Finite Group Databasebeta

← #10592430⋊C3 #13141615

Subgroups — #10592430⋊C3 #13141615

Order 1728.

708 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic, complemented
3C2×#3744⋊C3C212yesmaximal, complemented
4H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
5#63648⋊C3C212yesmaximal, complemented
6#10592430C311yesmaximal, characteristic, complemented
7C3×#4731—31nomaximal
8C23×(C3⋊D3)⋊C3—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)—24no—
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×#634⋊C3C2211yescharacteristic, complemented
14C2×C8×H(3,𝔽3)C2212yescomplemented
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
17#3750⋊C3—21no—
18#3744⋊C3—24no—
19C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
20C22×C3×D12—31no—
21#64133C2×C311yescharacteristic, complemented
22C2×#3744C2×C312yescomplemented
23C3×#924—31no—
24C2×C3×#204—32no—
25C32×C22⋊C8—21no—
26C32×C22⋊C8D311yescharacteristic, complemented
27C32×C22⋊C8C2×C311yescharacteristic, complemented
28#63648C2×C312yescomplemented
29C3×#821—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×(C3⋊D3)⋊C3—24no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—42no—
34C22×(C3⋊D3)⋊C3—42no—
35C2×C4×H(3,𝔽3)—22no—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×(C3⋊Dic3)⋊C3—21no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C4⋊((C3⋊D3)⋊C3)—24no—
41C4⋊((C3⋊D3)⋊C3)—22no—
42C4⋊((C3⋊D3)⋊C3)D414yes—
43#634⋊C3—22no—
44C8×H(3,𝔽3)—22no—
45C3×#450—91no—
46#4731—31no—
47#4731—91no—
48C23×C3⋊D3—21no—
49C23×C3×D3—61no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)—24no—
55C2×C4⋊(C3⋊D3)C22×C312yes—
56C2×#634C22×C311yescharacteristic, complemented
57C2×C3×D12—32no—
58C2×C3×D12—64no—
59C2×C3×D12—61no—
60C2×C3×Dic6—31no—
61C2×C32×C8—22no—
62C2×C32×C8C2×D312yescomplemented
63C2×C32×C8C22×C312yescomplemented
64#3757—21no—
65C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
66#3750—21no—
67#3744—24no—
68C3×#217—61no—
69C3×C4⋊Dic3—31no—
70C3×Dic3⋊C4—61no—
71C3×#204—64no—
72C22×H(3,𝔽3)—21no—
73C22×H(3,𝔽3)—21no—
74C22×H(3,𝔽3)SD1612yescomplemented
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C2×(C3⋊D3)⋊C3—24no—
77C2×(C3⋊D3)⋊C3—48no—
78C2×(C3⋊D3)⋊C3—42no—
79C2×(C3⋊D3)⋊C3—42no—
80C4×H(3,𝔽3)—21no—
81C4×H(3,𝔽3)C2×D412yes—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84C22×C3×D4—91no—
85C22×D12—31no—
86C22×D12—91no—
87C2×C3×SD16—92no—
88C3×C22⋊Q8—91no—
89#924—31no—
90#924—91no—
91C2×#204—32no—
92C2×#204—92no—
93C3×#102—92no—
94C3×C22⋊C8—31no—
95C3×C22⋊C8—31no—
96C3×C22⋊C8—61no—
97C3×C22⋊C8C3×D311yescharacteristic
98#821—32no—
99#821—92no—
100C23×C32—21no—
101C23×C32D1211yescharacteristic
102C23×C32C3×D411yescharacteristic
103C22×C3⋊D3—24no—
104C22×C3⋊D3—22no—
105C22×C3⋊D3—42no—
106C22×C3⋊D3—42no—
107C22×C3×D3—64no—
108C22×C3×D3—62no—
109C22×C3×D3—122no—
110C22×C3×D3—122no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—42no—
116C2×C32×C4D1211yescharacteristic
117C2×C32×C4C3×D411yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C23×C311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C4⋊(C3⋊D3)—24no—
