Finite Group Databasebeta

← #10592903⋊C3 #13143500

Subgroups — #10592903⋊C3 #13143500

Order 1728.

410 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 86 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
1H(3,𝔽3)×#137C211yesmaximal, characteristic
2#64160⋊C3C211yesmaximal, characteristic
3#11721689C211yesmaximal, characteristic
4C2×#3756⋊C3C211yesmaximal, characteristic, complemented
5#63663⋊C3C211yesmaximal, characteristic, complemented
6#63651⋊C3C211yesmaximal, characteristic, complemented
7#63649⋊C3C211yesmaximal, characteristic, complemented
8#10592903C311yesmaximal, characteristic, complemented
9C3×#5204—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13Q16×H(3,𝔽3)—21no—
14SD16×H(3,𝔽3)—21no—
15M16×H(3,𝔽3)C2211yescharacteristic
16#3756⋊C3—21no—
17#3756⋊C3—21no—
18#3756⋊C3—21no—
19#3754⋊C3—21no—
20#3748⋊C3—21no—
21#3748⋊C3C2211yescharacteristic
22#3748⋊C3C2211yescharacteristic
23C2×(C3⋊(C3⋊C8))⋊C3—21no—
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
25C32×#137—21no—
26C32×#137D311yescharacteristic
27C32×#137C2×C311yescharacteristic
28#64160C2×C311yescharacteristic
29#64157C2×C311yescharacteristic
30C2×#3756C2×C311yescharacteristic, complemented
31C3×#951—31no—
32C3×#948—31no—
33C2×C3×#216—31no—
34#63663C2×C311yescharacteristic, complemented
35#63651C2×C311yescharacteristic, complemented
36#63649C2×C311yescharacteristic, complemented
37C3×#836—31no—
38C3×#824—31no—
39C3×#822—31no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)D411yescharacteristic
43D4×H(3,𝔽3)—21no—
44D4×H(3,𝔽3)D411yescharacteristic
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×(C3⋊Dic3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51C8×H(3,𝔽3)—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
56C3×#473—91no—
57#5204—31no—
58#5204—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C22×C311yescharacteristic
62C2×C32×Q8—21no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C22×C311yescharacteristic
65C2×#634C22×C311yescharacteristic
66C2×C3×Dic6—31no—
67C32×Q16—22no—
68C32×Q16—21no—
69C32×Q16—21no—
70C32×SD16—22no—
71C32×SD16—21no—
72C32×SD16—21no—
73C32×M16—21no—
74C32×M16C2×D311yescharacteristic
75C32×M16C22×C311yescharacteristic
76#3756—21no—
77#3756—21no—
78#3756—21no—
79#3754—21no—
80#3748—21no—
81#3748C22×C311yescharacteristic
82#3748C22×C311yescharacteristic
83C2×C3⋊(C3⋊C8)—21no—
84C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
85C3×#216—61no—
86C3×#216—61no—
87C3×#216—61no—
88C3×#214—61no—
89C3×#208—31no—
90C3×#208—31no—
91C3×#208—61no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—61no—
94C22×H(3,𝔽3)—21no—
95C22×H(3,𝔽3)#4111yescharacteristic
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)—41no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100(C3⋊Dic3)⋊C3—41no—
101C3×#137—31no—
102C3×#137—31no—
103C3×#137—61no—
104C3×#137—91no—
105C3×#137C3×D311yescharacteristic
106C2×C3×Q16—91no—
107C3×#131—91no—
108#951—31no—
109#951—91no—
110#948—31no—
111#948—91no—
112C2×#216—31no—
113C2×#216—91no—
114C3×#108—91no—
115C3×#101—91no—
116C3×#101—91no—
117#836—31no—
118#836—91no—
119#824—31no—
120#824—91no—
121#822—31no—
122#822—91no—
123C32×Q8—22no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×Q8—41no—
130C32×Q8#6711yescharacteristic
131C32×Q8C3×D411yescharacteristic
132C32×D4—21no—
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—41no—
136C32×D4#6711yescharacteristic
137C32×D4C3×D411yescharacteristic
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—41no—
144C2×C32×C4—41no—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C23×C311yescharacteristic
147C2×C3⋊Dic3—21no—
148#634—21no—
149#634—21no—
150C2×C3×Dic3—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C32×C8—22no—
154C32×C8—21no—
155C32×C8—21no—
156C3⋊(C3⋊C8)—21no—
157C3⋊(C3⋊C8)—21no—
158C3⋊(C3⋊C8)C3×D411yescharacteristic
159C3⋊(C3⋊C8)C3×D411yescharacteristic
160C3×C3⋊C8—31no—
161C3×C3⋊C8—31no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164#473—91noSylow
165C2×H(3,𝔽3)—21no—
166C2×H(3,𝔽3)—41no—
167C2×H(3,𝔽3)#12111yescharacteristic
168C3×#41—31no—
169C3×#41—31no—
170C3×#41—61no—
171C3×#41C2×C3×D311yescharacteristic
172C2×C3×Q8—31no—
173C2×C3×Q8—31no—
174C2×C3×Q8—61no—
175C2×C3×Q8—91no—
176C2×C3×Q8C2×C3×D311yescharacteristic
177C2×Dic6—31no—
178C2×Dic6—91no—
179C3×Q16—21no—
180C3×Q16—61no—
181C3×Q16—61no—
182C3×Q16—62no—
183C3×Q16—181no—
184C3×Q16—181no—
185C3×Q16—181no—
186C3×SD16—21no—
187C3×SD16—61no—
188C3×SD16—61no—
189C3×SD16—62no—
