Finite Group Databasebeta

← #10592727⋊C3 #13142799

Subgroups — #10592727⋊C3 #13142799

Order 1728.

462 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic
3C2×#3756⋊C3C211yesmaximal, characteristic, complemented
4#64084⋊C3C211yesmaximal, characteristic
5#63636⋊C3C211yesmaximal, characteristic
6#63637⋊C3C211yesmaximal, characteristic, complemented
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
8#10592727C311yesmaximal, characteristic, complemented
9C3×#5028—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C42×H(3,𝔽3)—21no—
16C42×H(3,𝔽3)C2211yescharacteristic
17#3755⋊C3—22no—
18#3756⋊C3—22no—
19#3752⋊C3—21no—
20#3752⋊C3—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C32×C4×Q8—21no—
24C32×C4×Q8D311yescharacteristic, complemented
25C32×C4×Q8C2×C311yescharacteristic, complemented
26C2×#3756C2×C311yescharacteristic, complemented
27C2×#3755C2×C311yescharacteristic
28#64084C2×C311yescharacteristic
29C2×C3×#216—31no—
30C2×C3×#215—31no—
31C3×#875—31no—
32#63637C2×C311yescharacteristic, complemented
33#63636C2×C311yescharacteristic
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×#810—31no—
36C3×#809—31no—
37C3×C4⋊(C3⋊C8)—31no—
38C22×(C3⋊D3)⋊C3—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)D412yes—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)D411yescharacteristic
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53C3×#464—91no—
54#5028—31no—
55#5028—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C22×C311yescharacteristic
59C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
60C2×#634C22×C311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×Dic6—31no—
63C32×C4⋊C4—22no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C22×C311yescharacteristic
69C32×C42—22no—
70C32×C42—21no—
71C32×C42—21no—
72C32×C42—21no—
73C32×C42C2×D311yescharacteristic
74C32×C42C22×C311yescharacteristic
75#3756—22no—
76#3755—22no—
77#3752—21no—
78#3752—21no—
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C3×#216—62no—
82C3×#215—62no—
83C3×(C2×D3)⋊C4—61no—
84C3×(C2×D3)⋊C4—61no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C4×H(3,𝔽3)—22no—
92C4×H(3,𝔽3)—21no—
93C4×H(3,𝔽3)—21no—
94C4×H(3,𝔽3)—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—41no—
98C2×C3×Q16—91no—
99C2×C3×SD16—91no—
100C3×#124—91no—
101C3×C4×Q8—31no—
102C3×C4×Q8—31no—
103C3×C4×Q8—61no—
104C3×C4×Q8C3×D311yescharacteristic
105C2×#216—31no—
106C2×#216—91no—
107C2×#215—31no—
108C2×#215—91no—
109#875—31no—
110#875—91no—
111C3×C4⋊C8—91no—
112C3×#103—91no—
113C3×#102—91no—
114#810—31no—
115#810—91no—
116#809—31no—
117#809—91no—
118C4⋊(C3⋊C8)—31no—
119C4⋊(C3⋊C8)—91no—
120C22×C3⋊D3—21no—
121C22×C3×D3—61no—
122C32×Q8—22no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—22no—
126C32×Q8D1212yes—
127C32×Q8C3×D412yes—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—22no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4#6711yescharacteristic
138C2×C32×C4#6711yescharacteristic
139C2×C32×C4C3×D411yescharacteristic
140C2×C32×C4C3×D411yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C23×C311yescharacteristic
143C2×C3⋊Dic3—21no—
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)—21no—
146#634—21no—
147#634—21no—
148C2×C3×Dic3—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)—21no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157#464—91noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)#13511yescharacteristic
160C2×H(3,𝔽3)#13711yescharacteristic
161(C3⋊D3)⋊C3—81no—
162C2×C3×Q8—31no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8—91no—
166C2×C3×Q8C2×C3×D311yescharacteristic
167C2×C3×D4—91no—
168C2×D12—31no—
169C2×D12—91no—
170C2×Dic6—31no—
171C2×Dic6—91no—
172C3×Q16—182no—
173C3×SD16—182no—
174C2×C3×C8—91no—
175C2×C3×C8—91no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—31no—
178C3×C4⋊C4—31no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—61no—
181C3×C4⋊C4—62no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4C2×C3×D311yescharacteristic
184C3×C22⋊C4—181no—
185C3×C22⋊C4—181no—
186C3×C42—21no—
187C3×C42—31no—
188C3×C42—31no—
189C3×C42—61no—
190C3×C42—61no—
191C3×C42—62no—
192C3×C42—61no—
193C3×C42C2×C3×D311yescharacteristic
194#216—62no—
195#216—182no—
196#215—62no—
197#215—182no—
198(C2×D3)⋊C4—61no—
199(C2×D3)⋊C4—61no—
200(C2×D3)⋊C4—181no—
201(C2×D3)⋊C4—181no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—91no—
206C22×C32—21no—
207C22×C32C2×#6711yescharacteristic
