Finite Group Databasebeta

← #10592503⋊C3 #13141917

Subgroups — #10592503⋊C3 #13141917

Order 1728.

595 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88722C211yesmaximal, characteristic
2C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#64101⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#103C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592503C311yesmaximal, characteristic, complemented
9C3×#4804—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88722—22no—
12#88722C414yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C4×(C3⋊Dic3)⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C8×(C3⋊D3)⋊C3—22no—
24C2×#3832C2×C311yescharacteristic
25C2×C3×#239—31no—
26C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
27#64101C2×C311yescharacteristic, complemented
28C2×C3×D3×C8—31no—
29C3×#892—31no—
30C32×#103—21no—
31C32×#103D311yescharacteristic, complemented
32C32×#103C2×C311yescharacteristic, complemented
33#63662C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×#835—31no—
37C3×#821—31no—
38C3×#809—31no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)C2×C412yes—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4⋊((C3⋊D3)⋊C3)—24no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)C2×C412yes—
51C4×(C3⋊D3)⋊C3—24no—
52C4×(C3⋊D3)⋊C3C2×C412yes—
53C4×(C3⋊D3)⋊C3D412yes—
54C8×H(3,𝔽3)—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56C3×#416—91no—
57#4804—31no—
58#4804—91no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C22×C311yescharacteristic
62#3832—22no—
63#3832C3×C414yescomplemented
64C2×C4⋊(C3⋊D3)—21no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C4×C3⋊D3—21no—
67C2×C4×C3⋊D3C22×C311yescharacteristic
68C3×#239—34no—
69C3×#239—62no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—61no—
74C2×C32×C8—21no—
75C2×C32×C8C2×D311yescharacteristic
76C2×C32×C8C22×C311yescharacteristic
77C32×C4⋊C4—21no—
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C4⋊C4C22×C311yescharacteristic
80#3752—21no—
81C4⋊(C3⋊Dic3)C22×C311yescharacteristic
82C4×C3⋊Dic3—21no—
83C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
84C8×C3⋊D3—22no—
85C3×(C2×D3)⋊C4—61no—
86C3×C4⋊Dic3—31no—
87C3×C4×Dic3—61no—
88C2×C3×C3⋊C8—31no—
89C3×D3×C8—62no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—22no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3C22⋊C412yes—
97C4×H(3,𝔽3)—22no—
98C4×H(3,𝔽3)—41no—
99C4×H(3,𝔽3)C22×C411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101(C3⋊Dic3)⋊C3—41no—
102(C3⋊Dic3)⋊C3C22⋊C412yes—
103C2×C3×#41—91no—
104C2×#239—31no—
105C2×#239—91no—
106C22×C3×C8—91no—
107C3×#117—91no—
108C2×D3×C8—31no—
109C2×D3×C8—91no—
110#892—31no—
111#892—91no—
112C3×#103—31no—
113C3×#103—31no—
114C3×#103—61no—
115C3×#103—91no—
116C3×#103C3×D311yescharacteristic
117C3×#102—91no—
118C3×#102—91no—
119#835—31no—
120#835—91no—
121#821—31no—
122#821—91no—
123#809—31no—
124#809—91no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3C3×D411yescharacteristic
127C22×C3×D3—31no—
128C22×C3×D3—61no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—22no—
132C32×Q8—41no—
133C32×Q8C4×D312yes—
134C32×Q8C2×C3×C412yes—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—41no—
141C2×C32×C4—41no—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C23×C311yescharacteristic
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3C3×D411yescharacteristic
