Finite Group Databasebeta

← #10596287⋊C3 #13153662

Subgroups — #10596287⋊C3 #13153662

Order 1728.

1101 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic
3#64156⋊C3C212yesmaximal, complemented
4#64155⋊C3C214yesmaximal, complemented
5#64153⋊C3C212yesmaximal, complemented
6#64138⋊C3C214yesmaximal, complemented
7#64135⋊C3C211yesmaximal, characteristic, complemented
8#10596287C311yesmaximal, characteristic, complemented
9C3×#5816—31nomaximal
10C2×Q8×H(3,𝔽3)—24no—
11C2×Q8×H(3,𝔽3)C2212yes—
12C2×Q8×H(3,𝔽3)C2212yes—
13C22×C4×H(3,𝔽3)C2212yes—
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16#3828⋊C3—24no—
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20#3757⋊C3C2212yes—
21#3752⋊C3C2218yes—
22C4⋊((C3⋊Dic3)⋊C3)C2212yes—
23#3750⋊C3C2218yes—
24C4×(C3⋊Dic3)⋊C3C2212yes—
25C22×C32×Q8—21no—
26C22×C32×Q8D311yescharacteristic, complemented
27C22×C32×Q8C2×C311yescharacteristic, complemented
28C2×#3828C2×C311yescharacteristic
29C2×C3×#235—31no—
30#64156C2×C312yescomplemented
31#64155C2×C314yescomplemented
32#64153C2×C312yescomplemented
33#64138C2×C314yescomplemented
34#64135C2×C311yescharacteristic, complemented
35C3×#947—32no—
36C3×#946—34no—
37C3×#944—32no—
38C3×#929—34no—
39C3×#926—31no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×(C3⋊D3)⋊C3C2312yes—
42Q8×H(3,𝔽3)—24no—
43Q8×H(3,𝔽3)—24no—
44C2×C4×H(3,𝔽3)—24no—
45C2×C4×H(3,𝔽3)D414yes—
46C2×C4×H(3,𝔽3)C2314yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49#638⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3C2312yes—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C4⋊((C3⋊D3)⋊C3)—22no—
53C4×(C3⋊D3)⋊C3—24no—
54#634⋊C3—22no—
55C3×#536—91no—
56#5816—31no—
57#5816—91no—
58C2×C32×Q8—28no—
59C2×C32×Q8—22no—
60C2×C32×Q8—22no—
61C2×C32×Q8—24no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D312yes—
64C2×C32×Q8C2×D312yes—
65C2×C32×Q8C22×C312yes—
66C2×C32×Q8C22×C312yes—
67C22×C32×C4—22no—
68C22×C32×C4—21no—
69C22×C32×C4C2×D312yes—
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C22×C312yes—
72C22×C32×C4C22×C311yescharacteristic
73C2×#638C22×C312yes—
74#3828—24no—
75C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
76C2×C4×C3⋊D3C22×C312yes—
77C2×#634C22×C311yescharacteristic
78C2×C3×#67—32no—
79C3×#235—64no—
80C2×C3×D12—31no—
81C2×C3×C4×D3—32no—
82C2×C3×Dic6—31no—
83#3757C22×C312yes—
84#3752C22×C318yes—
85C4⋊(C3⋊Dic3)C22×C312yes—
86#3750C22×C318yes—
87C4×C3⋊Dic3C22×C312yes—
88C3×#217—32no—
89C3×(C2×D3)⋊C4—38no—
90C3×C4⋊Dic3—32no—
91C3×Dic3⋊C4—38no—
92C3×C4×Dic3—32no—
93C22×H(3,𝔽3)—22no—
94C22×H(3,𝔽3)C2×D412yes—
95C22×H(3,𝔽3)C2411yescharacteristic
96C2×(C3⋊D3)⋊C3—22no—
97C2×(C3⋊D3)⋊C3—24no—
98C4×H(3,𝔽3)—24no—
99C4×H(3,𝔽3)C2×D412yes—
100C4×H(3,𝔽3)C2×D412yes—
101(C3⋊Dic3)⋊C3—22no—
102(C3⋊Dic3)⋊C3—24no—
103C2×C3×#41—91no—
104C22×C3×Q8—31no—
105C22×C3×Q8—31no—
106C22×C3×Q8—61no—
107C22×C3×Q8C3×D311yescharacteristic
108C2×#235—31no—
109C2×#235—91no—
110C3×C4⋊Q8—92no—
111C3×#124—92no—
112C3×#123—94no—
113C3×C22⋊Q8—94no—
114C3×#117—91no—
115#947—32no—
116#947—92no—
117#946—34no—
118#946—94no—
119#944—32no—
120#944—92no—
121#929—34no—
122#929—94no—
123#926—31no—
124#926—91no—
125C23×C32—21no—
