Finite Group Databasebeta

← #10592863⋊C3 #13143366

Subgroups — #10592863⋊C3 #13143366

Order 1728.

488 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×H(3,𝔽3)C211yesmaximal, characteristic
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic
6#63645⋊C3C211yesmaximal, characteristic, complemented
7#63642⋊C3C211yesmaximal, characteristic, complemented
8#10592863C311yesmaximal, characteristic, complemented
9C3×#5164—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3—21no—
13SD16×H(3,𝔽3)C414yescomplemented
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×C32×SD16—21no—
23C2×C32×SD16D311yescharacteristic
24C2×C32×SD16C2×C311yescharacteristic
25Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
26D4×C3⋊Dic3C2×C311yescharacteristic
27C3×Q8×Dic3—31no—
28C3×D4×Dic3—31no—
29#63662C2×C311yescharacteristic, complemented
30#63659C2×C311yescharacteristic
31#63645C2×C311yescharacteristic, complemented
32#63642C2×C311yescharacteristic, complemented
33C3×#835—31no—
34C3×#832—31no—
35C3×#818—31no—
36C3×#815—31no—
37C23×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)—21no—
39Q8×H(3,𝔽3)C2×C412yescomplemented
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)C2×C412yes—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—42no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C8×H(3,𝔽3)C2×C412yescomplemented
50(C3⋊(C3⋊C8))⋊C3—21no—
51C3×#440—91no—
52#5164—31no—
53#5164—91no—
54C2×C32×Q8—21no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C32×Q8C22×C311yescharacteristic
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C22×C311yescharacteristic
60C22×C3⋊Dic3—21no—
61C22×C3×Dic3—61no—
62C32×SD16—24no—
63C32×SD16Dic314yescomplemented
64C32×SD16C3×C414yescomplemented
65C2×C32×C8—21no—
66C2×C32×C8C2×D311yescharacteristic
67C2×C32×C8C22×C311yescharacteristic
68#3757—21no—
69C4⋊(C3⋊Dic3)—21no—
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71C4⋊(C3⋊Dic3)C22×C311yescharacteristic
72C4×C3⋊Dic3—21no—
73C4×C3⋊Dic3C22×C311yescharacteristic
74C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
75C3×#217—61no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—61no—
79C3×C4×Dic3—31no—
80C3×C4×Dic3—61no—
81C2×C3×C3⋊C8—31no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)—41no—
84C22×H(3,𝔽3)—41no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C4×H(3,𝔽3)—22no—
87C4×H(3,𝔽3)C22×C411yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92(C3⋊Dic3)⋊C3—41no—
93C2×C3×SD16—31no—
94C2×C3×SD16—31no—
95C2×C3×SD16—61no—
96C2×C3×SD16C3×D311yescharacteristic
97C3×C4×Q8—91no—
98C3×C4×D4—91no—
99Q8×Dic3—31no—
100Q8×Dic3—91no—
101D4×Dic3—31no—
102D4×Dic3—91no—
103C3×#107—91no—
104C3×#103—91no—
105C3×#102—91no—
106C3×#97—91no—
107#835—31no—
108#835—91no—
109#832—31no—
110#832—91no—
111#818—31no—
112#818—91no—
113#815—31no—
114#815—91no—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32—41no—
118C32×Q8—21no—
119C32×Q8—21no—
120C32×Q8—22no—
121C32×Q8—41no—
122C32×Q8C2×Dic312yescomplemented
123C32×Q8C2×C3×C412yescomplemented
124C32×D4—21no—
125C32×D4—21no—
126C32×D4—22no—
127C32×D4—41no—
128C32×D4C2×Dic312yes—