123C4⋊(C3⋊D3)—22no—
124C4⋊(C3⋊D3)C3×D414yes—
125#634—22no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—34no—
129C3×D12—64no—
130C3×D12—62no—
131C3×Dic6—62no—
132C32×C8—22no—
133C32×C8—22no—
134C32×C8—42no—
135#450—91noSylow
136C2×H(3,𝔽3)—22no—
137C2×H(3,𝔽3)#12011yescharacteristic
138C2×H(3,𝔽3)C2×SD1611yescharacteristic
139C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
140(C3⋊D3)⋊C3—44no—
141C24×C3—181no—
142C23×D3—61no—
143C23×D3—181no—
144C2×C3×Q8—91no—
145C2×C3×D4—92no—
146C2×C3×D4—181no—
147C2×C3×D4—184no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—61no—
151C22×C3×C4C2×C3×D311yescharacteristic
152C2×D12—32no—
153C2×D12—64no—
154C2×D12—61no—
155C2×D12—92no—
156C2×D12—184no—
157C2×D12—181no—
158C2×Dic6—31no—
159C2×Dic6—91no—
160C3×SD16—184no—
161C2×C3×C8—32no—
162C2×C3×C8—32no—
163C2×C3×C8—62no—
164C2×C3×C8C2×C3×D312yes—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—181no—
167C3×C22⋊C4—181no—
168#217—61no—
169#217—181no—
170C4⋊Dic3—31no—
171C4⋊Dic3—91no—
172Dic3⋊C4—61no—
173Dic3⋊C4—181no—
174#204—64no—
175#204—184no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—22no—
181C22×C32—21no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32#20412yescomplemented
185C22×C32C3×SD1612yescomplemented
186C22×C32C2×D1211yescharacteristic
187C22×C32C2×C3×D411yescharacteristic
188C2×C3⋊D3—24no—
189C2×C3⋊D3—48no—
190C2×C3⋊D3—42no—
191C2×C3⋊D3—42no—
192C2×C3×D3—64no—
193C2×C3×D3—128no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—22no—
199C32×C4—41no—
200C32×C4D3×D412yes—
201C32×C4C2×C3×D412yes—
202C3⋊Dic3—41no—
203C3⋊Dic3—41no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C22×D4—91no—
207C2×SD16—92no—
208C22⋊Q8—91no—
209#102—92no—
210C22⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
211H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
212C23×C3—31no—
213C23×C3—31no—
214C23×C3—61no—
215C23×C3—182no—
216C23×C3—184no—
217C23×C3—362no—
218C23×C3—362no—
219C23×C3C3×D1211yescharacteristic
220C22×D3—64no—
221C22×D3—62no—
222C22×D3—122no—
223C22×D3—122no—
224C22×D3—182no—
225C22×D3—184no—
226C22×D3—362no—
227C22×D3—362no—
228C3×Q8—182no—
229C3×D4—94no—
230C3×D4—182no—
231C3×D4—184no—
232C2×C3×C4—22no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—122no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4C3×D1211yescharacteristic
245C2×C3×C4C22×C3×D311yescharacteristic
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—181no—
249C2×Dic3—181no—
250D12—34no—
251D12—64no—
252D12—62no—
253D12—94no—
254D12—182no—
255D12—184no—
256Dic6—62no—
257Dic6—182no—
258C3×C8—22no—
259C3×C8—62no—
260C3×C8—62no—
261C3×C8—122no—
262C2×C32—22no—
263C2×C32—22no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—42no—
268C2×C32#88211yescharacteristic
269C2×C32C2×#20411yescharacteristic
270C2×C32#90811yescharacteristic
271C2×C32C3×#12011yescharacteristic
272C2×C32C2×C3×SD1611yescharacteristic
273C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
274C3⋊D3—44no—
275C3×D3—124no—
276C24—181no—
