Finite Group Databasebeta

← #10596206⋊C3 #13153331

Subgroups — #10596206⋊C3 #13153331

Order 1728.

754 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic
3#64155⋊C3C212yesmaximal, complemented
4#64153⋊C3C211yesmaximal, characteristic, complemented
5#64107⋊C3C212yesmaximal, complemented
6#64105⋊C3C212yesmaximal, complemented
7#64103⋊C3C212yesmaximal
8#64098⋊C3C212yesmaximal, complemented
9#11721387C212yesmaximal, complemented
10#10596206C311yesmaximal, characteristic, complemented
11C3×#5735—31nomaximal
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C2×Q8×H(3,𝔽3)C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×#634⋊C3C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2212yes—
18C4⋊C4×H(3,𝔽3)C2212yes—
19C42×H(3,𝔽3)C2211yescharacteristic
20#3752⋊C3C2212yes—
21#3752⋊C3C2212yes—
22#3752⋊C3C2212yes—
23#3752⋊C3C2212yes—
24#3752⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2212yes—
26#3750⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
33C32×C4⋊Q8—21no—
34C32×C4⋊Q8D311yescharacteristic, complemented
35C32×C4⋊Q8C2×C311yescharacteristic, complemented
36#64156C2×C311yescharacteristic
37#64155C2×C312yescomplemented
38#64153C2×C311yescharacteristic, complemented
39#64107C2×C312yescomplemented
40#64105C2×C312yescomplemented
41#64103C2×C312yes—
42#64098C2×C312yescomplemented
43#64085C2×C312yescomplemented
44C3×#947—31no—
45C3×#946—32no—
46C3×#944—31no—
47C3×#898—32no—
48C3×#896—32no—
49C3×#894—32no—
50C3×#889—32no—
51C3×#876—32no—
52C22×(C3⋊D3)⋊C3C2312yes—
53Q8×H(3,𝔽3)—41no—
54Q8×H(3,𝔽3)—41no—
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×C4×H(3,𝔽3)C2312yes—
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C2×C4×H(3,𝔽3)C2311yescharacteristic
59C2×C4×H(3,𝔽3)C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C4⋊((C3⋊D3)⋊C3)—41no—
64C4×(C3⋊D3)⋊C3—42no—
65#634⋊C3—41no—
66C3×#563—91no—
67#5735—31no—
68#5735—91no—
69C2×C32×Q8—21no—
70C2×C32×Q8—21no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C311yescharacteristic
74C2×C32×Q8C22×C311yescharacteristic
75C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
76C2×C4×C3⋊D3C22×C312yes—
77C2×#634C22×C311yescharacteristic
78C2×C3×D12—31no—
79C2×C3×C4×D3—32no—
80C2×C3×Dic6—31no—
81C32×C4⋊C4—22no—
82C32×C4⋊C4—22no—
83C32×C4⋊C4C2×D312yes—
84C32×C4⋊C4C2×D312yes—
85C32×C4⋊C4C22×C312yes—
86C32×C4⋊C4C22×C312yes—
87C32×C42—21no—
88C32×C42C2×D311yescharacteristic
89C32×C42C22×C311yescharacteristic
90#3752C22×C312yes—
91#3752C22×C312yes—
92#3752C22×C312yes—
93#3752C22×C312yes—
94#3752C22×C312yes—
95C4⋊(C3⋊Dic3)C22×C312yes—
96#3750C22×C312yes—
97#3750C22×C312yes—
98#3750C22×C312yes—
99#3750C22×C312yes—
100#3750C22×C312yes—
101C4×C3⋊Dic3C22×C311yescharacteristic
102C4×C3⋊Dic3C22×C311yescharacteristic
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×C4⋊Dic3—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—31no—
116C22×H(3,𝔽3)C2411yescharacteristic
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3—24no—
119C4×H(3,𝔽3)—21no—
120C4×H(3,𝔽3)—21no—
121C4×H(3,𝔽3)—22no—
122C4×H(3,𝔽3)—21no—
123C4×H(3,𝔽3)—22no—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127C3×C4⋊Q8—31no—
128C3×C4⋊Q8—31no—
129C3×C4⋊Q8—61no—
130C3×C4⋊Q8—91no—
131C3×C4⋊Q8C3×D311yescharacteristic
132C3×#126—92no—
133C3×#126—92no—
134C3×#125—92no—
135C3×#124—91no—
136C3×#123—92no—
137C3×C22⋊Q8—92no—
138C3×C22⋊Q8—92no—
139#947—31no—
140#947—91no—
141#946—32no—
142#946—92no—
