Finite Group Databasebeta

← #13141062

Subgroups — #13141062

Order 1728.

426 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721869C211yesmaximal, characteristic, complemented
2#64158⋊C3C211yesmaximal, characteristic
3#64160⋊C3C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720321C211yesmaximal, characteristic, complemented
6#11720320C211yesmaximal, characteristic, complemented
7#11720316C211yesmaximal, characteristic, complemented
8#10591680—31nomaximal
9#10591679—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13#88538—21no—
14#88538C2211yescharacteristic
15C2×#6180—21no—
16C2×#6180C2211yescharacteristic
17#3755⋊C3—21no—
18#3756⋊C3—21no—
19#3753⋊C3—21no—
20#3754⋊C3—21no—
21#3748⋊C3C2211yescharacteristic
22#3748⋊C3C2211yescharacteristic
23#88431—21no—
24#88431—21no—
25#64160D311yescharacteristic, complemented
26#64158D311yescharacteristic
27C3×#951—31no—
28C3×#949—31no—
29C3×#948—31no—
30C3×#948—31no—
31C3×#948—61no—
32C2×#3717—31no—
33C2×#3716—31no—
34#63577—31no—
35#63577—31no—
36#63561—31no—
37#63560—31no—
38#63559—31no—
39#63558—31no—
40C22×(C3⋊D3)⋊C3—21no—
41Q8×H(3,𝔽3)D411yescharacteristic
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)—21no—
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#6180—21no—
52#6180D411yescharacteristic
53#6180D411yescharacteristic
54(C3⋊(C3⋊C8))⋊C3—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5203—91no—
57#5200—91no—
58#5141—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C2×D311yescharacteristic
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×#634C2×D311yescharacteristic
64C2×C3×D12—31no—
65C2×C3×Dic6—31no—
66#3756—21no—
67#3755—21no—
68#3754—21no—
69#3753—21no—
70#3748C2×D311yescharacteristic
71#3748C2×D311yescharacteristic
72C3×#216—61no—
73C3×#215—61no—
74C3×#214—61no—
75C3×#213—61no—
76C3×#208—31no—
77C3×#208—31no—
78C3×#208—31no—
79C3×#208—31no—
80C3×#208—61no—
81C3×#208—61no—
82C3×#208—61no—
83C3×#208—62no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—61no—
87C2×C3×C3⋊C8—61no—
88C2×C3×C3⋊C8—61no—
89C2×C3×C3⋊C8—62no—
90#3717—61no—
91#3717—61no—
92#3716—61no—
93#3716—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)C2×D411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101(C3⋊Dic3)⋊C3—41no—
102C3×#137—91no—
103C3×AGL(1,ℤ/8)—91no—
104C3×#131—91no—
105#951—31no—
106#951—91no—
107#949—31no—
108#949—91no—
109#948—31no—
110#948—31no—
111#948—61no—
112C2×#215—91no—
113C2×#214—91no—
114C2×#204—91no—
115#836—91no—
116#833—91no—
117#824—91no—
118#824—91no—
119#823—91no—
120#823—91no—
121#822—91no—
122#822—91no—
123#819—91no—
124C22×C3⋊D3—21no—
125C22×C3×D3—61no—
126C32×Q8—21no—
127C32×Q8#6711yescharacteristic
128C32×Q8#6711yescharacteristic
129C32×D4—22no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4#6711yescharacteristic
134C32×D4#6711yescharacteristic
135C2×C32×C4—22no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—21no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144#634—21no—
145#634—21no—
146C2×C3×Dic3—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3⋊(C3⋊C8)—21no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—31no—
154C3×C3⋊C8—31no—
155C3×C3⋊C8—31no—
156C3×C3⋊C8—31no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—61no—
159C3×C3⋊C8—62no—
160C3×C3⋊C8—61no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164#471—271noSylow
165C2×H(3,𝔽3)—21no—
166C2×H(3,𝔽3)—41no—
167C2×H(3,𝔽3)#12111yescharacteristic
168(C3⋊D3)⋊C3—81no—
169C3×#41—31no—
170C3×#41—31no—
171C3×#41—61no—
172C3×#41D3211yescharacteristic
173C2×C3×Q8—91no—
174C2×C3×D4—91no—
175C2×D12—31no—
176C2×D12—91no—
177C2×Dic6—31no—
178C2×Dic6—91no—
179C3×Q16—181no—
180C3×SD16—181no—
181C3×SD16—181no—
182C3×D8—181no—
183C3×M16—91no—
184C3×M16—91no—
185C3×M16—91no—
186C3×M16—181no—
187C2×C3×C8—91no—
188C2×C3×C8—181no—
189#216—61no—
190#216—181no—
191#215—61no—
192#215—181no—
193#215—181no—
194#215—181no—
195#214—61no—
196#214—181no—
197#214—181no—
198#214—181no—
199#213—61no—
200#213—181no—
201#208—31no—
202#208—31no—
203#208—31no—
204#208—31no—
205#208—61no—
206#208—61no—
207#208—61no—
208#208—62no—
209#208—91no—
210#208—91no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—61no—
214C2×C3⋊C8—61no—
