Finite Group Databasebeta

← #10592485⋊C3 #13141773

Subgroups — #10592485⋊C3 #13141773

Order 1728.

540 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 94 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
1#64149⋊C3C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3C2×#3745⋊C3C211yesmaximal, characteristic, complemented
4#64102⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592485C311yesmaximal, characteristic, complemented
9C3×#4786—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18#3753⋊C3—22no—
19#3752⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3745⋊C3—22no—
23#64149C2×C311yescharacteristic, complemented
24C2×#3753C2×C311yescharacteristic, complemented
25C2×#3745C2×C311yescharacteristic, complemented
26#64102C2×C311yescharacteristic, complemented
27C3×#940—31no—
28C2×C3×#213—31no—
29C2×C3×D24—31no—
30C3×#893—31no—
31C32×#102—21no—
32C32×#102D311yescharacteristic, complemented
33C32×#102C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×Dic3⋊C8—31no—
37C3×#809—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—21no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#638⋊C3—42no—
46#638⋊C3—42no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)D412yescomplemented
53C4×(C3⋊D3)⋊C3—42no—
54C8×H(3,𝔽3)—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56C3×#459—91no—
57#4786—31no—
58#4786—91no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C22×C311yescharacteristic
62C2×#638—21no—
63C2×#638—21no—
64C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C4×C3⋊D3—21no—
67C2×C3×#67—61no—
68C2×C3×#67—61no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×C4×D3—61no—
72C2×C32×C8—21no—
73C2×C32×C8C2×D311yescharacteristic
74C2×C32×C8C22×C311yescharacteristic
75C32×C4⋊C4—21no—
76C32×C4⋊C4C2×D311yescharacteristic
77C32×C4⋊C4C22×C311yescharacteristic
78#3753—22no—
79#3752—21no—
80C4×C3⋊Dic3C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82#3745—22no—
83C3×#213—62no—
84C3×(C2×D3)⋊C4—61no—
85C3×C4×Dic3—31no—
86C2×C3×C3⋊C8—31no—
87C3×D24—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—22no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—41no—
98C4×H(3,𝔽3)—41no—
99C4×H(3,𝔽3)C2×D411yescharacteristic
100C4×H(3,𝔽3)#4111yescharacteristic
101(C3⋊Dic3)⋊C3—21no—
102(C3⋊Dic3)⋊C3D812yescomplemented
103C2×C3×D8—91no—
104C2×C3×D8—91no—
105C3×#127—91no—
106C3×C4×D4—91no—
107#940—31no—
108#940—91no—
109C2×#213—31no—
110C2×#213—91no—
111C2×D24—31no—
112C2×D24—91no—
113#893—31no—
114#893—91no—
115C3×C4⋊C8—91no—
116C3×#102—31no—
117C3×#102—31no—
118C3×#102—61no—
119C3×#102—91no—
120C3×#102C3×D311yescharacteristic
121Dic3⋊C8—31no—
122Dic3⋊C8—91no—
123#809—31no—
124#809—91no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—41no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—21no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C32×D4—22no—
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—41no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C23×C311yescharacteristic
144#638—42no—
145#638—42no—
146C2×C3⋊Dic3C3×D411yescharacteristic
147C2×C3⋊Dic3C3×D411yescharacteristic
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)C3×D412yescomplemented
152C4×C3⋊D3—42no—
153C3×#67—122no—
154C3×#67—122no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—31no—
157C3×D12—32no—
158C3×D12—61no—
159C3×D12—61no—
