Finite Group Databasebeta

← #13153721

Subgroups — #13153721

Order 1728.

1179 conjugacy classes of subgroups.

The subgroup lattice has 415 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×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#64147⋊C3C213yesmaximal, complemented
5#64145⋊C3C213yesmaximal, complemented
6C4×#638⋊C3C213yesmaximal, complemented
7#64133⋊C3C213yesmaximal, complemented
8#10596302C311yesmaximal, characteristic, complemented
9C3×#5831—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—
15Q8×(C3⋊D3)⋊C3—24no—
16C2×C4×(C3⋊D3)⋊C3C2213yes—
17C2×#634⋊C3C2213yes—
18#3757⋊C3C2216yes—
19#3757⋊C3C2213yes—
20#3752⋊C3C2211yescharacteristic
21C4⋊((C3⋊Dic3)⋊C3)C2213yes—
22#3750⋊C3C2213yes—
23C4×(C3⋊Dic3)⋊C3C2213yes—
24C2×C32×#41—21no—
25C2×C32×#41D311yescharacteristic, complemented
26C2×C32×#41C2×C311yescharacteristic, complemented
27C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
28C2×C3×D3×Q8—31no—
29Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
30#64147C2×C313yescomplemented
31#64145C2×C313yescomplemented
32C4×#638C2×C313yescomplemented
33#64133C2×C313yescomplemented
34C3×Q8×Dic3—31no—
35C3×#938—33no—
36C3×#936—33no—
37C3×C4×#67—33no—
38C3×#924—33no—
39C23×H(3,𝔽3)C2313yes—
40C22×(C3⋊D3)⋊C3C2311yescharacteristic
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—
47C2×(C3⋊Dic3)⋊C3C2313yes—
48C2×(C3⋊Dic3)⋊C3C2313yes—
49C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
50C4×(C3⋊D3)⋊C3—26no—
51#634⋊C3—26no—
52C3×#548—91no—
53#5831—31no—
54#5831—91no—
55C32×#41—28no—
56C32×#41—24no—
57C32×#41—24no—
58C2×C32×Q8—21no—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×Q8C22×C311yescharacteristic
61C2×C32×D4—23no—
62C2×C32×D4C2×D313yes—
63C2×C32×D4C22×C313yes—
64C22×C32×C4—23no—
65C22×C32×C4C2×D313yes—
66C22×C32×C4C22×C313yes—
67C2×#638C22×C313yes—
68Q8×C3⋊D3—24no—
69C2×C4×C3⋊D3C22×C313yes—
70C2×#634C22×C313yes—
71C2×C3×#67—33no—
72C3×D3×Q8—64no—
73C2×C3×C4×D3—33no—
74C2×C3×Dic6—33no—
75#3757C22×C316yes—
76#3757C22×C313yes—
77#3752C22×C311yescharacteristic
78C4⋊(C3⋊Dic3)C22×C313yes—
79#3750C22×C313yes—
80C4×C3⋊Dic3C22×C313yes—
81C3×#217—36no—
82C3×#217—33no—
83C3×(C2×D3)⋊C4—31no—
84C3×C4⋊Dic3—33no—
85C3×Dic3⋊C4—33no—
86C3×C4×Dic3—33no—
87C22×H(3,𝔽3)—23no—
88C22×H(3,𝔽3)—23no—
89C22×H(3,𝔽3)—23no—
90C22×H(3,𝔽3)C2411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3#4112yescomplemented
94C4×H(3,𝔽3)—21no—
95C4×H(3,𝔽3)C2×D416yes—
96(C3⋊Dic3)⋊C3—23no—
97(C3⋊Dic3)⋊C3—23no—
98(C3⋊Dic3)⋊C3—21no—
99C2×C3×#41—31no—
100C2×C3×#41—31no—
101C2×C3×#41—61no—
102C2×C3×#41C3×D311yescharacteristic
103C22×C3×Q8—91no—
104C2×D3×Q8—31no—
105C2×D3×Q8—91no—
106C3×#124—93no—
107C3×C22⋊Q8—93no—
108C3×#121—93no—
109C3×C4×Q8—91no—
110C3×C4×D4—93no—
111Q8×Dic3—31no—
112Q8×Dic3—91no—
113#938—33no—
114#938—93no—
115#936—33no—
116#936—93no—
117C4×#67—33no—
118C4×#67—93no—
119#924—33no—
120#924—93no—
121C23×C32—23no—
122C23×C32C22×D313yes—
123C23×C32C23×C313yes—
124C22×C3⋊D3C23×C311yescharacteristic
125C22×C3×D3—31no—
126C32×Q8—24no—
127C32×Q8#6714yescomplemented
128C32×Q8C3×D414yescomplemented
129C32×D4—212no—
130C32×D4—26no—
131C32×D4—26no—
132C2×C32×C4—212no—
133C2×C32×C4—26no—
134C2×C32×C4—26no—