190C3×SD16—181no—
191C3×M16—31no—
192C3×M16—31no—
193C3×M16—61no—
194C3×M16—91no—
195C3×M16—91no—
196C3×M16—181no—
197C3×M16C2×C3×D311yescharacteristic
198C2×C3×C8—91no—
199C2×C3×C8—181no—
200#216—61no—
201#216—61no—
202#216—61no—
203#216—181no—
204#216—181no—
205#216—181no—
206#214—61no—
207#214—181no—
208#208—31no—
209#208—31no—
210#208—61no—
211#208—91no—
212#208—91no—
213#208—181no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—61no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—181no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—41no—
222C22×C32#23711yescharacteristic
223C22×C32C3×#4111yescharacteristic
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—41no—
229C32×C4—42no—
230C32×C4—41no—
231C32×C4—41no—
232C32×C4D3×D411yescharacteristic
233C32×C4C2×#6711yescharacteristic
234C32×C4C2×C3×D411yescharacteristic
235C32×C4C2×C3×D411yescharacteristic
236C3⋊Dic3—41no—
237C3×Dic3—121no—
238#137—91no—
239#137(C3⋊D3)⋊C311yescharacteristic
240C2×Q16—91no—
241#131—91no—
242#108—91no—
243#101—91no—
244#101—91no—
245H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
246C3×Q8—21no—
247C3×Q8—21no—
248C3×Q8—31no—
249C3×Q8—31no—
250C3×Q8—61no—
251C3×Q8—61no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8—62no—
255C3×Q8—61no—
256C3×Q8—121no—
257C3×Q8—181no—
258C3×Q8—181no—
259C3×Q8C3×#6711yescharacteristic
260C3×D4—21no—
261C3×D4—31no—
262C3×D4—31no—
263C3×D4—61no—
264C3×D4—61no—
265C3×D4—61no—
266C3×D4—121no—
267C3×D4C3×#6711yescharacteristic
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—121no—
278C2×C3×C4—121no—
279C2×C3×C4—181no—
280C2×C3×C4C22×C3×D311yescharacteristic
281C2×Dic3—61no—
282C2×Dic3—181no—
283Dic6—61no—
284Dic6—61no—
285Dic6—181no—
286Dic6—181no—
287C3×C8—21no—
288C3×C8—61no—
289C3×C8—61no—
290C3×C8—62no—
291C3×C8—91no—
292C3×C8—91no—
293C3×C8—181no—
294C3×C8—181no—
295C3⋊C8—31no—
296C3⋊C8—31no—
297C3⋊C8—61no—
298C3⋊C8—61no—
299C3⋊C8—91no—
300C3⋊C8—91no—
301C3⋊C8—181no—
302C3⋊C8—181no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32—81no—
310C2×C32#93911yescharacteristic
311C2×C32C3×#12111yescharacteristic
312#41C2×(C3⋊D3)⋊C311yescharacteristic
313C2×Q8—91no—
314C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
315Q16—21no—
316Q16—181no—
317Q16—181no—
318Q16—181no—
319SD16—21no—
320SD16—181no—
321M16—91no—
322M16—91no—
323M16—181no—
324M16C2×(C3⋊D3)⋊C311yescharacteristic
325C2×C8—91no—
326C2×C8—181no—
327C22×C3—21no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—61no—
331C22×C3—61no—
332C22×C3—61no—
333C22×C3—121no—
334C22×C3C3×#23711yescharacteristic
335C3×C4—21no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—41no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—121no—
346C3×C4—121no—
347C3×C4—121no—
348C3×C4—122no—
349C3×C4—361no—
350C3×C4C3×D3×D411yescharacteristic
351C3×C4C2×C3×#6711yescharacteristic
352Dic3—121no—
353Dic3—361no—
354C32—21no—
355C32#520411yescharacteristic, complemented
356C32C3×#47311yescharacteristic, complemented
357Q8—21no—
358Q8—21no—
359Q8—181no—
360Q8—181no—
361Q8#638⋊C311yescharacteristic
362D4—21no—
363D4#638⋊C311yescharacteristic
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—181no—
367C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
368C8—21no—
369C8—91no—
370C8—91no—
371C8—181no—
372C8—181no—
373C2×C3—21no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—41no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—121no—
381C2×C3—121no—
382C2×C3—121no—
383C2×C3—241no—
384C2×C3C3×#93911yescharacteristic
385C22—21no—
386C22#3830⋊C311yescharacteristic
387C4—21no—
388C4—41no—
389C4—361no—
390C4D4×(C3⋊D3)⋊C311yescharacteristic
391C4C2×#638⋊C311yescharacteristic
392C3—31no—
393C3—31no—
394C3—61no—
395C3C3×#520411yescharacteristic
396C2—21no—
397C2—41no—
398C2#64148⋊C311yescharacteristic
subgroupG/H
centerC2#64148⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#137C2
socleC2×C3C3×#939
prime radical#41×H(3,𝔽3)C22
hypercenter#137(C3⋊D3)⋊C3
hypocenterC32C3×#473
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
4. C2×#3756⋊C3C2semidirect
5. #63663⋊C3C2semidirect
6. #63651⋊C3C2semidirect
7. #63649⋊C3C2semidirect
8. #10592903C3semidirect
30. C2×#3756C2×C3semidirect
34. #63663C2×C3semidirect
35. #63651C2×C3semidirect
36. #63649C2×C3semidirect
245. H(3,𝔽3)#473semidirect
355. C32#5204semidirect
356. C32C3×#473semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1372
#64160⋊C32
#117216892
C2×#3756⋊C32
#63663⋊C32
#63651⋊C32
#63649⋊C32
#105929033
C3×#52043