160C3×D12—61no—
161C3×C4×D3—122no—
162C32×C8—21no—
163C32×C8—21no—
164C32×C8—41no—
165C3⋊(C3⋊C8)—21no—
166C3×C3⋊C8—61no—
167#459—91noSylow
168C2×H(3,𝔽3)—81no—
169C2×H(3,𝔽3)#12111yescharacteristic
170C2×H(3,𝔽3)C2×D811yescharacteristic
171C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
172(C3⋊D3)⋊C3—42no—
173(C3⋊D3)⋊C3—81no—
174C2×C3×D4—31no—
175C2×C3×D4—31no—
176C2×C3×D4—61no—
177C2×C3×D4—91no—
178C2×C3×D4—91no—
179C2×C3×D4—181no—
180C2×C3×D4—181no—
181C2×C3×D4C2×C3×D311yescharacteristic
182C22×C3×C4—181no—
183C2×#67—61no—
184C2×#67—61no—
185C2×#67—181no—
186C2×#67—181no—
187C2×D12—31no—
188C2×D12—31no—
189C2×D12—91no—
190C2×D12—91no—
191C2×C4×D3—61no—
192C2×C4×D3—181no—
193C3×D8—182no—
194C3×D8—182no—
195C2×C3×C8—31no—
196C2×C3×C8—31no—
197C2×C3×C8—61no—
198C2×C3×C8—91no—
199C2×C3×C8C2×C3×D311yescharacteristic
200C3×C4⋊C4—31no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4C2×C3×D311yescharacteristic
204C3×C22⋊C4—181no—
205C3×C42—91no—
206#213—62no—
207#213—182no—
208(C2×D3)⋊C4—61no—
209(C2×D3)⋊C4—181no—
210C4×Dic3—31no—
211C4×Dic3—91no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—91no—
214D24—62no—
215D24—182no—
216C22×C32—21no—
217C22×C32—42no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—81no—
225C22×C32—81no—
226C22×C32D3×D411yescharacteristic
227C22×C32C2×C3×D411yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3—41no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—41no—
234C2×C3×D3—62no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—41no—
243C32×C4—41no—
244C32×C4—81no—
245C32×C4#23511yescharacteristic
246C32×C4D3×D411yescharacteristic
247C32×C4C2×C3×D411yescharacteristic
248C32×C4C3×#4111yescharacteristic
249C3⋊Dic3—21no—
250C3⋊Dic3C3×D812yescomplemented
251C3×Dic3—32no—
252C3×Dic3—61no—
253C2×D8—91no—
254C2×D8—91no—
255#127—91no—
256C4×D4—91no—
257C4⋊C8—91no—
258#102—91no—
259#102(C3⋊D3)⋊C311yescharacteristic, complemented
260H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
261C23×C3—21no—
262C23×C3—61no—
263C23×C3—61no—
264C23×C3—121no—
265C23×C3—181no—
266C23×C3—181no—
267C22×D3—61no—
268C22×D3—61no—
269C22×D3—181no—
270C22×D3—181no—
271C3×D4—21no—
272C3×D4—21no—
273C3×D4—61no—
274C3×D4—61no—
275C3×D4—61no—
276C3×D4—61no—
277C3×D4—62no—
278C3×D4—92no—
279C3×D4—121no—
280C3×D4—181no—
281C3×D4—181no—
282C3×D4—181no—
283C3×D4—362no—
284C3×D4—362no—
285C2×C3×C4—21no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—91no—
292C2×C3×C4—91no—
293C2×C3×C4—121no—
294C2×C3×C4—362no—
295C2×C3×C4C22×C3×D311yescharacteristic
296#67—122no—
297#67—122no—
298#67—362no—
299#67—362no—
300C2×Dic3—31no—
301C2×Dic3—31no—
302C2×Dic3—91no—
303C2×Dic3—91no—
304D12—32no—
305D12—61no—
306D12—61no—
307D12—61no—
308D12—92no—
309D12—181no—
310D12—181no—
311D12—181no—
312C4×D3—122no—
313C4×D3—362no—
314C3×C8—21no—
315C3×C8—61no—
316C3×C8—61no—
317C3×C8—121no—
318C3×C8—181no—
319C3⋊C8—61no—
320C3⋊C8—181no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—82no—
325C2×C32—81no—
326C2×C32—81no—
327C2×C32#88411yescharacteristic
328C2×C32D3×D811yescharacteristic
329C2×C32#91411yescharacteristic
330C2×C32C3×#12111yescharacteristic
331C2×C32C2×C3×D811yescharacteristic
332C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
333C3⋊D3—42no—
334C3⋊D3—81no—
335C3×D3—122no—
336C3×D3—241no—
337C2×D4—91no—
338C2×D4—91no—
339C2×D4—181no—
340C2×D4—181no—