135C2×C32×C4—21no—
136C2×C32×C4—23no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D313yes—
139C2×C32×C4C23×C311yescharacteristic
140C2×C32×C4C23×C313yes—
141#638—26no—
142C2×C3⋊Dic3C23×C311yescharacteristic
143C2×C3⋊Dic3C23×C313yes—
144C2×C3⋊Dic3C23×C313yes—
145C4×C3⋊D3—26no—
146#634—26no—
147C3×#67—66no—
148C2×C3×Dic3—31no—
149C2×C3×Dic3—33no—
150C2×C3×Dic3—33no—
151C3×C4×D3—66no—
152C3×Dic6—66no—
153#548—91noSylow
154C2×H(3,𝔽3)—43no—
155C2×H(3,𝔽3)C22×D411yescharacteristic
156C2×H(3,𝔽3)C2×#4111yescharacteristic
157C2×H(3,𝔽3)2−1+411yescharacteristic
158(C3⋊D3)⋊C3—22no—
159C3×#41—24no—
160C3×#41—64no—
161C3×#41—64no—
162C3×#41—68no—
163C2×C3×Q8—31no—
164C2×C3×Q8—31no—
165C2×C3×Q8—61no—
166C2×C3×Q8—93no—
167C2×C3×Q8—184no—
168C2×C3×Q8C2×C3×D311yescharacteristic
169C2×C3×D4—33no—
170C2×C3×D4—33no—
171C2×C3×D4—63no—
172C2×C3×D4—93no—
173C2×C3×D4C2×C3×D313yes—
174C22×C3×C4—33no—
175C22×C3×C4—33no—
176C22×C3×C4—63no—
177C22×C3×C4—93no—
178C22×C3×C4C2×C3×D313yes—
179C2×#67—33no—
180C2×#67—93no—
181D3×Q8—64no—
182D3×Q8—184no—
183C2×C4×D3—33no—
184C2×C4×D3—93no—
185C2×Dic6—33no—
186C2×Dic6—93no—
187C3×C4⋊C4—93no—
188C3×C4⋊C4—93no—
189C3×C22⋊C4—91no—
190C3×C22⋊C4—93no—
191C3×C22⋊C4—96no—
192C3×C42—93no—
193#217—36no—
194#217—33no—
195#217—96no—
196#217—93no—
197(C2×D3)⋊C4—31no—
198(C2×D3)⋊C4—91no—
199C4⋊Dic3—33no—
200C4⋊Dic3—93no—
201Dic3⋊C4—33no—
202Dic3⋊C4—93no—
203C4×Dic3—33no—
204C4×Dic3—93no—
205C22×C32—26no—
206C22×C32—23no—
207C22×C32—23no—
208C22×C32—23no—
209C22×C32—23no—
210C22×C32—23no—
211C22×C32—23no—
212C22×C32—21no—
213C22×C32—43no—
214C22×C32—43no—
215C22×C32C23×D311yescharacteristic
216C22×C32C24×C311yescharacteristic
217C2×C3⋊D3—21no—
218C2×C3⋊D3—21no—
219C2×C3⋊D3C3×#4112yescomplemented
220C2×C3×D3—32no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
223C32×C4—22no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—26no—
227C32×C4C2×#6716yes—
228C32×C4C2×C3×D416yes—
229C3⋊Dic3—23no—
230C3⋊Dic3—23no—
231C3⋊Dic3—21no—
232C3×Dic3—63no—
233C3×Dic3—61no—
234C3×Dic3—63no—
235C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
236C22×Q8—91no—
237#124—93no—
238C22⋊Q8—93no—
239#121—93no—
240C4×Q8—91no—
241C4×D4—93no—
242H(3,𝔽3)#54811yescharacteristic, complemented, Sylow
243C23×C3—33no—
244C23×C3—33no—
245C23×C3—63no—
246C23×C3—91no—
247C23×C3C22×C3×D313yes—
248C22×D3—31no—
249C22×D3—91no—
250C3×Q8—34no—
251C3×Q8—34no—
252C3×Q8—64no—
253C3×Q8—186no—
254C3×Q8C3×#6714yes—
255C3×D4—26no—
256C3×D4—66no—
257C3×D4—66no—
258C3×D4—612no—
259C3×D4—186no—
260C2×C3×C4—26no—
261C2×C3×C4—33no—
262C2×C3×C4—33no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—66no—
266C2×C3×C4—66no—
267C2×C3×C4—612no—
268C2×C3×C4—61no—
269C2×C3×C4—63no—
270C2×C3×C4—91no—
271C2×C3×C4—93no—
272C2×C3×C4—93no—
273C2×C3×C4—186no—
274C2×C3×C4C22×C3×D311yescharacteristic
275C2×C3×C4C22×C3×D313yes—
276#67—66no—
277#67—186no—
278C2×Dic3—31no—
279C2×Dic3—33no—
280C2×Dic3—33no—
281C2×Dic3—91no—
282C2×Dic3—93no—
283C2×Dic3—93no—
284C4×D3—66no—
285C4×D3—186no—
286Dic6—66no—
287Dic6—186no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—46no—
292C2×C32—43no—
293C2×C32—43no—
294C2×C32D3×#4111yescharacteristic
295C2×C32#101011yescharacteristic