126C23×C32C22×D311yescharacteristic
127C23×C32C23×C311yescharacteristic
128C22×C3⋊D3C23×C312yes—
129C22×C3×D3—32no—
130C32×Q8—28no—
131C32×Q8—28no—
132C32×Q8—24no—
133C32×Q8—24no—
134C32×Q8—24no—
135C32×Q8—24no—
136C2×C32×C4—28no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4#6714yes—
144C2×C32×C4C3×D414yes—
145C2×C32×C4C22×D314yes—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C23×C314yes—
149C2×C32×C4C23×C311yescharacteristic
150C2×C32×C4C23×C311yescharacteristic
151#638—24no—
152C2×C3⋊Dic3C23×C314yes—
153C2×C3⋊Dic3C23×C312yes—
154C4⋊(C3⋊D3)—22no—
155C4×C3⋊D3—24no—
156#634—22no—
157C3×#67—64no—
158C2×C3×Dic3—34no—
159C2×C3×Dic3—32no—
160C3×D12—62no—
161C3×C4×D3—64no—
162C3×Dic6—62no—
163#536—91noSylow
164C2×H(3,𝔽3)—22no—
165C2×H(3,𝔽3)C22×D411yescharacteristic
166C2×H(3,𝔽3)2−1+412yes—
167(C3⋊D3)⋊C3—42no—
168C3×#41—184no—
169C2×C3×Q8—24no—
170C2×C3×Q8—32no—
171C2×C3×Q8—32no—
172C2×C3×Q8—32no—
173C2×C3×Q8—32no—
174C2×C3×Q8—64no—
175C2×C3×Q8—64no—
176C2×C3×Q8—68no—
177C2×C3×Q8—62no—
178C2×C3×Q8—62no—
179C2×C3×Q8—91no—
180C2×C3×Q8C2×C3×D312yes—
181C2×C3×Q8C2×C3×D312yes—
182C2×C3×D4—91no—
183C2×C3×D4—92no—
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—62no—
189C22×C3×C4—61no—
190C22×C3×C4—92no—
191C22×C3×C4C2×C3×D312yes—
192C22×C3×C4C2×C3×D311yescharacteristic
193C2×#67—32no—
194C2×#67—92no—
195#235—64no—
196#235—184no—
197C2×D12—31no—
198C2×D12—91no—
199C2×C4×D3—32no—
200C2×C4×D3—92no—
201C2×Dic6—31no—
202C2×Dic6—91no—
203C3×C4⋊C4—98no—
204C3×C4⋊C4—92no—
205C3×C22⋊C4—98no—
206C3×C22⋊C4—92no—
207C3×C42—92no—
208#217—32no—
209#217—92no—
210(C2×D3)⋊C4—38no—
211(C2×D3)⋊C4—98no—
212C4⋊Dic3—32no—
213C4⋊Dic3—92no—
214Dic3⋊C4—38no—
215Dic3⋊C4—98no—
216C4×Dic3—32no—
217C4×Dic3—92no—
218C22×C32—24no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32C2×#6712yes—
224C22×C32C2×C3×D412yes—
225C22×C32C23×D311yescharacteristic
226C22×C32C24×C311yescharacteristic
227C2×C3⋊D3—22no—
228C2×C3⋊D3—24no—
229C2×C3×D3—64no—
230C2×C3×D3—62no—
231C32×C4—28no—
232C32×C4—24no—
233C32×C4—24no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4C2×#6712yes—
237C32×C4C2×#6712yes—
238C32×C4C2×C3×D412yes—
239C32×C4C2×C3×D412yes—
240C3⋊Dic3—22no—
241C3⋊Dic3—24no—
242C3×Dic3—64no—
243C3×Dic3—62no—
244C2×#41—91no—
245C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
246C4⋊Q8—92no—
247#124—92no—
248#123—94no—
249C22⋊Q8—94no—
250#117—91no—
251H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
252C23×C3—31no—
253C23×C3—31no—
254C23×C3—61no—
255C23×C3—92no—
256C23×C3C22×C3×D311yescharacteristic
257C22×D3—32no—
258C22×D3—92no—
259C3×Q8—24no—
260C3×Q8—24no—
261C3×Q8—64no—
262C3×Q8—64no—
263C3×Q8—64no—
264C3×Q8—64no—
265C3×Q8—68no—
266C3×Q8—68no—
267C3×Q8—182no—
268C3×D4—182no—
269C3×D4—184no—
270C2×C3×C4—24no—
271C2×C3×C4—34no—
272C2×C3×C4—34no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—34no—
278C2×C3×C4—34no—
279C2×C3×C4—64no—
280C2×C3×C4—64no—
281C2×C3×C4—68no—
282C2×C3×C4—64no—
283C2×C3×C4—64no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—94no—
287C2×C3×C4—92no—
288C2×C3×C4—184no—
289C2×C3×C4C3×#6714yes—
290C2×C3×C4C22×C3×D314yes—
291C2×C3×C4C22×C3×D311yescharacteristic
