Finite Group Databasebeta

← #10592704⋊C3 #13142698

Subgroups — #10592704⋊C3 #13142698

Order 1728.

615 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4#63632⋊C3C214yesmaximal, complemented
5#10592704C311yesmaximal, characteristic, complemented
6C3×#5005—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#638⋊C3—21no—
9#3828⋊C3—21no—
10#3828⋊C3—21no—
11#3828⋊C3C412yescomplemented
12#3828⋊C3C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
14C2×C4×(C3⋊D3)⋊C3—21no—
15C2×#634⋊C3C411yescharacteristic, complemented
16C4⋊C4×H(3,𝔽3)—21no—
17C22⋊C4×H(3,𝔽3)—21no—
18C42×H(3,𝔽3)C2212yescomplemented
19#3748⋊C3—21no—
20#3748⋊C3C2212yescomplemented
21C2×(C3⋊(C3⋊C8))⋊C3—21no—
22C2×#3828C2×C311yescharacteristic
23C2×C3×#235—31no—
24C32×#117—21no—
25C32×#117D311yescharacteristic, complemented
26C32×#117C2×C311yescharacteristic, complemented
27C2×#3748C2×C311yescharacteristic, complemented
28C2×C3×#208—31no—
29#63632C2×C314yescomplemented
30C3×#805—34no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×(C3⋊D3)⋊C3—21no—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)C2×C411yescharacteristic
35C2×C4×H(3,𝔽3)C2×C411yescharacteristic
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40#638⋊C3—22no—
41#638⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)C2×C412yes—
45C4×(C3⋊D3)⋊C3—22no—
46C4×(C3⋊D3)⋊C3—22no—
47#634⋊C3—21no—
48#634⋊C3C2×C412yes—
49(C3⋊(C3⋊C8))⋊C3—22no—
50C3×#421—91no—
51#5005—31no—
52#5005—91no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C2×#638—21no—
57#3828—21no—
58#3828—21no—
59#3828C3×C412yescomplemented
60#3828C22×C312yes—
61C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
62C2×C4×C3⋊D3—21no—
63C2×#634C3×C411yescharacteristic, complemented
64C2×C3×#67—61no—
65C3×#235—32no—
66C3×#235—32no—
67C3×#235—61no—
68C3×#235—61no—
69C2×C3×D12—31no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—31no—
72C32×C4⋊C4—22no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—21no—
75C32×C22⋊C4—22no—
76C32×C22⋊C4—21no—
77C32×C22⋊C4—21no—
78C32×C42—22no—
79C32×C42C2×D312yescomplemented
80C32×C42C22×C312yescomplemented
81#3748—21no—
82#3748C22×C312yescomplemented
83C2×C3⋊(C3⋊C8)—21no—
84C3×#208—32no—
85C3×#208—61no—
86C2×C3×C3⋊C8—61no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)C22⋊C411yescharacteristic
89C22×H(3,𝔽3)C22⋊C411yescharacteristic
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3—42no—
93C4×H(3,𝔽3)—42no—
94C4×H(3,𝔽3)C22⋊C411yescharacteristic
95C4×H(3,𝔽3)C22⋊C411yescharacteristic
96C4×H(3,𝔽3)C22×C411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98(C3⋊Dic3)⋊C3—22no—
99C2×C3×#41—91no—
100C2×#235—31no—
101C2×#235—91no—
102C2×C3×M16—91no—
103C3×#117—31no—
104C3×#117—31no—
105C3×#117—61no—
106C3×#117C3×D311yescharacteristic
107C2×#208—31no—
108C2×#208—91no—
109C3×#104—94no—
110#805—34no—
111#805—94no—
112C23×C32—21no—
113C23×C32#6711yescharacteristic
114C23×C32C3×D411yescharacteristic
115C22×C3⋊D3—21no—
116C22×C3×D3—61no—
117C2×C32×C4—24no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4C4×D311yescharacteristic
127C2×C32×C4C4×D311yescharacteristic
128C2×C32×C4D1211yescharacteristic
129C2×C32×C4D1211yescharacteristic
130C2×C32×C4#6711yescharacteristic
131C2×C32×C4C2×C3×C411yescharacteristic
132C2×C32×C4C2×C3×C411yescharacteristic
133C2×C32×C4C3×D411yescharacteristic
134C2×C32×C4C3×D411yescharacteristic
135C2×C32×C4C3×D411yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C23×C311yescharacteristic
138#638—22no—
139#638—22no—