215C2×C3⋊C8—61no—
216C2×C3⋊C8—62no—
217#204—181no—
218#204—181no—
219C22×C32—22no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32#23711yescharacteristic
224C22×C32#23711yescharacteristic
225C2×C3⋊D3—41no—
226C2×C3⋊D3—81no—
227C2×C3×D3—121no—
228C2×C3×D3—241no—
229C32×C4—22no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4D3×D411yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4C2×#6711yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C3⋊Dic3—41no—
239C3×Dic3—121no—
240#137—91no—
241AGL(1,ℤ/8)—91no—
242C2×SD16—271no—
243#131—91no—
244#108—271no—
245#101—271no—
246#100—271no—
247H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
248C23×C3—181no—
249C22×D3—61no—
250C22×D3—181no—
251C3×Q8—31no—
252C3×Q8—31no—
253C3×Q8—61no—
254C3×Q8—181no—
255C3×Q8—181no—
256C3×Q8#62511yescharacteristic
257C3×D4—21no—
258C3×D4—31no—
259C3×D4—31no—
260C3×D4—61no—
261C3×D4—61no—
262C3×D4—62no—
263C3×D4—61no—
264C3×D4—181no—
265C3×D4—181no—
266C3×D4#62511yescharacteristic
267C2×C3×C4—21no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—62no—
273C2×C3×C4—61no—
274C2×C3×C4—181no—
275C2×C3×C4C2×D3211yescharacteristic
276C2×Dic3—61no—
277C2×Dic3—181no—
278D12—61no—
279D12—61no—
280D12—181no—
281D12—181no—
282Dic6—61no—
283Dic6—61no—
284Dic6—181no—
285Dic6—181no—
286C3×C8—91no—
287C3×C8—91no—
288C3×C8—181no—
289C3×C8—181no—
290C3×C8—181no—
291C3⋊C8—31no—
292C3⋊C8—31no—
293C3⋊C8—31no—
294C3⋊C8—31no—
295C3⋊C8—61no—
296C3⋊C8—61no—
297C3⋊C8—61no—
298C3⋊C8—61no—
299C3⋊C8—62no—
300C3⋊C8—61no—
301C3⋊C8—61no—
302C3⋊C8—181no—
303C3⋊C8—181no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32—42no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32#93911yescharacteristic
312C2×C32#93911yescharacteristic
313C3⋊D3—81no—
314C3×D3—241no—
315#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
316C2×Q8—91no—
317C2×D4—91no—
318Q16—181no—
319SD16—181no—
320SD16—181no—
321SD16—541no—
322SD16—541no—
323D8—181no—
324M16—91no—
325M16—91no—
326M16—91no—
327M16—181no—
328C2×C8—91no—
329C2×C8—181no—
330C22×C3—21no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—62no—
336C22×C3—61no—
337C22×C3—361no—
338C22×C3—721no—
339C22×C3#380511yescharacteristic
340C2×D3—121no—
341C2×D3—241no—
342C2×D3—361no—
343C2×D3—721no—
344C3×C4—21no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—31no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—62no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—361no—
355C3×C4C2×#62511yescharacteristic
356C3×C4#381111yescharacteristic
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#520011yescharacteristic, complemented
361C32#520311yescharacteristic, complemented
362C23—181no—
363Q8—181no—
364Q8—181no—
365Q8#619811yescharacteristic
366D4—21no—
367D4—181no—
368D4—181no—
369D4#619811yescharacteristic
370C2×C4—21no—
371C2×C4—181no—
372C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
373C8—91no—
374C8—91no—
375C8—181no—
376C8—181no—
377C8—181no—
378C2×C3—21no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—41no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—61no—
385C2×C3—121no—
386C2×C3—121no—
387C2×C3—121no—
388C2×C3—122no—
389C2×C3—721no—
390C2×C3#6398011yescharacteristic
391D3—241no—
392D3—721no—
393C22—21no—
394C22—361no—
395C22—721no—
396C22#8867211yescharacteristic
397C4—21no—
398C4—361no—
399C4C2×#619811yescharacteristic
400C4#8867711yescharacteristic
401C3—31no—
402C3—31no—
403C3—61no—
404C3#1059168111yescharacteristic
405C2—21no—
406C2—41no—
407C2—721no—
408C2#1172131811yescharacteristic
subgroupG/H
centerC2#11721318
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3#63980
prime radical#64158⋊C3C2
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#471
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11721869C2semidirect
3. #64160⋊C3C2semidirect
4. C2×#88431C2semidirect
5. #11720321C2semidirect
6. #11720320C2semidirect
7. #11720316C2semidirect
25. #64160D3semidirect
247. H(3,𝔽3)#471semidirect
360. C32#5200semidirect
361. C32#5203semidirect

Maximal subgroups

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

subgroupindex
#117218692
#64158⋊C32
#64160⋊C32
C2×#884312
#117203212
#117203202
#117203162
#105916803
#105916793