292C2×C3×C4C22×C3×D311yescharacteristic
293#67—64no—
294#67—184no—
295C2×Dic3—34no—
296C2×Dic3—32no—
297C2×Dic3—94no—
298C2×Dic3—92no—
299D12—62no—
300D12—182no—
301C4×D3—64no—
302C4×D3—184no—
303Dic6—62no—
304Dic6—182no—
305C2×C32—24no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32#100712yes—
311C2×C32C22×#6711yescharacteristic
312C2×C32C22×C3×D411yescharacteristic
313C2×C32C3×2−1+412yes—
314C3⋊D3—42no—
315C3×D3—122no—
316#41—184no—
317C2×Q8—24no—
318C2×Q8—91no—
319C2×Q8C2×(C3⋊D3)⋊C312yes—
320C2×Q8C2×(C3⋊D3)⋊C312yes—
321C2×D4—92no—
322C2×D4—91no—
323C22×C4—92no—
324C22×C4C2×(C3⋊D3)⋊C312yes—
325C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
326C4⋊C4—92no—
327C4⋊C4—98no—
328C22⋊C4—92no—
329C22⋊C4—98no—
330C42—92no—
331C22×C3—22no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—32no—
335C22×C3—32no—
336C22×C3—62no—
337C22×C3—62no—
338C22×C3—64no—
339C22×C3—61no—
340C22×C3—62no—
341C22×C3—182no—
342C22×C3—184no—
343C22×C3C2×C3×#6712yes—
344C22×C3C23×C3×D311yescharacteristic
345C2×D3—64no—
346C2×D3—62no—
347C2×D3—182no—
348C2×D3—184no—
349C3×C4—24no—
350C3×C4—32no—
351C3×C4—32no—
352C3×C4—32no—
353C3×C4—32no—
354C3×C4—64no—
355C3×C4—64no—
356C3×C4—68no—
357C3×C4—62no—
358C3×C4—62no—
359C3×C4—182no—
360C3×C4—184no—
361C3×C4C2×C3×#6712yes—
362C3×C4C2×C3×#6712yes—
363Dic3—64no—
364Dic3—62no—
365Dic3—182no—
366Dic3—184no—
367C32—21no—
368C32#581611yescharacteristic, complemented
369C32C3×#53611yescharacteristic, complemented
370C23—92no—
371C23C22×(C3⋊D3)⋊C311yescharacteristic
372Q8—24no—
373Q8—24no—
374Q8—182no—
375D4—184no—
376D4—182no—
377C2×C4—24no—
378C2×C4—94no—
379C2×C4—92no—
380C2×C4—184no—
381C2×C4#638⋊C314yes—
382C2×C4C22×(C3⋊D3)⋊C314yes—
383C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
384C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
385C2×C3—22no—
386C2×C3—32no—
387C2×C3—32no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—62no—
391C2×C3—62no—
392C2×C3—64no—
393C2×C3—62no—
394C2×C3—61no—
395C2×C3—362no—
396C2×C3C3×#100712yes—
397C2×C3C22×C3×#6711yescharacteristic
398D3—122no—
399D3—362no—
400C22—22no—
401C22—184no—
402C22—182no—
403C22C2×#638⋊C312yes—
404C22C23×(C3⋊D3)⋊C311yescharacteristic
405C4—24no—
406C4—184no—
407C4—182no—
408C4C2×#638⋊C312yes—
409C4C2×#638⋊C312yes—
410C3—31no—
411C3—31no—
412C3—61no—
413C3C3×#581611yescharacteristic
414C2—22no—
415C2—362no—
416C2#1172262712yes—
417C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
prime radicalC2×#3828⋊C3C2
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×H(3,𝔽3)C2semidirect
3. #64156⋊C3C2semidirect
4. #64155⋊C3C2semidirect
5. #64153⋊C3C2semidirect
6. #64138⋊C3C2semidirect
7. #64135⋊C3C2semidirect
8. #10596287C3semidirect
26. C22×C32×Q8D3semidirect
27. C22×C32×Q8C2×C3semidirect
30. #64156C2×C3semidirect
31. #64155C2×C3semidirect
32. #64153C2×C3semidirect
33. #64138C2×C3semidirect
34. #64135C2×C3semidirect
245. C22×Q8(C3⋊D3)⋊C3semidirect
251. H(3,𝔽3)#536semidirect
368. C32#5816semidirect
369. C32C3×#536semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C2×#3828⋊C32
#64156⋊C32
#64155⋊C32
#64153⋊C32
#64138⋊C32
#64135⋊C32
#105962873
C3×#58163