146C4⋊(C3⋊D3)—24no—
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)C2×C3×C412yes—
149C4×C3⋊D3—24no—
150C4×C3⋊D3C2×C3×C412yes—
151C4×C3⋊D3C3×D412yes—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—61no—
154C3×D12—32no—
155C3×D12—64no—
156C3×D12—61no—
157C3×C4×D3—32no—
158C3×C4×D3—32no—
159C3×C4×D3—64no—
160C32×C8—21no—
161C32×C8—21no—
162C32×C8—41no—
163C3⋊(C3⋊C8)—21no—
164C3×C3⋊C8—61no—
165#416—91noSylow
166C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
167C2×H(3,𝔽3)#13511yescharacteristic
168C2×H(3,𝔽3)#13511yescharacteristic
169(C3⋊D3)⋊C3—22no—
170(C3⋊D3)⋊C3—42no—
171C3×#41—94no—
172C3×#41—182no—
173C2×C3×Q8—31no—
174C2×C3×Q8—31no—
175C2×C3×Q8—61no—
176C2×C3×Q8C2×C3×D311yescharacteristic
177C2×C3×D4—91no—
178C2×C3×D4—181no—
179C22×C3×C4—91no—
180C22×C3×C4—181no—
181#239—34no—
182#239—62no—
183#239—94no—
184#239—182no—
185C2×D12—31no—
186C2×D12—61no—
187C2×D12—91no—
188C2×D12—181no—
189C2×C4×D3—31no—
190C2×C4×D3—61no—
191C2×C4×D3—91no—
192C2×C4×D3—181no—
193C2×C3×C8—31no—
194C2×C3×C8—31no—
195C2×C3×C8—61no—
196C2×C3×C8—91no—
197C2×C3×C8—182no—
198C2×C3×C8C2×C3×D311yescharacteristic
199C3×C4⋊C4—31no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—91no—
203C3×C4⋊C4C2×C3×D311yescharacteristic
204C3×C22⋊C4—181no—
205C3×C42—181no—
206(C2×D3)⋊C4—61no—
207(C2×D3)⋊C4—181no—
208C4⋊Dic3—31no—
209C4⋊Dic3—91no—
210C4×Dic3—61no—
211C4×Dic3—181no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—91no—
214D3×C8—62no—
215D3×C8—182no—
216C22×C32—21no—
217C22×C32D3×D411yescharacteristic
218C22×C32C2×C3×D411yescharacteristic
219C2×C3⋊D3—21no—
220C2×C3⋊D3—21no—
221C2×C3⋊D3—22no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3C3×C22⋊C412yes—
225C2×C3×D3—32no—
226C2×C3×D3—61no—
227C2×C3×D3—61no—
228C2×C3×D3—62no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—42no—
236C32×C4—41no—
237C32×C4—41no—
238C32×C4—81no—
239C32×C4C2×C4×D311yescharacteristic
240C32×C4D3×D411yescharacteristic
241C32×C4C22×C3×C411yescharacteristic
242C32×C4C2×C3×D411yescharacteristic
243C3⋊Dic3—41no—
244C3⋊Dic3C3×C22⋊C412yes—
245C3×Dic3—32no—
246C3×Dic3—121no—
247C2×#41—91no—
248C22×C8—91no—
249#117—91no—
250#103—91no—
251#103(C3⋊D3)⋊C311yescharacteristic, complemented
252#102—91no—
253#102—91no—
254H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
255C23×C3—91no—
256C23×C3—181no—
257C22×D3—31no—
258C22×D3—61no—
259C22×D3—91no—
260C22×D3—181no—
261C3×Q8—21no—
262C3×Q8—32no—
263C3×Q8—32no—
264C3×Q8—61no—
265C3×Q8—61no—
266C3×Q8—62no—
267C3×Q8—121no—
268C3×Q8C3×C4×D312yes—
269C3×D4—92no—
270C3×D4—181no—
271C3×D4—184no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—92no—
282C2×C3×C4—92no—
283C2×C3×C4—91no—
284C2×C3×C4—121no—
285C2×C3×C4—121no—
286C2×C3×C4—184no—
287C2×C3×C4—181no—
288C2×C3×C4C22×C3×D311yescharacteristic
289C2×Dic3—31no—
290C2×Dic3—61no—
291C2×Dic3—91no—
292C2×Dic3—181no—
293D12—32no—
294D12—64no—
295D12—61no—
296D12—92no—
297D12—181no—
298D12—184no—
299C4×D3—32no—
300C4×D3—32no—
301C4×D3—64no—
302C4×D3—92no—
303C4×D3—92no—
304C4×D3—184no—
305C3×C8—21no—
306C3×C8—61no—
307C3×C8—61no—
308C3×C8—121no—
309C3×C8—181no—
310C3⋊C8—61no—
311C3⋊C8—181no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32D3×C22⋊C411yescharacteristic
316C2×C32#91611yescharacteristic
317C2×C32#91911yescharacteristic
318C2×C32C2×C3×C22⋊C411yescharacteristic
