Finite Group Databasebeta

← #10592902⋊C3 #13143501

Subgroups — #10592902⋊C3 #13143501

Order 1728.

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

normal subgroups only

This group has 424 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, complemented
2#64158⋊C3C211yesmaximal, characteristic
3#11721689C211yesmaximal, characteristic, complemented
4C2×#3755⋊C3C211yesmaximal, characteristic, complemented
5#63663⋊C3C211yesmaximal, characteristic, complemented
6#63650⋊C3C211yesmaximal, characteristic, complemented
7#63649⋊C3C211yesmaximal, characteristic, complemented
8#10592902C311yesmaximal, characteristic, complemented
9C3×#5203—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13Q16×H(3,𝔽3)—21no—
14SD16×H(3,𝔽3)—21no—
15M16×H(3,𝔽3)C2211yescharacteristic
16#3755⋊C3—21no—
17#3755⋊C3—21no—
18#3755⋊C3—21no—
19#3753⋊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, complemented
27C32×#137C2×C311yescharacteristic, complemented
28#64158C2×C311yescharacteristic
29#64157C2×C311yescharacteristic, complemented
30C2×#3755C2×C311yescharacteristic, complemented
31C3×#949—31no—
32C3×#948—31no—
33C2×C3×#215—31no—
34#63663C2×C311yescharacteristic, complemented
35#63650C2×C311yescharacteristic, complemented
36#63649C2×C311yescharacteristic, complemented
37C3×#836—31no—
38C3×#823—31no—
39C3×#822—31no—
40C22×(C3⋊D3)⋊C3—21no—
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)D411yescharacteristic
44D4×H(3,𝔽3)—21no—
45D4×H(3,𝔽3)D411yescharacteristic
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊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×#471—91no—
57#5203—31no—
58#5203—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C22×C311yescharacteristic
62C2×C32×Q8—21no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C22×C311yescharacteristic
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C3×D12—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#3755—21no—
77#3755—21no—
78#3755—21no—
79#3753—21no—
80#3748—21no—
81#3748C22×C311yescharacteristic
82#3748C22×C311yescharacteristic
83C2×C3⋊(C3⋊C8)—21no—
84C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
85C3×#215—61no—
86C3×#215—61no—
87C3×#215—61no—
88C3×#213—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
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—81no—
98C4×H(3,𝔽3)—21no—
99C4×H(3,𝔽3)—41no—
100C4×H(3,𝔽3)C2×D411yescharacteristic
101C4×H(3,𝔽3)C2×D411yescharacteristic
102C3×#137—31no—
103C3×#137—31no—
104C3×#137—61no—
105C3×#137C3×D311yescharacteristic
106C3×AGL(1,ℤ/8)—91no—
107C2×C3×SD16—91no—
108C3×#131—91no—
109#949—31no—
110#949—91no—
111#948—31no—
112#948—91no—
113C2×#215—31no—
114C2×#215—91no—
115C3×#108—91no—
116C3×#101—91no—
117C3×#100—91no—
118#836—31no—
119#836—91no—
120#823—31no—
121#823—91no—
122#822—31no—
123#822—91no—
124C22×C3⋊D3—21no—
125C22×C3×D3—61no—
126C32×Q8—22no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—41no—
133C32×Q8#6711yescharacteristic
134C32×Q8C3×D411yescharacteristic
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—41no—
139C32×D4#6711yescharacteristic
140C32×D4C3×D411yescharacteristic
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C23×C311yescharacteristic
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)—21no—
152C3×D12—61no—
153C3×D12—61no—
154C32×C8—22no—
155C32×C8—21no—
156C32×C8—21no—
157C3⋊(C3⋊C8)—21no—
158C3⋊(C3⋊C8)—21no—
159C3⋊(C3⋊C8)C3×D411yescharacteristic
160C3⋊(C3⋊C8)C3×D411yescharacteristic
161C3×C3⋊C8—31no—
162C3×C3⋊C8—31no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165#471—91noSylow
166C2×H(3,𝔽3)—21no—
167C2×H(3,𝔽3)—41no—
168C2×H(3,𝔽3)#12111yescharacteristic
169(C3⋊D3)⋊C3—81no—
170C3×#41—31no—
171C3×#41—31no—
172C3×#41—61no—
173C3×#41C2×C3×D311yescharacteristic
174C2×C3×Q8—31no—
175C2×C3×Q8—31no—
176C2×C3×Q8—61no—
177C2×C3×Q8C2×C3×D311yescharacteristic
178C2×C3×D4—91no—
179C2×D12—31no—
180C2×D12—91no—
181C3×Q16—21no—
182C3×Q16—61no—
183C3×Q16—61no—
184C3×Q16—62no—
185C3×SD16—21no—
186C3×SD16—61no—
187C3×SD16—61no—
188C3×SD16—62no—
189C3×SD16—181no—
190C3×SD16—181no—
191C3×SD16—181no—
192C3×D8—181no—
193C3×M16—31no—
194C3×M16—31no—
195C3×M16—61no—
196C3×M16—91no—
197C3×M16—91no—
198C3×M16—181no—
199C3×M16C2×C3×D311yescharacteristic
200C2×C3×C8—91no—
201C2×C3×C8—181no—
202#215—61no—
203#215—61no—
204#215—61no—
205#215—181no—
206#215—181no—
207#215—181no—
208#213—61no—
209#213—181no—
210#208—31no—
211#208—31no—