208C22×C32C2×C3×D411yescharacteristic
209C2×C3⋊D3—41no—
210C2×C3⋊D3—41no—
211C2×C3⋊D3—41no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C32×C4—22no—
216C32×C4—24no—
217C32×C4—22no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—22no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—41no—
226C32×C4—42no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4C2×D1211yescharacteristic
230C32×C4#23511yescharacteristic
231C32×C4C2×C3×D411yescharacteristic
232C32×C4C3×#4111yescharacteristic
233C3⋊Dic3—41no—
234C3×Dic3—121no—
235C2×Q16—91no—
236C2×SD16—91no—
237#124—91no—
238C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
239C4⋊C8—91no—
240#103—91no—
241#102—91no—
242H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
243C23×C3—181no—
244C22×D3—61no—
245C22×D3—181no—
246C3×Q8—21no—
247C3×Q8—32no—
248C3×Q8—32no—
249C3×Q8—61no—
250C3×Q8—61no—
251C3×Q8—62no—
252C3×Q8—62no—
253C3×Q8—181no—
254C3×Q8—181no—
255C3×Q8C3×D1212yes—
256C3×D4—181no—
257C3×D4—181no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—62no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—62no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—181no—
276C2×C3×C4C3×#6711yescharacteristic
277C2×C3×C4C3×#6711yescharacteristic
278C2×C3×C4C22×C3×D311yescharacteristic
279C2×Dic3—61no—
280C2×Dic3—181no—
281D12—61no—
282D12—61no—
283D12—181no—
284D12—181no—
285Dic6—61no—
286Dic6—61no—
287Dic6—181no—
288Dic6—181no—
289C3×C8—181no—
290C3×C8—181no—
291C3⋊C8—61no—
292C3⋊C8—61no—
293C3⋊C8—181no—
294C3⋊C8—181no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32#93011yescharacteristic
299C2×C32#94211yescharacteristic
300C2×C32#95011yescharacteristic
301C2×C32C3×#12111yescharacteristic
302C2×C32C3×#13511yescharacteristic
303C2×C32C3×#13711yescharacteristic
304C3⋊D3—81no—
305C3×D3—241no—
306C2×Q8—91no—
307C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
308C2×D4—91no—
309Q16—182no—
310SD16—182no—
311C2×C8—91no—
312C2×C8—91no—
313C4⋊C4—21no—
314C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
315C22⋊C4—181no—
316C22⋊C4—181no—
317C42—21no—
318C42C2×(C3⋊D3)⋊C311yescharacteristic
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—61no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3—361no—
325C22×C3C2×C3×#6711yescharacteristic
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—361no—
330C2×D3—361no—
331C2×D3—361no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—41no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—62no—
345C3×C4—62no—
346C3×C4—64no—
347C3×C4—62no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—121no—
351C3×C4—121no—
352C3×C4—121no—
353C3×C4—122no—
354C3×C4—361no—
355C3×C4C2×C3×D1211yescharacteristic
356C3×C4C3×#23511yescharacteristic
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#502811yescharacteristic, complemented
361C32C3×#46411yescharacteristic, complemented
362C23—181no—
363Q8—21no—
364Q8—181no—
365Q8—181no—
366Q8C4⋊((C3⋊D3)⋊C3)12yes—
367D4—181no—
368D4—181no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—181no—
372C2×C4#638⋊C311yescharacteristic
373C2×C4#638⋊C311yescharacteristic
374C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
375C8—181no—
376C8—181no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—61no—
386C2×C3—721no—
387C2×C3C3×#93011yescharacteristic
388C2×C3C3×#94211yescharacteristic
389C2×C3C3×#95011yescharacteristic
390D3—241no—
391D3—721no—
392C22—361no—
393C22—361no—
394C22—361no—
395C22C2×#638⋊C311yescharacteristic
396C4—22no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—361no—
401C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
402C4#3828⋊C311yescharacteristic
403C3—31no—
404C3—31no—
405C3—61no—
406C3C3×#502811yescharacteristic
407C2—721no—
408C2#64139⋊C311yescharacteristic
409C2#64151⋊C311yescharacteristic
410C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×H(3,𝔽3)C2semidirect
3. C2×#3756⋊C3C2semidirect
6. #63637⋊C3C2semidirect
7. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
8. #10592727C3semidirect
24. C32×C4×Q8D3semidirect
25. C32×C4×Q8C2×C3semidirect
26. C2×#3756C2×C3semidirect
32. #63637C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
238. C4×Q8(C3⋊D3)⋊C3semidirect
242. H(3,𝔽3)#464semidirect
360. C32#5028semidirect
361. C32C3×#464semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
C2×#3755⋊C32
C2×#3756⋊C32
#64084⋊C32
#63636⋊C32
#63637⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927273
C3×#50283