341C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
342C22×C4—181no—
343D8—182no—
344D8—182no—
345C2×C8—91no—
346C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
347C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
348C22⋊C4—181no—
349C42—91no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—61no—
356C22×C3—121no—
357C22×C3—121no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3—122no—
363C22×C3—182no—
364C22×C3—241no—
365C22×C3—241no—
366C22×C3—361no—
367C22×C3—361no—
368C22×C3—361no—
369C22×C3—361no—
370C22×C3—361no—
371C22×C3C3×D3×D411yescharacteristic
372C2×D3—62no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—121no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—182no—
379C2×D3—361no—
380C2×D3—361no—
381C2×D3—361no—
382C2×D3—361no—
383C2×D3—361no—
384C3×C4—31no—
385C3×C4—31no—
386C3×C4—31no—
387C3×C4—31no—
388C3×C4—41no—
389C3×C4—61no—
390C3×C4—61no—
391C3×C4—92no—
392C3×C4—121no—
393C3×C4—121no—
394C3×C4—181no—
395C3×C4—241no—
396C3×C4C3×#23511yescharacteristic
397C3×C4C3×D3×D411yescharacteristic
398Dic3—32no—
399Dic3—61no—
400Dic3—92no—
401Dic3—181no—
402C32—21no—
403C32#478611yescharacteristic, complemented
404C32C3×#45911yescharacteristic, complemented
405C23—21no—
406C23—181no—
407C23—181no—
408D4—21no—
409D4—21no—
410D4—92no—
411D4—181no—
412D4—181no—
413D4—181no—
414D4—362no—
415D4—362no—
416C2×C4—21no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—362no—
420C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
421C8—21no—
422C8—181no—
423C2×C3—31no—
424C2×C3—31no—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—61no—
430C2×C3—61no—
431C2×C3—61no—
432C2×C3—81no—
433C2×C3—241no—
434C2×C3—241no—
435C2×C3—242no—
436C2×C3—362no—
437C2×C3—721no—
438C2×C3C3×#88411yescharacteristic
439C2×C3C3×D3×D811yescharacteristic
440C2×C3C3×#91411yescharacteristic
441D3—122no—
442D3—241no—
443D3—362no—
444D3—721no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—182no—
449C22—361no—
450C22—361no—
451C22—361no—
452C22—361no—
453C22—361no—
454C22D4×(C3⋊D3)⋊C311yescharacteristic
455C4—41no—
456C4—92no—
457C4—181no—
458C4#3828⋊C311yescharacteristic
459C4D4×(C3⋊D3)⋊C311yescharacteristic
460C3—31no—
461C3—31no—
462C3—61no—
463C3C3×#478611yescharacteristic
464C2—81no—
465C2—362no—
466C2—721no—
467C2#64093⋊C311yescharacteristic
468C2D8×(C3⋊D3)⋊C311yescharacteristic
469C2#64123⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#459
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. #64149⋊C3C2semidirect
2. C2×#3753⋊C3C2semidirect
3. C2×#3745⋊C3C2semidirect
4. #64102⋊C3C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #63641⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592485C3semidirect
23. #64149C2×C3semidirect
24. C2×#3753C2×C3semidirect
25. C2×#3745C2×C3semidirect
26. #64102C2×C3semidirect
32. C32×#102D3semidirect
33. C32×#102C2×C3semidirect
34. #63641C2×C3semidirect
35. #63636C2×C3semidirect
52. C4⋊((C3⋊D3)⋊C3)D4semidirect
102. (C3⋊Dic3)⋊C3D8semidirect
151. C4⋊(C3⋊D3)C3×D4semidirect
250. C3⋊Dic3C3×D8semidirect
259. #102(C3⋊D3)⋊C3semidirect
260. H(3,𝔽3)#459semidirect
403. C32#4786semidirect
404. C32C3×#459semidirect

Maximal subgroups

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

subgroupindex
#64149⋊C32
C2×#3753⋊C32
C2×#3745⋊C32
#64102⋊C32
H(3,𝔽3)×#1022
#63641⋊C32
#63636⋊C32
#105924853
C3×#47863