143#944—31no—
144#944—91no—
145#898—32no—
146#898—92no—
147#896—32no—
148#896—92no—
149#894—32no—
150#894—92no—
151#889—32no—
152#889—92no—
153#876—32no—
154#876—92no—
155C22×C3⋊D3C23×C312yes—
156C22×C3×D3—32no—
157C32×Q8—41no—
158C32×Q8—41no—
159C32×Q8—41no—
160C32×Q8—41no—
161C32×Q8—42no—
162C32×Q8—42no—
163C2×C32×C4—22no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D312yes—
173C2×C32×C4C23×C311yescharacteristic
174C2×C32×C4C23×C311yescharacteristic
175C2×C32×C4C23×C312yes—
176C2×C32×C4C23×C311yescharacteristic
177C2×C32×C4C23×C312yes—
178C2×C3⋊Dic3C23×C312yes—
179C2×C3⋊Dic3C23×C312yes—
180C2×C3⋊Dic3C23×C312yes—
181C4⋊(C3⋊D3)—41no—
182C4×C3⋊D3—42no—
183#634—41no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C3×D12—121no—
188C3×C4×D3—122no—
189C3×Dic6—121no—
190#563—91noSylow
191C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
192C2×H(3,𝔽3)2−1+412yes—
193(C3⋊D3)⋊C3—42no—
194C2×C3×Q8—31no—
195C2×C3×Q8—31no—
196C2×C3×Q8—31no—
197C2×C3×Q8—31no—
198C2×C3×Q8—61no—
199C2×C3×Q8—61no—
200C2×C3×Q8—91no—
201C2×C3×Q8C2×C3×D311yescharacteristic
202C2×C3×Q8C2×C3×D311yescharacteristic
203C2×C3×D4—91no—
204C22×C3×C4—92no—
205C2×D12—31no—
206C2×D12—91no—
207C2×C4×D3—32no—
208C2×C4×D3—92no—
209C2×Dic6—31no—
210C2×Dic6—91no—
211C3×C4⋊C4—32no—
212C3×C4⋊C4—32no—
213C3×C4⋊C4—32no—
214C3×C4⋊C4—32no—
215C3×C4⋊C4—62no—
216C3×C4⋊C4—62no—
217C3×C4⋊C4—92no—
218C3×C4⋊C4—92no—
219C3×C4⋊C4—92no—
220C3×C4⋊C4—92no—
221C3×C4⋊C4—92no—
222C3×C4⋊C4—92no—
223C3×C4⋊C4C2×C3×D312yes—
224C3×C4⋊C4C2×C3×D312yes—
225C3×C22⋊C4—92no—
226C3×C22⋊C4—92no—
227C3×C22⋊C4—92no—
228C3×C22⋊C4—92no—
229C3×C22⋊C4—92no—
230C3×C42—31no—
231C3×C42—31no—
232C3×C42—61no—
233C3×C42—91no—
234C3×C42—91no—
235C3×C42C2×C3×D311yescharacteristic
236(C2×D3)⋊C4—32no—
237(C2×D3)⋊C4—32no—
238(C2×D3)⋊C4—32no—
239(C2×D3)⋊C4—32no—
240(C2×D3)⋊C4—32no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246C4⋊Dic3—32no—
247C4⋊Dic3—92no—
248Dic3⋊C4—32no—
249Dic3⋊C4—32no—
250Dic3⋊C4—32no—
251Dic3⋊C4—32no—
252Dic3⋊C4—32no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258C4×Dic3—31no—
259C4×Dic3—31no—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C22×C32—21no—
263C22×C32C23×D311yescharacteristic
264C22×C32C24×C311yescharacteristic
265C2×C3⋊D3—22no—
266C2×C3⋊D3—24no—
267C2×C3×D3—62no—
268C2×C3×D3—64no—
269C32×C4—24no—
270C32×C4—22no—
271C32×C4—21no—
272C32×C4—22no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—22no—
276C32×C4—21no—
277C32×C4—22no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—22no—
281C32×C4—41no—
282C32×C4—42no—
283C32×C4—41no—
284C3⋊Dic3—22no—
285C3⋊Dic3—22no—
286C3⋊Dic3—22no—
287C3×Dic3—62no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290C4⋊Q8—91no—
291C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
292#126—92no—
293#126—92no—
294#125—92no—
295#124—91no—
296#123—92no—
297C22⋊Q8—92no—
298C22⋊Q8—92no—
299H(3,𝔽3)#56311yescharacteristic, complemented, Sylow
300C23×C3—92no—
301C22×D3—32no—
302C22×D3—92no—
303C3×Q8—41no—
304C3×Q8—41no—
305C3×Q8—121no—
306C3×Q8—121no—
307C3×Q8—121no—
308C3×Q8—121no—
309C3×Q8—122no—
310C3×Q8—122no—
311C3×Q8—361no—
312C3×D4—361no—
313C2×C3×C4—32no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—32no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—32no—
321C2×C3×C4—31no—
322C2×C3×C4—32no—
323C2×C3×C4—62no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—62no—
328C2×C3×C4—92no—
