Finite Group Databasebeta

← #13153725

Subgroups — #13153725

Order 1728.

1256 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
1C2×#41×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88722C211yesmaximal, characteristic, complemented
3Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#64149⋊C3C213yesmaximal, complemented
5#64147⋊C3C213yesmaximal, complemented
6#64139⋊C3C213yesmaximal, complemented
7C4×#638⋊C3C213yesmaximal, complemented
8#10596305C311yesmaximal, characteristic, complemented
9C3×#5834—31nomaximal
10#41×H(3,𝔽3)—24no—
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×D4×H(3,𝔽3)C2213yes—
13C22×C4×H(3,𝔽3)C2213yes—
14C2×#638⋊C3C2213yes—
15C2×#638⋊C3C2216yes—
16#88722—24no—
17C2×C4⋊((C3⋊D3)⋊C3)C2213yes—
18C2×C4×(C3⋊D3)⋊C3C2213yes—
19#3757⋊C3C2213yes—
20#3752⋊C3C2213yes—
21C4⋊((C3⋊Dic3)⋊C3)C2213yes—
22#3750⋊C3C2211yescharacteristic
23C4×(C3⋊Dic3)⋊C3C2213yes—
24C2×C32×#41—21no—
25C2×C32×#41D311yescharacteristic, complemented
26C2×C32×#41C2×C311yescharacteristic, complemented
27C2×#3832C2×C311yescharacteristic, complemented
28C2×C3×#239—31no—
29Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
30#64149C2×C313yescomplemented
31#64147C2×C313yescomplemented
32#64139C2×C313yescomplemented
33C4×#638C2×C313yescomplemented
34C3×Q8×Dic3—31no—
35C3×#940—33no—
36C3×#938—33no—
37C3×#930—33no—
38C3×C4×#67—33no—
39C23×H(3,𝔽3)C2313yes—
40C22×(C3⋊D3)⋊C3C2313yes—
41Q8×H(3,𝔽3)D414yescomplemented
42D4×H(3,𝔽3)—26no—
43C2×C4×H(3,𝔽3)—26no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×C4×H(3,𝔽3)C2313yes—
46#638⋊C3—26no—
47#638⋊C3—46no—
48C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
49C2×(C3⋊Dic3)⋊C3C2313yes—
50C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—26no—
52C4×(C3⋊D3)⋊C3—26no—
53C3×#550—91no—
54#5834—31no—
55#5834—91no—
56C32×#41—28no—
57C32×#41—24no—
58C32×#41—24no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C22×C311yescharacteristic
62C2×C32×D4—23no—
63C2×C32×D4C2×D313yes—
64C2×C32×D4C22×C313yes—
65C22×C32×C4—23no—
66C22×C32×C4C2×D313yes—
67C22×C32×C4C22×C313yes—
68C2×#638C22×C313yes—
69C2×#638C22×C316yes—
70#3832—24no—
71C2×C4⋊(C3⋊D3)C22×C313yes—
72C2×C4×C3⋊D3C22×C313yes—
73C2×C3×#67—33no—
74C2×C3×#67—36no—
75C3×#239—64no—
76C2×C3×D12—33no—
77C2×C3×C4×D3—33no—
78#3757C22×C313yes—
79#3752C22×C313yes—
80C4⋊(C3⋊Dic3)C22×C313yes—
81#3750C22×C311yescharacteristic
82C4×C3⋊Dic3C22×C313yes—
83C3×#217—33no—
84C3×(C2×D3)⋊C4—33no—
85C3×C4⋊Dic3—33no—
86C3×Dic3⋊C4—31no—
87C3×C4×Dic3—33no—
88C22×H(3,𝔽3)—23no—
89C22×H(3,𝔽3)—23no—
90C22×H(3,𝔽3)—23no—
91C22×H(3,𝔽3)C2411yescharacteristic
92C2×(C3⋊D3)⋊C3—23no—
93C2×(C3⋊D3)⋊C3—23no—
94C2×(C3⋊D3)⋊C3—23no—
95C4×H(3,𝔽3)—21no—
96C4×H(3,𝔽3)C2×D416yes—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—23no—
99(C3⋊Dic3)⋊C3#4112yescomplemented
100C2×C3×#41—31no—
101C2×C3×#41—31no—
102C2×C3×#41—61no—
103C2×C3×#41—91no—
104C2×C3×#41C3×D311yescharacteristic
105C2×#239—31no—
106C2×#239—91no—
107C3×#127—93no—
108C3×#121—93no—
109C3×#121—93no—
110C3×C4×Q8—91no—
111C3×C4×D4—93no—
112Q8×Dic3—31no—
113Q8×Dic3—91no—
114#940—33no—
115#940—93no—
116#938—33no—
117#938—93no—
118#930—33no—
119#930—93no—
120C4×#67—33no—
121C4×#67—93no—
122C23×C32—23no—
123C23×C32C22×D313yes—
124C23×C32C23×C313yes—
125C22×C3⋊D3C23×C313yes—
126C22×C3×D3—33no—
127C32×Q8—24no—
128C32×Q8#6714yescomplemented
129C32×Q8C3×D414yescomplemented