140C2×C3⋊Dic3—21no—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)C2×C3×C412yes—
143C4×C3⋊D3—22no—
144C4×C3⋊D3—22no—
145#634—21no—
146#634C2×C3×C412yes—
147C3×#67—62no—
148C3×#67—62no—
149C2×C3×Dic3—61no—
150C3×D12—32no—
151C3×D12—61no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×Dic6—32no—
155C3×Dic6—61no—
156C3⋊(C3⋊C8)—22no—
157C3×C3⋊C8—62no—
158#421—91noSylow
159C2×H(3,𝔽3)—21no—
160C2×H(3,𝔽3)—21no—
161C2×H(3,𝔽3)—21no—
162C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
163(C3⋊D3)⋊C3—42no—
164C3×#41—92no—
165C3×#41—92no—
166C3×#41—181no—
167C3×#41—181no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C2×C3×D4—181no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—61no—
174C22×C3×C4—181no—
175C22×C3×C4C2×C3×D311yescharacteristic
176C2×#67—61no—
177C2×#67—181no—
178#235—32no—
179#235—32no—
180#235—61no—
181#235—61no—
182#235—92no—
183#235—92no—
184#235—181no—
185#235—181no—
186C2×D12—31no—
187C2×D12—91no—
188C2×C4×D3—61no—
189C2×C4×D3—181no—
190C2×Dic6—31no—
191C2×Dic6—91no—
192C3×M16—92no—
193C3×M16—181no—
194C2×C3×C8—181no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4—62no—
199C3×C22⋊C4—21no—
200C3×C22⋊C4—61no—
201C3×C22⋊C4—61no—
202C3×C22⋊C4—62no—
203C3×C42—32no—
204C3×C42—32no—
205C3×C42—62no—
206C3×C42C2×C3×D312yes—
207#208—32no—
208#208—61no—
209#208—92no—
210#208—181no—
211C2×C3⋊C8—61no—
212C2×C3⋊C8—181no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—42no—
219C22×C32(C2×D3)⋊C411yescharacteristic
220C22×C32(C2×D3)⋊C411yescharacteristic
221C22×C32C3×C22⋊C411yescharacteristic
222C22×C32C3×C22⋊C411yescharacteristic
223C22×C32C2×#6711yescharacteristic
224C22×C32C2×C3×D411yescharacteristic
225C2×C3⋊D3—22no—
226C2×C3⋊D3—42no—
227C2×C3×D3—62no—
228C2×C3×D3—122no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—44no—
234C32×C4—42no—
235C32×C4—42no—
236C32×C4(C2×D3)⋊C411yescharacteristic
237C32×C4(C2×D3)⋊C411yescharacteristic
238C32×C4C3×C22⋊C411yescharacteristic
239C32×C4C3×C22⋊C411yescharacteristic
240C32×C4C2×C4×D311yescharacteristic
241C32×C4C2×D1211yescharacteristic
242C32×C4C22×C3×C411yescharacteristic
243C32×C4C2×C3×D411yescharacteristic
244C3⋊Dic3—22no—
245C3×Dic3—62no—
246C2×#41—91no—
247C2×M16—91no—
248#117(C3⋊D3)⋊C311yescharacteristic, complemented
249#104—94no—
250H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—61no—
254C23×C3—181no—
255C23×C3C3×#6711yescharacteristic
256C22×D3—61no—
257C22×D3—181no—
258C3×Q8—92no—
259C3×Q8—181no—
260C3×D4—92no—
261C3×D4—181no—
262C3×D4—182no—
263C3×D4—182no—
264C2×C3×C4—22no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—62no—
278C2×C3×C4—62no—
279C2×C3×C4—64no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—182no—
287C2×C3×C4—182no—
288C2×C3×C4—181no—
289C2×C3×C4C3×C4×D311yescharacteristic
290C2×C3×C4C3×C4×D311yescharacteristic
291C2×C3×C4C3×D1211yescharacteristic
292C2×C3×C4C3×D1211yescharacteristic
293C2×C3×C4C3×#6711yescharacteristic
294C2×C3×C4C22×C3×D311yescharacteristic
295#67—62no—
296#67—62no—
297#67—182no—
298#67—182no—
299C2×Dic3—61no—
300C2×Dic3—181no—
301D12—32no—
302D12—61no—
303D12—92no—
304D12—181no—
305C4×D3—62no—
306C4×D3—62no—
307C4×D3—182no—
308C4×D3—182no—
309Dic6—32no—
310Dic6—61no—
311Dic6—92no—
312Dic6—181no—
313C3×C8—182no—
314C3⋊C8—62no—
315C3⋊C8—182no—
316C2×C32—22no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—41no—
325C2×C32—41no—
326C2×C32C2×(C2×D3)⋊C411yescharacteristic
327C2×C32C2×C3×C22⋊C411yescharacteristic
328C3⋊D3—42no—
329C3×D3—122no—
330#41—92no—
331#41—92no—
332#41—181no—