319C2×C32C3×#13511yescharacteristic
320C2×C32C3×#13511yescharacteristic
321C3⋊D3—22no—
322C3⋊D3—42no—
323C3×D3—62no—
324C3×D3—122no—
325#41—94no—
326#41—182no—
327C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
328C2×D4—91no—
329C2×D4—181no—
330C22×C4—91no—
331C22×C4—181no—
332C2×C8—91no—
333C2×C8—182no—
334C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
335C4⋊C4—91no—
336C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
337C22⋊C4—181no—
338C42—181no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—61no—
342C22×C3—92no—
343C22×C3—181no—
344C22×C3—181no—
345C22×C3—182no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3C3×D3×D411yescharacteristic
349C2×D3—32no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—62no—
353C2×D3—92no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—181no—
357C2×D3—181no—
358C2×D3—182no—
359C2×D3—361no—
360C2×D3—361no—
361C3×C4—22no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—31no—
365C3×C4—31no—
366C3×C4—41no—
367C3×C4—62no—
368C3×C4—62no—
369C3×C4—61no—
370C3×C4—61no—
371C3×C4—92no—
372C3×C4—122no—
373C3×C4—121no—
374C3×C4—121no—
375C3×C4—241no—
376C3×C4—361no—
377C3×C4C2×C3×C4×D311yescharacteristic
378C3×C4C3×D3×D411yescharacteristic
379Dic3—32no—
380Dic3—92no—
381Dic3—121no—
382Dic3—361no—
383C32—21no—
384C32#480411yescharacteristic, complemented
385C32C3×#41611yescharacteristic, complemented
386C23—91no—
387C23—181no—
388Q8—21no—
389Q8C4×(C3⋊D3)⋊C312yes—
390D4—92no—
391D4—184no—
392D4—181no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—91no—
396C2×C4—92no—
397C2×C4—92no—
398C2×C4—184no—
399C2×C4—181no—
400C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
401C8—21no—
402C8—181no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—61no—
410C2×C3—61no—
411C2×C3—61no—
412C2×C3—182no—
413C2×C3—362no—
414C2×C3C3×D3×C22⋊C411yescharacteristic
415C2×C3C3×#91611yescharacteristic
416C2×C3C3×#91911yescharacteristic
417D3—62no—
418D3—122no—
419D3—182no—
420D3—362no—
421C22—92no—
422C22—181no—
423C22—181no—
424C22—182no—
425C22—361no—
426C22—361no—
427C22D4×(C3⋊D3)⋊C311yescharacteristic
428C4—22no—
429C4—41no—
430C4—92no—
431C4—361no—
432C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
433C4D4×(C3⋊D3)⋊C311yescharacteristic
434C3—31no—
435C3—31no—
436C3—61no—
437C3C3×#480411yescharacteristic
438C2—182no—
439C2—362no—
440C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
441C2#64125⋊C311yescharacteristic
442C2#64128⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#103C2
socleC22×C3C3×D3×D4
prime radicalC2×#88722C2
cosocleC2×C32×C4C23×C3
hypercenter#103(C3⋊D3)⋊C3
hypocenterC32C3×#416
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×C8×(C3⋊D3)⋊C3C2semidirect
3. #64101⋊C3C2semidirect
4. H(3,𝔽3)×#103C2semidirect
5. #63662⋊C3C2semidirect
6. #63648⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592503C3semidirect
12. #88722C4semidirect
26. C2×C8×C3⋊D3C2×C3semidirect
27. #64101C2×C3semidirect
31. C32×#103D3semidirect
32. C32×#103C2×C3semidirect
33. #63662C2×C3semidirect
34. #63648C2×C3semidirect
35. #63636C2×C3semidirect
63. #3832C3×C4semidirect
251. #103(C3⋊D3)⋊C3semidirect
254. H(3,𝔽3)#416semidirect
384. C32#4804semidirect
385. C32C3×#416semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×C8×(C3⋊D3)⋊C32
#64101⋊C32
H(3,𝔽3)×#1032
#63662⋊C32
#63648⋊C32
#63636⋊C32
#105925033
C3×#48043