130C32×D4—212no—
131C32×D4—26no—
132C32×D4—26no—
133C2×C32×C4—212no—
134C2×C32×C4—26no—
135C2×C32×C4—26no—
136C2×C32×C4—21no—
137C2×C32×C4—23no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D313yes—
140C2×C32×C4C23×C311yescharacteristic
141C2×C32×C4C23×C313yes—
142#638—26no—
143#638—46no—
144C2×C3⋊Dic3C23×C313yes—
145C2×C3⋊Dic3C23×C311yescharacteristic
146C2×C3⋊Dic3C23×C311yescharacteristic
147C4⋊(C3⋊D3)—26no—
148C4×C3⋊D3—26no—
149C3×#67—66no—
150C3×#67—126no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—33no—
153C2×C3×Dic3—31no—
154C3×D12—66no—
155C3×C4×D3—66no—
156#550—91noSylow
157C2×H(3,𝔽3)—43no—
158C2×H(3,𝔽3)C22×D411yescharacteristic
159C2×H(3,𝔽3)C2×#4111yescharacteristic
160C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
161(C3⋊D3)⋊C3—43no—
162C3×#41—24no—
163C3×#41—64no—
164C3×#41—64no—
165C3×#41—68no—
166C3×#41—184no—
167C2×C3×Q8—31no—
168C2×C3×Q8—31no—
169C2×C3×Q8—61no—
170C2×C3×Q8C2×C3×D311yescharacteristic
171C2×C3×D4—33no—
172C2×C3×D4—33no—
173C2×C3×D4—63no—
174C2×C3×D4—93no—
175C2×C3×D4—93no—
176C2×C3×D4—96no—
177C2×C3×D4C2×C3×D313yes—
178C22×C3×C4—33no—
179C22×C3×C4—33no—
180C22×C3×C4—63no—
181C22×C3×C4—93no—
182C22×C3×C4C2×C3×D313yes—
183C2×#67—33no—
184C2×#67—36no—
185C2×#67—93no—
186C2×#67—96no—
187#239—64no—
188#239—184no—
189C2×D12—33no—
190C2×D12—93no—
191C2×C4×D3—33no—
192C2×C4×D3—93no—
193C3×C4⋊C4—91no—
194C3×C4⋊C4—93no—
195C3×C22⋊C4—93no—
196C3×C22⋊C4—93no—
197C3×C42—93no—
198#217—33no—
199#217—93no—
200(C2×D3)⋊C4—33no—
201(C2×D3)⋊C4—93no—
202C4⋊Dic3—33no—
203C4⋊Dic3—93no—
204Dic3⋊C4—31no—
205Dic3⋊C4—91no—
206C4×Dic3—33no—
207C4×Dic3—93no—
208C22×C32—26no—
209C22×C32—23no—
210C22×C32—23no—
211C22×C32—23no—
212C22×C32—23no—
213C22×C32—23no—
214C22×C32—23no—
215C22×C32—21no—
216C22×C32—43no—
217C22×C32—43no—
218C22×C32C23×D311yescharacteristic
219C22×C32C24×C311yescharacteristic
220C2×C3⋊D3—23no—
221C2×C3⋊D3—23no—
222C2×C3⋊D3—23no—
223C2×C3×D3—63no—
224C2×C3×D3—63no—
225C2×C3×D3—63no—
226C32×C4—22no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—26no—
230C32×C4C2×#6716yes—
231C32×C4C2×C3×D416yes—
232C3⋊Dic3—21no—
233C3⋊Dic3—23no—
234C3⋊Dic3C3×#4112yescomplemented
235C3×Dic3—32no—
236C3×Dic3—63no—
237C3×Dic3—61no—
238C2×#41—91no—
239C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
240#127—93no—
241#121—93no—
242#121—93no—
243C4×Q8—91no—
244C4×D4—93no—
245H(3,𝔽3)#55011yescharacteristic, complemented, Sylow
246C23×C3—33no—
247C23×C3—33no—
248C23×C3—63no—
249C23×C3—93no—
250C23×C3C22×C3×D313yes—
251C22×D3—33no—
252C22×D3—93no—
253C3×Q8—34no—
254C3×Q8—34no—
255C3×Q8—64no—
256C3×Q8C3×#6714yes—
257C3×D4—26no—
258C3×D4—66no—
259C3×D4—66no—
260C3×D4—612no—
261C3×D4—186no—
262C3×D4—186no—
263C3×D4—366no—
264C2×C3×C4—26no—
265C2×C3×C4—33no—
266C2×C3×C4—33no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—66no—
270C2×C3×C4—66no—
271C2×C3×C4—612no—
272C2×C3×C4—61no—
273C2×C3×C4—63no—
274C2×C3×C4—91no—
275C2×C3×C4—93no—
276C2×C3×C4—91no—
277C2×C3×C4—186no—
278C2×C3×C4C22×C3×D311yescharacteristic
279C2×C3×C4C22×C3×D313yes—
280#67—66no—
281#67—126no—
282#67—186no—
283#67—366no—
284C2×Dic3—31no—
285C2×Dic3—33no—
286C2×Dic3—31no—
287C2×Dic3—91no—
288C2×Dic3—93no—
289C2×Dic3—91no—
290D12—66no—
291D12—186no—
292C4×D3—66no—
293C4×D3—186no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—46no—
298C2×C32—43no—
