Finite Group Databasebeta

← #10592436⋊C3 #13141620

Subgroups — #10592436⋊C3 #13141620

Order 1728.

421 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64139⋊C3C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#63645⋊C3C211yesmaximal, characteristic, complemented
7#63644⋊C3C211yesmaximal, characteristic, complemented
8#10592436C311yesmaximal, characteristic, complemented
9C3×#4737—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#3757⋊C3—21no—
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4×(C3⋊Dic3)⋊C3—21no—
21#64139C2×C311yescharacteristic
22#64135C2×C311yescharacteristic, complemented
23C3×#930—31no—
24C3×#926—31no—
25C32×C22⋊C8—21no—
26C32×C22⋊C8D311yescharacteristic, complemented
27C32×C22⋊C8C2×C311yescharacteristic, complemented
28#63648C2×C311yescharacteristic, complemented
29#63648C2×C311yescharacteristic, complemented
30#63645C2×C311yescharacteristic, complemented
31#63644C2×C311yescharacteristic, complemented
32C3×#821—31no—
33C3×#821—31no—
34C3×#818—31no—
35C3×#817—31no—
36C23×H(3,𝔽3)D411yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#638⋊C3—42no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C8×H(3,𝔽3)—21no—
49C8×H(3,𝔽3)—21no—
50C3×#482—91no—
51#4737—31no—
52#4737—91no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C2×#638—21no—
57C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×D12—31no—
60C2×C32×C8—21no—
61C2×C32×C8—21no—
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C2×D311yescharacteristic
64C2×C32×C8C22×C311yescharacteristic
65C2×C32×C8C22×C311yescharacteristic
66#3757—21no—
67#3752—21no—
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71C4×C3⋊Dic3—21no—
72C3×#217—61no—
73C3×(C2×D3)⋊C4—61no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—31no—
77C3×C4×Dic3—61no—
78C22×H(3,𝔽3)—21no—
79C22×H(3,𝔽3)—21no—
80C22×H(3,𝔽3)—21no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)—21no—
86C4×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—41no—
91C3×#121—91no—
92C3×#117—91no—
93#930—31no—
94#930—91no—
95#926—31no—
96#926—91no—
97C3×#107—91no—
98C3×#106—91no—
99C3×#102—91no—
100C3×#102—91no—
101C3×C22⋊C8—31no—
102C3×C22⋊C8—31no—
103C3×C22⋊C8—61no—
104C3×C22⋊C8C3×D311yescharacteristic
105#821—31no—
106#821—31no—
107#821—91no—
108#821—91no—
109#818—31no—
110#818—91no—
111#817—31no—
112#817—91no—
113C23×C32—21no—
114C23×C32D1211yescharacteristic
115C23×C32C3×D411yescharacteristic
116C22×C3⋊D3—21no—
117C22×C3×D3—61no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—41no—
125C2×C32×C4—41no—
126C2×C32×C4D1211yescharacteristic
127C2×C32×C4C3×D411yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C23×C311yescharacteristic
130#638—42no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136C3×#67—122no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C3×D12—61no—
141C3×D12—61no—
142C32×C8—21no—
143C32×C8—21no—
144C32×C8—21no—
145C32×C8—21no—
146C32×C8—41no—
147C32×C8—41no—
148#482—91noSylow
149C2×H(3,𝔽3)—41no—
150C2×H(3,𝔽3)#12311yescharacteristic
151C2×H(3,𝔽3)#13511yescharacteristic
152C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
153(C3⋊D3)⋊C3—81no—
154C2×C3×D4—91no—
155C2×C3×D4—181no—
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—61no—
159C22×C3×C4C2×C3×D311yescharacteristic
160C2×#67—61no—
161C2×#67—181no—
162C2×D12—31no—
163C2×D12—91no—
164C2×C3×C8—31no—
165C2×C3×C8—31no—
166C2×C3×C8—31no—
167C2×C3×C8—31no—
168C2×C3×C8—61no—
169C2×C3×C8—61no—
170C2×C3×C8C2×C3×D311yescharacteristic
171C2×C3×C8C2×C3×D311yescharacteristic
172C3×C4⋊C4—91no—
173C3×C4⋊C4—91no—
174C3×C4⋊C4—91no—
175C3×C22⋊C4—181no—
176C3×C22⋊C4—181no—
177C3×C42—181no—
178#217—61no—
179#217—181no—
180(C2×D3)⋊C4—61no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—31no—
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—91no—
186C4⋊Dic3—91no—
187C4⋊Dic3—91no—
188C4×Dic3—61no—
189C4×Dic3—181no—
190C22×C32—22no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32C2×D1211yescharacteristic
201C22×C32C2×C3×D411yescharacteristic
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—41no—
213C32×C4#23711yescharacteristic
214C32×C4#23711yescharacteristic
215C32×C4C3×#4111yescharacteristic
216C32×C4C3×#4111yescharacteristic
217C3⋊Dic3—41no—
218C3⋊Dic3—41no—