329C2×C3×C4—92no—
330C2×C3×C4—92no—
331C2×C3×C4—362no—
332C2×C3×C4C22×C3×D311yescharacteristic
333C2×C3×C4C22×C3×D311yescharacteristic
334C2×C3×C4C22×C3×D312yes—
335C2×C3×C4C22×C3×D311yescharacteristic
336C2×C3×C4C22×C3×D312yes—
337C2×Dic3—32no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—92no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343D12—121no—
344D12—361no—
345C4×D3—122no—
346C4×D3—362no—
347Dic6—121no—
348Dic6—361no—
349C2×C32—21no—
350C2×C32—22no—
351C2×C32#100411yescharacteristic
352C2×C32#100712yes—
353C2×C32C3×H(5,𝔽2)11yescharacteristic
354C2×C32C3×2−1+412yes—
355C3⋊D3—42no—
356C3×D3—122no—
357C2×Q8—91no—
358C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
359C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
360C2×D4—91no—
361C22×C4—92no—
362C4⋊C4—92no—
363C4⋊C4—92no—
364C4⋊C4—92no—
365C4⋊C4—92no—
366C4⋊C4—92no—
367C4⋊C4—92no—
368C4⋊C4C2×(C3⋊D3)⋊C312yes—
369C4⋊C4C2×(C3⋊D3)⋊C312yes—
370C22⋊C4—92no—
371C22⋊C4—92no—
372C22⋊C4—92no—
373C22⋊C4—92no—
374C22⋊C4—92no—
375C42—91no—
376C42—91no—
377C42C2×(C3⋊D3)⋊C311yescharacteristic
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—61no—
381C22×C3—182no—
382C22×C3—184no—
383C22×C3C23×C3×D311yescharacteristic
384C2×D3—62no—
385C2×D3—64no—
386C2×D3—182no—
387C2×D3—184no—
388C3×C4—21no—
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—22no—
393C3×C4—61no—
394C3×C4—62no—
395C3×C4—61no—
396C3×C4—62no—
397C3×C4—61no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—62no—
402C3×C4—62no—
403C3×C4—62no—
404C3×C4—64no—
405C3×C4—122no—
406C3×C4—121no—
407C3×C4—121no—
408C3×C4—182no—
409C3×C4—182no—
410C3×C4—182no—
411Dic3—62no—
412Dic3—62no—
413Dic3—62no—
414Dic3—182no—
415Dic3—182no—
416Dic3—182no—
417C32—21no—
418C32#573511yescharacteristic, complemented
419C32C3×#56311yescharacteristic, complemented
420C23—92no—
421Q8—41no—
422Q8—41no—
423Q8—361no—
424D4—361no—
425C2×C4—92no—
426C2×C4—92no—
427C2×C4—92no—
428C2×C4—362no—
429C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
430C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
431C2×C4C22×(C3⋊D3)⋊C312yes—
432C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
433C2×C4C22×(C3⋊D3)⋊C312yes—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—32no—
437C2×C3—32no—
438C2×C3—61no—
439C2×C3—62no—
440C2×C3—362no—
441C2×C3C3×#100411yescharacteristic
442C2×C3C3×#100712yes—
443D3—122no—
444D3—362no—
445C22—182no—
446C22—184no—
447C22C23×(C3⋊D3)⋊C311yescharacteristic
448C4—22no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—22no—
453C4—182no—
454C4—182no—
455C4—182no—
456C3—31no—
457C3—31no—
458C3—61no—
459C3C3×#573511yescharacteristic
460C2—362no—
461C2#64361⋊C311yescharacteristic
462C2#1172262712yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #64155⋊C3C2semidirect
4. #64153⋊C3C2semidirect
5. #64107⋊C3C2semidirect
6. #64105⋊C3C2semidirect
8. #64098⋊C3C2semidirect
9. #11721387C2semidirect
10. #10596206C3semidirect
34. C32×C4⋊Q8D3semidirect
35. C32×C4⋊Q8C2×C3semidirect
37. #64155C2×C3semidirect
38. #64153C2×C3semidirect
39. #64107C2×C3semidirect
40. #64105C2×C3semidirect
42. #64098C2×C3semidirect
43. #64085C2×C3semidirect
291. C4⋊Q8(C3⋊D3)⋊C3semidirect
299. H(3,𝔽3)#563semidirect
418. C32#5735semidirect
419. C32C3×#563semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#64156⋊C32
#64155⋊C32
#64153⋊C32
#64107⋊C32
#64105⋊C32
#64103⋊C32
#64098⋊C32
#117213872
#105962063
C3×#57353