Q814yes—
226C2×C3×D3—34no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C32×C4—22no—
230C32×C4—24no—
231C32×C4—21no—
232C32×C4—24no—
233C32×C4—21no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—44no—
237C32×C4D3×Q812yes—
238C32×C4#23912yes—
239C32×C4C2×C3×Q812yes—
240C32×C4C3×#4112yes—
241C3⋊Dic3—22no—
242C3⋊Dic3—22no—
243C3⋊Dic3—22no—
244C3×Dic3—62no—
245C3×Dic3—62no—
246C3×Dic3—62no—
247C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
248#125—92no—
249C22⋊Q8—94no—
250C22⋊Q8—92no—
251C4×Q8—91no—
252C4×D4—92no—
253C4×D4—91no—
254C2×C4⋊C4—92no—
255H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
256C23×C3—92no—
257C22×D3—32no—
258C22×D3—92no—
259C3×Q8—42no—
260C3×Q8—122no—
261C3×Q8—122no—
262C3×Q8—124no—
263C3×Q8—182no—
264C3×D4—94no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—34no—
271C2×C3×C4—31no—
272C2×C3×C4—34no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—64no—
277C2×C3×C4—92no—
278C2×C3×C4—92no—
279C2×C3×C4—92no—
280C2×C3×C4—184no—
281C2×C3×C4—364no—
282C2×C3×C4C22×C3×D311yescharacteristic
283C2×C3×C4C22×C3×D314yes—
284C2×C3×C4C22×C3×D311yescharacteristic
285C2×C3×C4C22×C3×D311yescharacteristic
286C2×Dic3—32no—
287C2×Dic3—32no—
288C2×Dic3—32no—
289C2×Dic3—92no—
290C2×Dic3—92no—
291C2×Dic3—92no—
292D12—34no—
293D12—94no—
294C4×D3—64no—
295C4×D3—124no—
296C4×D3—184no—
297C4×D3—364no—
298Dic6—62no—
299Dic6—182no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32#100411yescharacteristic
304C2×C32C2×D3×Q811yescharacteristic
305C2×C32C2×#23911yescharacteristic
306C2×C32C22×C3×Q811yescharacteristic
307C2×C32C2×C3×#4111yescharacteristic
308C2×C32C3×H(5,𝔽2)11yescharacteristic
309C3⋊D3—24no—
310C3×D3—64no—
311C2×Q8—91no—
312C2×Q8C2×(C3⋊D3)⋊C312yes—
313C2×D4—91no—
314C22×C4—92no—
315C22×C4—94no—
316C4⋊C4—91no—
317C4⋊C4—94no—
318C4⋊C4—91no—
319C4⋊C4—92no—
320C4⋊C4—94no—
321C4⋊C4C2×(C3⋊D3)⋊C312yes—
322C4⋊C4C2×(C3⋊D3)⋊C312yes—
323C22⋊C4—94no—
324C22⋊C4—92no—
325C42—92no—
326C42C2×(C3⋊D3)⋊C311yescharacteristic
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—61no—
330C22×C3—94no—
331C22×C3—182no—
332C22×C3—182no—
333C22×C3C23×C3×D311yescharacteristic
334C2×D3—34no—
335C2×D3—62no—
336C2×D3—62no—
337C2×D3—94no—
338C2×D3—182no—
339C2×D3—182no—
340C3×C4—24no—
341C3×C4—21no—
342C3×C4—32no—
343C3×C4—32no—
344C3×C4—32no—
345C3×C4—32no—
346C3×C4—64no—
347C3×C4—64no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—62no—
351C3×C4—62no—
352C3×C4—62no—
353C3×C4—124no—
354C3×C4—182no—
355C3×C4—182no—
356C3×C4—182no—
357C3×C4C3×D3×Q812yes—
358C3×C4C3×#23912yes—
359Dic3—62no—
360Dic3—62no—
361Dic3—62no—
362Dic3—182no—
363Dic3—182no—
364Dic3—182no—
365C32—21no—
366C32#573011yescharacteristic, complemented
367C32C3×#55411yescharacteristic, complemented
368C23—92no—
369Q8—42no—
370Q8—182no—
371D4—94no—
372C2×C4—92no—
373C2×C4—92no—
374C2×C4—92no—
375C2×C4—184no—
376C2×C4—364no—
377C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
378C2×C4C22×(C3⋊D3)⋊C314yes—
379C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
380C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—61no—
390C2×C3—184no—
391C2×C3C3×#100411yescharacteristic
392C2×C3C2×C3×D3×Q811yescharacteristic
393C2×C3C2×C3×#23911yescharacteristic
394D3—64no—
395D3—184no—
396C22—94no—
397C22—182no—
398C22—182no—
399C22C23×(C3⋊D3)⋊C311yescharacteristic
400C4—24no—
401C4—21no—
402C4—182no—
403C4—182no—
404C4—182no—
405C4Q8×(C3⋊D3)⋊C312yes—
406C4#8872212yes—
407C3—31no—
408C3—31no—
409C3—61no—
410C3C3×#573011yescharacteristic
411C2—184no—
412C2#64361⋊C311yescharacteristic
413C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
414C2C2×#8872211yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
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. H(3,𝔽3)×C4⋊Q8C2semidirect
2. #64155⋊C3C2semidirect
3. #64106⋊C3C2semidirect
5. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
6. #64099⋊C3C2semidirect
8. C4×#634⋊C3C2semidirect
9. #10596201C3semidirect
28. C32×C4⋊Q8D3semidirect
29. C32×C4⋊Q8C2×C3semidirect
30. #64155C2×C3semidirect
31. #64106C2×C3semidirect
33. C4⋊C4×C3⋊D3C2×C3semidirect
34. #64099C2×C3semidirect
36. C4×#634C2×C3semidirect
53. C4⋊((C3⋊D3)⋊C3)Q8semidirect
152. C4⋊(C3⋊D3)C3×Q8semidirect
247. C4⋊Q8(C3⋊D3)⋊C3semidirect
255. H(3,𝔽3)#554semidirect
366. C32#5730semidirect
367. C32C3×#554semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#64155⋊C32
#64106⋊C32
#64102⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64099⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#105962013
C3×#57303