219C3⋊Dic3—41no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223#121—91no—
224#117—91no—
225#107—91no—
226#106—91no—
227#102—91no—
228#102—91no—
229C22⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
230H(3,𝔽3)#48211yescharacteristic, complemented, Sylow
231C23×C3—31no—
232C23×C3—31no—
233C23×C3—61no—
234C23×C3—181no—
235C23×C3C3×D1211yescharacteristic
236C22×D3—61no—
237C22×D3—181no—
238C3×D4—181no—
239C3×D4—181no—
240C3×D4—362no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—121no—
254C2×C3×C4—121no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4—181no—
258C2×C3×C4C3×D1211yescharacteristic
259C2×C3×C4C22×C3×D311yescharacteristic
260#67—122no—
261#67—362no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267C2×Dic3—181no—
268D12—61no—
269D12—61no—
270D12—181no—
271D12—181no—
272C3×C8—21no—
273C3×C8—21no—
274C3×C8—61no—
275C3×C8—61no—
276C3×C8—61no—
277C3×C8—61no—
278C3×C8—121no—
279C3×C8—121no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—42no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32#88611yescharacteristic
287C2×C32#90411yescharacteristic
288C2×C32#90811yescharacteristic
289C2×C32C3×#12311yescharacteristic
290C2×C32C3×#13511yescharacteristic
291C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
292C3⋊D3—81no—
293C3×D3—241no—
294C2×D4—91no—
295C2×D4—181no—
296C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
297C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
298C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
299C4⋊C4—91no—
300C4⋊C4—91no—
301C4⋊C4—91no—
302C22⋊C4—181no—
303C22⋊C4—181no—
304C42—181no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—31no—
309C22×C3—31no—
310C22×C3—61no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—61no—
315C22×C3—61no—
316C22×C3—62no—
317C22×C3—61no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3C2×C3×D1211yescharacteristic
324C2×D3—121no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—361no—
328C2×D3—361no—
329C2×D3—361no—
330C3×C4—21no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—31no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—61no—
339C3×C4—121no—
340C3×C4—361no—
341C3×C4—361no—
342C3×C4—361no—
343C3×C4C3×#23711yescharacteristic
344C3×C4C3×#23711yescharacteristic
345Dic3—121no—
346Dic3—121no—
347Dic3—121no—
348Dic3—361no—
349Dic3—361no—
350Dic3—361no—
351C32—21no—
352C32#473711yescharacteristic, complemented
353C32C3×#48211yescharacteristic, complemented
354C23—181no—
355C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
356D4—181no—
357D4—181no—
358D4—362no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—181no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
365C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
366C8—21no—
367C8—21no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—41no—
375C2×C3—61no—
376C2×C3—61no—
377C2×C3—61no—
378C2×C3—121no—
379C2×C3—121no—
380C2×C3—122no—
381C2×C3—721no—
382C2×C3C3×#88611yescharacteristic
383C2×C3C3×#90411yescharacteristic
384C2×C3C3×#90811yescharacteristic
385D3—241no—
386D3—721no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—361no—
391C22—361no—
392C22—361no—
393C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
394C4—21no—
395C4—361no—
396C4—361no—
397C4—361no—
398C4#3830⋊C311yescharacteristic
399C4#3830⋊C311yescharacteristic
400C3—31no—
401C3—31no—
402C3—61no—
403C3C3×#473711yescharacteristic
404C2—41no—
405C2—721no—
406C2#64095⋊C311yescharacteristic
407C2#64113⋊C311yescharacteristic
408C2#64117⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radical#64139⋊C3C2
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #64135⋊C3C2semidirect
3. H(3,𝔽3)×C22⋊C8C2semidirect
4. #63648⋊C3C2semidirect
5. #63648⋊C3C2semidirect
6. #63645⋊C3C2semidirect
7. #63644⋊C3C2semidirect
8. #10592436C3semidirect
22. #64135C2×C3semidirect
26. C32×C22⋊C8D3semidirect
27. C32×C22⋊C8C2×C3semidirect
28. #63648C2×C3semidirect
29. #63648C2×C3semidirect
30. #63645C2×C3semidirect
31. #63644C2×C3semidirect
229. C22⋊C8(C3⋊D3)⋊C3semidirect
230. H(3,𝔽3)#482semidirect
352. C32#4737semidirect
353. C32C3×#482semidirect

Maximal subgroups

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

subgroupindex
#64139⋊C32
#64135⋊C32
H(3,𝔽3)×C22⋊C82
#63648⋊C32
#63645⋊C32
#63644⋊C32
#105924363
C3×#47373