129C32×D4C2×C3×C412yes—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—41no—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C23×C311yescharacteristic
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—42no—
139C2×C3⋊Dic3C3×D411yescharacteristic
140C2×C3⋊Dic3C3×D411yescharacteristic
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—122no—
146C32×C8—22no—
147C32×C8C2×Dic312yescomplemented
148C32×C8C2×C3×C412yescomplemented
149C3⋊(C3⋊C8)—21no—
150C3×C3⋊C8—61no—
151#440—91noSylow
152C2×H(3,𝔽3)—42no—
153C2×H(3,𝔽3)C4×D411yescharacteristic
154C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
155C2×H(3,𝔽3)#13711yescharacteristic
156C2×C3×Q8—31no—
157C2×C3×Q8—31no—
158C2×C3×Q8—61no—
159C2×C3×Q8C2×C3×D311yescharacteristic
160C2×C3×D4—31no—
161C2×C3×D4—31no—
162C2×C3×D4—61no—
163C2×C3×D4C2×C3×D311yescharacteristic
164C22×C3×C4—181no—
165C22×Dic3—61no—
166C22×Dic3—181no—
167C3×SD16—34no—
168C3×SD16—34no—
169C3×SD16—64no—
170C3×SD16C3×Dic314yes—
171C2×C3×C8—31no—
172C2×C3×C8—31no—
173C2×C3×C8—61no—
174C2×C3×C8—91no—
175C2×C3×C8C2×C3×D311yescharacteristic
176C3×C4⋊C4—91no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C22⋊C4—181no—
180C3×C42—91no—
181C3×C42—181no—
182#217—61no—
183#217—181no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—61no—
187C4⋊Dic3—91no—
188C4⋊Dic3—91no—
189C4⋊Dic3—181no—
190C4×Dic3—31no—
191C4×Dic3—61no—
192C4×Dic3—91no—
193C4×Dic3—181no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—91no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—21no—
199C22×C32—42no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—81no—
205C22×C32—81no—
206C22×C32D3×D411yescharacteristic
207C22×C32C2×C3×D411yescharacteristic
208C32×C4—22no—
209C32×C4—22no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—42no—
213C32×C4#23711yescharacteristic
214C32×C4C22×Dic311yescharacteristic
215C32×C4C22×C3×C411yescharacteristic
216C32×C4C3×#4111yescharacteristic
217C3⋊Dic3—21no—
218C3⋊Dic3—21no—
219C3⋊Dic3—41no—
220C3⋊Dic3—41no—
221C3×Dic3—61no—
222C3×Dic3—61no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C2×SD16(C3⋊D3)⋊C311yescharacteristic
226C4×Q8—91no—
227C4×D4—91no—
228#107—91no—
229#103—91no—
230#102—91no—
231#97—91no—
232H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
233C23×C3—21no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—121no—
237C3×Q8—21no—
238C3×Q8—32no—
239C3×Q8—32no—
240C3×Q8—61no—
241C3×Q8—61no—
242C3×Q8—62no—
243C3×Q8—121no—
244C3×Q8C2×C3×Dic312yes—
245C3×D4—21no—
246C3×D4—32no—
247C3×D4—32no—
248C3×D4—61no—
249C3×D4—61no—
250C3×D4—62no—
251C3×D4—121no—
252C3×D4C2×C3×Dic312yes—
253C2×C3×C4—21no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—121no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4—362no—
265C2×C3×C4C22×C3×D311yescharacteristic
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—91no—
271C2×Dic3—91no—
272C2×Dic3—122no—
273C2×Dic3—181no—
274C2×Dic3—181no—
275C2×Dic3—362no—
276C3×C8—32no—
277C3×C8—32no—
278C3×C8—62no—
279C3×C8—181no—
280C3×C8C2×C3×Dic312yes—
281C3⋊C8—61no—
282C3⋊C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—42no—
287C2×C32—42no—
288C2×C32—82no—
289C2×C32#91411yescharacteristic
290C2×C32#91511yescharacteristic
291C2×C32D4×Dic311yescharacteristic
292C2×C32C3×C4×D411yescharacteristic