277C2×Q8—91no—
278C2×D4—92no—
279C2×D4—184no—
280C2×D4—181no—
281C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
282SD16—184no—
283C2×C8C2×(C3⋊D3)⋊C312yescomplemented
284C4⋊C4—91no—
285C4⋊C4—181no—
286C22⋊C4—181no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—61no—
294C22×C3—61no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—62no—
299C22×C3—121no—
300C22×C3—121no—
301C22×C3—184no—
302C22×C3—362no—
303C22×C3—362no—
304C22×C3—368no—
305C22×C3C3×#20412yes—
306C22×C3C2×C3×D1211yescharacteristic
307C2×D3—64no—
308C2×D3—128no—
309C2×D3—122no—
310C2×D3—122no—
311C2×D3—184no—
312C2×D3—362no—
313C2×D3—362no—
314C2×D3—368no—
315C3×C4—21no—
316C3×C4—32no—
317C3×C4—32no—
318C3×C4—61no—
319C3×C4—61no—
320C3×C4—62no—
321C3×C4—121no—
322C3×C4—361no—
323C3×C4—361no—
324C3×C4C3×D3×D412yes—
325Dic3—121no—
326Dic3—121no—
327Dic3—361no—
328Dic3—361no—
329C32—21no—
330C32#473111yescharacteristic, complemented
331C32C3×#45011yescharacteristic, complemented
332C23—184no—
333C23—182no—
334C23—362no—
335C23—362no—
336C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
337Q8—182no—
338D4—94no—
339D4—184no—
340D4—182no—
341C2×C4—22no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
345C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
346C8—22no—
347C2×C3—22no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—62no—
355C2×C3—62no—
356C2×C3—61no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—122no—
360C2×C3—364no—
361C2×C3C3×#88211yescharacteristic
362C2×C3C2×C3×#20411yescharacteristic
363C2×C3C3×#90811yescharacteristic
364D3—124no—
365D3—364no—
366C22—21no—
367C22—21no—
368C22—184no—
369C22—368no—
370C22—362no—
371C22—362no—
372C22#3744⋊C312yescomplemented
373C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
374C4—21no—
375C4—361no—
376C4—361no—
377C4D4×(C3⋊D3)⋊C312yes—
378C3—31no—
379C3—31no—
380C3—61no—
381C3C3×#473111yescharacteristic
382C2—22no—
383C2—364no—
384C2#64091⋊C311yescharacteristic
385C2C2×#3744⋊C311yescharacteristic
386C2#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 radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#450
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. #64133⋊C3C2semidirect
3. C2×#3744⋊C3C2semidirect
4. H(3,𝔽3)×C22⋊C8C2semidirect
5. #63648⋊C3C2semidirect
6. #10592430C3semidirect
13. C2×#634⋊C3C22semidirect
14. C2×C8×H(3,𝔽3)C22semidirect
16. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. #64133C2×C3semidirect
22. C2×#3744C2×C3semidirect
26. C32×C22⋊C8D3semidirect
27. C32×C22⋊C8C2×C3semidirect
28. #63648C2×C3semidirect
56. C2×#634C22×C3semidirect
62. C2×C32×C8C2×D3semidirect
63. C2×C32×C8C22×C3semidirect
65. C4⋊(C3⋊Dic3)C22×C3semidirect
74. C22×H(3,𝔽3)SD16semidirect
184. C22×C32#204semidirect
185. C22×C32C3×SD16semidirect
210. C22⋊C8(C3⋊D3)⋊C3semidirect
211. H(3,𝔽3)#450semidirect
283. C2×C8C2×(C3⋊D3)⋊C3semidirect
330. C32#4731semidirect
331. C32C3×#450semidirect
372. C22#3744⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
#64133⋊C32
C2×#3744⋊C32
H(3,𝔽3)×C22⋊C82
#63648⋊C32
#105924303
C3×#47313