299C2×C32—43no—
300C2×C32D3×#4111yescharacteristic
301C2×C32#100911yescharacteristic
302C2×C32C22×#6711yescharacteristic
303C2×C32C22×C3×D411yescharacteristic
304C2×C32C2×C3×#4111yescharacteristic
305C2×C32C3×H(5,𝔽2)11yescharacteristic
306C3⋊D3—43no—
307C3×D3—123no—
308#41—24no—
309#41—184no—
310C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
311C2×D4—93no—
312C2×D4—96no—
313C2×D4—93no—
314C2×D4C2×(C3⋊D3)⋊C313yes—
315C22×C4—93no—
316C22×C4C2×(C3⋊D3)⋊C313yes—
317C4⋊C4—91no—
318C4⋊C4—93no—
319C22⋊C4—93no—
320C22⋊C4—93no—
321C42—93no—
322C22×C3—23no—
323C22×C3—23no—
324C22×C3—23no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—63no—
328C22×C3—63no—
329C22×C3—63no—
330C22×C3—63no—
331C22×C3—63no—
332C22×C3—63no—
333C22×C3—66no—
334C22×C3—61no—
335C22×C3—123no—
336C22×C3—123no—
337C22×C3—183no—
338C22×C3—183no—
339C22×C3—183no—
340C22×C3C23×C3×D311yescharacteristic
341C2×D3—63no—
342C2×D3—63no—
343C2×D3—63no—
344C2×D3—183no—
345C2×D3—183no—
346C2×D3—183no—
347C3×C4—21no—
348C3×C4—36no—
349C3×C4—36no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—62no—
353C3×C4—66no—
354C3×C4—92no—
355C3×C4—181no—
356C3×C4—183no—
357C3×C4C2×C3×#6716yes—
358Dic3—32no—
359Dic3—63no—
360Dic3—61no—
361Dic3—92no—
362Dic3—181no—
363Dic3—183no—
364C32—21no—
365C32#583411yescharacteristic, complemented
366C32C3×#55011yescharacteristic, complemented
367C23—93no—
368C23C22×(C3⋊D3)⋊C313yes—
369Q8#638⋊C314yescomplemented
370D4—26no—
371D4—186no—
372D4—186no—
373D4—366no—
374C2×C4—26no—
375C2×C4—91no—
376C2×C4—93no—
377C2×C4—91no—
378C2×C4—186no—
379C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
380C2×C4C22×(C3⋊D3)⋊C313yes—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—43no—
388C2×C3—61no—
389C2×C3—61no—
390C2×C3—61no—
391C2×C3—123no—
392C2×C3—123no—
393C2×C3—126no—
394C2×C3—363no—
395C2×C3C3×D3×#4111yescharacteristic
396C2×C3C3×#100911yescharacteristic
397C2×C3C22×C3×#6711yescharacteristic
398D3—123no—
399D3—363no—
400C22—23no—
401C22—23no—
402C22—23no—
403C22—183no—
404C22—183no—
405C22—183no—
406C22C23×(C3⋊D3)⋊C311yescharacteristic
407C4—21no—
408C4—92no—
409C4—183no—
410C4—181no—
411C4C2×#638⋊C316yes—
412C3—31no—
413C3—31no—
414C3—61no—
415C3C3×#583411yescharacteristic
416C2—43no—
417C2—363no—
418C2#41×(C3⋊D3)⋊C311yescharacteristic
419C2#1172263811yescharacteristic
420C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×#41×H(3,𝔽3)C2
hypercenterC2×#41(C3⋊D3)⋊C3
hypocenterC32C3×#550
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×H(3,𝔽3)C2semidirect
2. C2×#88722C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
4. #64149⋊C3C2semidirect
5. #64147⋊C3C2semidirect
6. #64139⋊C3C2semidirect
7. C4×#638⋊C3C2semidirect
8. #10596305C3semidirect
25. C2×C32×#41D3semidirect
26. C2×C32×#41C2×C3semidirect
27. C2×#3832C2×C3semidirect
29. Q8×C3⋊Dic3C2×C3semidirect
30. #64149C2×C3semidirect
31. #64147C2×C3semidirect
32. #64139C2×C3semidirect
33. C4×#638C2×C3semidirect
41. Q8×H(3,𝔽3)D4semidirect
99. (C3⋊Dic3)⋊C3#41semidirect
128. C32×Q8#67semidirect
129. C32×Q8C3×D4semidirect
234. C3⋊Dic3C3×#41semidirect
239. C2×#41(C3⋊D3)⋊C3semidirect
245. H(3,𝔽3)#550semidirect
365. C32#5834semidirect
366. C32C3×#550semidirect
369. Q8#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#41×H(3,𝔽3)2
C2×#887222
Q8×(C3⋊Dic3)⋊C32
#64149⋊C32
#64147⋊C32
#64139⋊C32
C4×#638⋊C32
#105963053
C3×#58343