212#208—61no—
213#208—91no—
214#208—91no—
215#208—181no—
216C2×C3⋊C8—31no—
217C2×C3⋊C8—61no—
218C2×C3⋊C8—91no—
219C2×C3⋊C8—181no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—41no—
224C22×C32#23711yescharacteristic
225C22×C32C3×#4111yescharacteristic
226C2×C3⋊D3—41no—
227C2×C3⋊D3—81no—
228C2×C3×D3—121no—
229C2×C3×D3—241no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—41no—
235C32×C4—42no—
236C32×C4—41no—
237C32×C4—41no—
238C32×C4D3×D411yescharacteristic
239C32×C4C2×#6711yescharacteristic
240C32×C4C2×C3×D411yescharacteristic
241C32×C4C2×C3×D411yescharacteristic
242#137(C3⋊D3)⋊C311yescharacteristic, complemented
243AGL(1,ℤ/8)—91no—
244C2×SD16—91no—
245#131—91no—
246#108—91no—
247#101—91no—
248#100—91no—
249H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
250C23×C3—181no—
251C22×D3—61no—
252C22×D3—181no—
253C3×Q8—21no—
254C3×Q8—21no—
255C3×Q8—31no—
256C3×Q8—31no—
257C3×Q8—61no—
258C3×Q8—61no—
259C3×Q8—61no—
260C3×Q8—61no—
261C3×Q8—62no—
262C3×Q8—61no—
263C3×Q8—121no—
264C3×Q8C3×#6711yescharacteristic
265C3×D4—21no—
266C3×D4—31no—
267C3×D4—31no—
268C3×D4—61no—
269C3×D4—61no—
270C3×D4—61no—
271C3×D4—121no—
272C3×D4—181no—
273C3×D4—181no—
274C3×D4C3×#6711yescharacteristic
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—121no—
285C2×C3×C4—121no—
286C2×C3×C4C22×C3×D311yescharacteristic
287D12—61no—
288D12—61no—
289D12—181no—
290D12—181no—
291C3×C8—21no—
292C3×C8—61no—
293C3×C8—61no—
294C3×C8—62no—
295C3×C8—91no—
296C3×C8—91no—
297C3×C8—181no—
298C3×C8—181no—
299C3⋊C8—31no—
300C3⋊C8—31no—
301C3⋊C8—61no—
302C3⋊C8—61no—
303C3⋊C8—91no—
304C3⋊C8—91no—
305C3⋊C8—181no—
306C3⋊C8—181no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—81no—
314C2×C32#93911yescharacteristic
315C2×C32C3×#12111yescharacteristic
316C3⋊D3—81no—
317C3×D3—241no—
318#41C2×(C3⋊D3)⋊C311yescharacteristic
319C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
320C2×D4—91no—
321Q16—21no—
322SD16—21no—
323SD16—181no—
324SD16—181no—
325SD16—181no—
326D8—181no—
327M16—91no—
328M16—91no—
329M16—181no—
330M16C2×(C3⋊D3)⋊C311yescharacteristic
331C2×C8—91no—
332C2×C8—181no—
333C22×C3—21no—
334C22×C3—31no—
335C22×C3—31no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—61no—
339C22×C3—121no—
340C22×C3—361no—
341C22×C3—721no—
342C22×C3C3×#23711yescharacteristic
343C2×D3—121no—
344C2×D3—241no—
345C2×D3—361no—
346C2×D3—721no—
347C3×C4—21no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—41no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—61no—
356C3×C4—61no—
357C3×C4—121no—
358C3×C4—121no—
359C3×C4—121no—
360C3×C4—122no—
361C3×C4C3×D3×D411yescharacteristic
362C3×C4C2×C3×#6711yescharacteristic
363C32—21no—
364C32#520311yescharacteristic, complemented
365C32C3×#47111yescharacteristic, complemented
366C23—181no—
367Q8—21no—
368Q8—21no—
369Q8#638⋊C311yescharacteristic
370D4—21no—
371D4—181no—
372D4—181no—
373D4#638⋊C311yescharacteristic
374C2×C4—21no—
375C2×C4—21no—
376C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
377C8—21no—
378C8—91no—
379C8—91no—
380C8—181no—
381C8—181no—
382C2×C3—21no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—41no—
386C2×C3—61no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—121no—
390C2×C3—121no—
391C2×C3—121no—
392C2×C3—241no—
393C2×C3—721no—
394C2×C3C3×#93911yescharacteristic
395D3—241no—
396D3—721no—
397C22—21no—
398C22—361no—
399C22—721no—
400C22#3830⋊C311yescharacteristic
401C4—21no—
402C4—41no—
403C4D4×(C3⋊D3)⋊C311yescharacteristic
404C4C2×#638⋊C311yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3C3×#520311yescharacteristic
409C2—21no—
410C2—41no—
411C2—721no—
412C2#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#64158⋊C3C2
hypercenter#137(C3⋊D3)⋊C3
hypocenterC32C3×#471
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
1. H(3,𝔽3)×#137C2semidirect
3. #11721689C2semidirect
4. C2×#3755⋊C3C2semidirect
5. #63663⋊C3C2semidirect
6. #63650⋊C3C2semidirect
7. #63649⋊C3C2semidirect
8. #10592902C3semidirect
26. C32×#137D3semidirect
27. C32×#137C2×C3semidirect
29. #64157C2×C3semidirect
30. C2×#3755C2×C3semidirect
34. #63663C2×C3semidirect
35. #63650C2×C3semidirect
36. #63649C2×C3semidirect
242. #137(C3⋊D3)⋊C3semidirect
249. H(3,𝔽3)#471semidirect
364. C32#5203semidirect
365. C32C3×#471semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1372
#64158⋊C32
#117216892
C2×#3755⋊C32
#63663⋊C32
#63650⋊C32
#63649⋊C32
#105929023
C3×#52033