333#41—181no—
334C2×Q8—91no—
335C2×D4—91no—
336C2×D4—181no—
337C22×C4—181no—
338C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
339M16—92no—
340M16—181no—
341C2×C8—181no—
342C4⋊C4—21no—
343C22⋊C4—21no—
344C42C2×(C3⋊D3)⋊C312yescomplemented
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—41no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—122no—
358C22×C3—182no—
359C22×C3—362no—
360C22×C3C3×(C2×D3)⋊C411yescharacteristic
361C22×C3C3×(C2×D3)⋊C411yescharacteristic
362C22×C3C2×C3×#6711yescharacteristic
363C2×D3—62no—
364C2×D3—122no—
365C2×D3—182no—
366C2×D3—362no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—31no—
373C3×C4—31no—
374C3×C4—31no—
375C3×C4—42no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—61no—
379C3×C4—61no—
380C3×C4—122no—
381C3×C4—122no—
382C3×C4—124no—
383C3×C4—182no—
384C3×C4C3×(C2×D3)⋊C411yescharacteristic
385C3×C4C3×(C2×D3)⋊C411yescharacteristic
386C3×C4C2×C3×C4×D311yescharacteristic
387C3×C4C2×C3×D1211yescharacteristic
388Dic3—62no—
389Dic3—182no—
390C32—21no—
391C32#500511yescharacteristic, complemented
392C32C3×#42111yescharacteristic, complemented
393C23—181no—
394C23#638⋊C311yescharacteristic
395Q8—92no—
396Q8—181no—
397D4—92no—
398D4—182no—
399D4—182no—
400D4—181no—
401C2×C4—22no—
402C2×C4—181no—
403C2×C4—182no—
404C2×C4—182no—
405C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
406C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
407C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
408C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
409C2×C4#638⋊C311yescharacteristic
410C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
411C8—182no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—31no—
416C2×C3—31no—
417C2×C3—61no—
418C2×C3—61no—
419C2×C3—61no—
420C2×C3—61no—
421C2×C3—61no—
422C2×C3—61no—
423C2×C3—62no—
424C2×C3—61no—
425C2×C3—121no—
426C2×C3—121no—
427C2×C3—362no—
428C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
429D3—122no—
430D3—362no—
431C22—41no—
432C22—182no—
433C22—362no—
434C22#3752⋊C311yescharacteristic
435C22#3752⋊C311yescharacteristic
436C22C2×#638⋊C311yescharacteristic
437C4—42no—
438C4—182no—
439C4#3752⋊C311yescharacteristic
440C4#3752⋊C311yescharacteristic
441C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
442C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
443C3—31no—
444C3—31no—
445C3—61no—
446C3C3×#500511yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—362no—
451C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC4C2×C4⋊((C3⋊D3)⋊C3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#117C2
socleC2×C3C2×C3×(C2×D3)⋊C4
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#421
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#117C2semidirect
3. C2×#3748⋊C3C2semidirect
4. #63632⋊C3C2semidirect
5. #10592704C3semidirect
11. #3828⋊C3C4semidirect
13. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
15. C2×#634⋊C3C4semidirect
18. C42×H(3,𝔽3)C22semidirect
20. #3748⋊C3C22semidirect
25. C32×#117D3semidirect
26. C32×#117C2×C3semidirect
27. C2×#3748C2×C3semidirect
29. #63632C2×C3semidirect
59. #3828C3×C4semidirect
61. C2×C4⋊(C3⋊D3)C3×C4semidirect
63. C2×#634C3×C4semidirect
79. C32×C42C2×D3semidirect
80. C32×C42C22×C3semidirect
82. #3748C22×C3semidirect
248. #117(C3⋊D3)⋊C3semidirect
250. H(3,𝔽3)#421semidirect
344. C42C2×(C3⋊D3)⋊C3semidirect
391. C32#5005semidirect
392. C32C3×#421semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
H(3,𝔽3)×#1172
C2×#3748⋊C32
#63632⋊C32
#105927043
C3×#50053