293C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
294C2×C32C3×#13711yescharacteristic
295C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
296C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
297C22×C4—181no—
298SD16(C3⋊Dic3)⋊C314yescomplemented
299C2×C8—91no—
300C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
301C4⋊C4—91no—
302C4⋊C4—91no—
303C4⋊C4—181no—
304C22⋊C4—181no—
305C42—91no—
306C42—181no—
307C22×C3—22no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—61no—
315C22×C3—122no—
316C22×C3—121no—
317C22×C3—121no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—241no—
321C22×C3—241no—
322C22×C3C3×D3×D411yescharacteristic
323C3×C4—22no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—62no—
329C3×C4—62no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—122no—
333C3×C4—181no—
334C3×C4—181no—
335C3×C4—361no—
336C3×C4—361no—
337C3×C4C3×#23711yescharacteristic
338C3×C4C22×C3×Dic311yescharacteristic
339Dic3—61no—
340Dic3—61no—
341Dic3—121no—
342Dic3—121no—
343Dic3—181no—
344Dic3—181no—
345Dic3—361no—
346Dic3—361no—
347C32—21no—
348C32#516411yescharacteristic, complemented
349C32C3×#44011yescharacteristic, complemented
350C23—21no—
351Q8—21no—
352Q8C2×(C3⋊Dic3)⋊C312yescomplemented
353D4—21no—
354D4C2×(C3⋊Dic3)⋊C312yes—
355C2×C4—21no—
356C2×C4—91no—
357C2×C4—91no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—362no—
361C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
362C8—181no—
363C8C2×(C3⋊Dic3)⋊C312yescomplemented
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—42no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—122no—
375C2×C3—122no—
376C2×C3—242no—
377C2×C3C3×#91411yescharacteristic
378C2×C3C3×#91511yescharacteristic
379C2×C3C3×D4×Dic311yescharacteristic
380C22—22no—
381C22—41no—
382C22—41no—
383C22D4×(C3⋊D3)⋊C311yescharacteristic
384C4—22no—
385C4—181no—
386C4—181no—
387C4—361no—
388C4—361no—
389C4#3830⋊C311yescharacteristic
390C4C22×(C3⋊Dic3)⋊C311yescharacteristic
391C3—31no—
392C3—31no—
393C3—61no—
394C3C3×#516411yescharacteristic
395C2—42no—
396C2#64123⋊C311yescharacteristic
397C2#64124⋊C311yescharacteristic
398C2D4×(C3⋊Dic3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×H(3,𝔽3)C22
cosocleC2×C32×C4C23×C3
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#440
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
2. Q8×(C3⋊Dic3)⋊C3C2semidirect
4. #63662⋊C3C2semidirect
6. #63645⋊C3C2semidirect
7. #63642⋊C3C2semidirect
8. #10592863C3semidirect
13. SD16×H(3,𝔽3)C4semidirect
25. Q8×C3⋊Dic3C2×C3semidirect
29. #63662C2×C3semidirect
31. #63645C2×C3semidirect
32. #63642C2×C3semidirect
39. Q8×H(3,𝔽3)C2×C4semidirect
49. C8×H(3,𝔽3)C2×C4semidirect
63. C32×SD16Dic3semidirect
64. C32×SD16C3×C4semidirect
122. C32×Q8C2×Dic3semidirect
123. C32×Q8C2×C3×C4semidirect
147. C32×C8C2×Dic3semidirect
148. C32×C8C2×C3×C4semidirect
232. H(3,𝔽3)#440semidirect
298. SD16(C3⋊Dic3)⋊C3semidirect
348. C32#5164semidirect
349. C32C3×#440semidirect
352. Q8C2×(C3⋊Dic3)⋊C3semidirect
363. C8C2×(C3⋊Dic3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
Q8×(C3⋊Dic3)⋊C32
D4×(C3⋊Dic3)⋊C32
#63662⋊C32
#63659⋊C32
#63645⋊C32
#63642⋊C32
#105928633
C3×#51643