296C2×C32C22×#6711yescharacteristic
297C2×C32C22×C3×D411yescharacteristic
298C2×C32C2×C3×#4111yescharacteristic
299C2×C32C3×2−1+411yescharacteristic
300C3⋊D3—22no—
301C3×D3—62no—
302#41—24no—
303C2×Q8—93no—
304C2×Q8—184no—
305C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
306C2×D4—93no—
307C2×D4C2×(C3⋊D3)⋊C313yes—
308C22×C4—93no—
309C22×C4C2×(C3⋊D3)⋊C313yes—
310C4⋊C4—93no—
311C4⋊C4—93no—
312C22⋊C4—93no—
313C22⋊C4—96no—
314C22⋊C4—91no—
315C42—93no—
316C22×C3—23no—
317C22×C3—23no—
318C22×C3—23no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—63no—
322C22×C3—63no—
323C22×C3—63no—
324C22×C3—63no—
325C22×C3—63no—
326C22×C3—63no—
327C22×C3—66no—
328C22×C3—61no—
329C22×C3—92no—
330C22×C3—123no—
331C22×C3—123no—
332C22×C3—181no—
333C22×C3—181no—
334C22×C3C23×C3×D311yescharacteristic
335C2×D3—32no—
336C2×D3—61no—
337C2×D3—61no—
338C2×D3—92no—
339C2×D3—181no—
340C2×D3—181no—
341C3×C4—21no—
342C3×C4—36no—
343C3×C4—36no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—62no—
347C3×C4—66no—
348C3×C4—183no—
349C3×C4—183no—
350C3×C4—181no—
351C3×C4C2×C3×#6716yes—
352Dic3—63no—
353Dic3—61no—
354Dic3—63no—
355Dic3—183no—
356Dic3—183no—
357Dic3—181no—
358C32—21no—
359C32#583111yescharacteristic, complemented
360C32C3×#54811yescharacteristic, complemented
361C23—91no—
362C23C22×(C3⋊D3)⋊C313yes—
363Q8—186no—
364Q8#638⋊C314yescomplemented
365D4—26no—
366D4—186no—
367C2×C4—26no—
368C2×C4—91no—
369C2×C4—93no—
370C2×C4—93no—
371C2×C4—186no—
372C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
373C2×C4C22×(C3⋊D3)⋊C313yes—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—43no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—123no—
385C2×C3—123no—
386C2×C3—126no—
387C2×C3—182no—
388C2×C3C3×D3×#4111yescharacteristic
389C2×C3C3×#101011yescharacteristic
390C2×C3C22×C3×#6711yescharacteristic
391D3—62no—
392D3—182no—
393C22—23no—
394C22—23no—
395C22—23no—
396C22—92no—
397C22—181no—
398C22—181no—
399C22C23×(C3⋊D3)⋊C311yescharacteristic
400C4—21no—
401C4—183no—
402C4—181no—
403C4—183no—
404C4C2×#638⋊C316yes—
405C3—31no—
406C3—31no—
407C3—61no—
408C3C3×#583111yescharacteristic
409C2—43no—
410C2—182no—
411C2#41×(C3⋊D3)⋊C311yescharacteristic
412C2#1172263911yescharacteristic
413C2C22×#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×#548
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×Q8×(C3⋊D3)⋊C3C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
4. #64147⋊C3C2semidirect
5. #64145⋊C3C2semidirect
6. C4×#638⋊C3C2semidirect
7. #64133⋊C3C2semidirect
8. #10596302C3semidirect
25. C2×C32×#41D3semidirect
26. C2×C32×#41C2×C3semidirect
27. C2×Q8×C3⋊D3C2×C3semidirect
29. Q8×C3⋊Dic3C2×C3semidirect
30. #64147C2×C3semidirect
31. #64145C2×C3semidirect
32. C4×#638C2×C3semidirect
33. #64133C2×C3semidirect
41. Q8×H(3,𝔽3)D4semidirect
93. C2×(C3⋊D3)⋊C3#41semidirect
127. C32×Q8#67semidirect
128. C32×Q8C3×D4semidirect
219. C2×C3⋊D3C3×#41semidirect
235. C2×#41(C3⋊D3)⋊C3semidirect
242. H(3,𝔽3)#548semidirect
359. C32#5831semidirect
360. C32C3×#548semidirect
364. 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×Q8×(C3⋊D3)⋊C32
Q8×(C3⋊Dic3)⋊C32
#64147⋊C32
#64145⋊C32
C4×#638⋊C32
#64133⋊C32
#105963023
C3×#58313