Finite Group Databasebeta

← #13141079

Subgroups — #13141079

Order 1728.

424 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 69 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#11721872C211yesmaximal, characteristic, complemented
2#64158⋊C3C211yesmaximal, characteristic
3#11721689C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5#11720327C211yesmaximal, characteristic, complemented
6#11720320C211yesmaximal, characteristic, complemented
7#11720319C211yesmaximal, characteristic, complemented
8#10591681—31nomaximal
9#10591680—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13#88550—21no—
14#88546—21no—
15#88538C2211yescharacteristic
16#3755⋊C3—21no—
17#3753⋊C3—21no—
18#3748⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20#3748⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88438—21no—
24#88438—21no—
25#64158D311yescharacteristic
26#64157D311yescharacteristic, complemented
27C3×#951—31no—
28C3×#951—31no—
29C3×#951—61no—
30C3×#949—31no—
31C3×#948—31no—
32C2×#3717—31no—
33C2×#3715—31no—
34#63578—31no—
35#63577—31no—
36#63560—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×#1084—21no—
47#6229—21no—
48#6229—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
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×C3×D12—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67#3755—21no—
68#3753—21no—
69#3748—21no—
70#3748C2×D311yescharacteristic
71#3748C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—21no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×#216—62no—
75C3×#216—61no—
76C3×#216—61no—
77C3×#215—61no—
78C3×#214—62no—
79C3×#214—61no—
80C3×#214—61no—
81C3×#213—61no—
82C3×#208—31no—
83C3×#208—31no—
84C3×#208—31no—
85C3×#208—31no—
86C3×#208—61no—
87C3×#208—61no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—61no—
90#3717—61no—
91#3717—61no—
92#3715—61no—
93#3715—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#1084—41no—
102C3×#137—91no—
103C3×AGL(1,ℤ/8)—91no—
104C3×#131—91no—
105#951—31no—
106#951—31no—
107#951—61no—
108#949—31no—
109#949—91no—
110#948—31no—
111#948—91no—
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×Q8D1211yescharacteristic
128C32×Q8#6711yescharacteristic
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—41no—
133C32×D4D1211yescharacteristic
134C32×D4#6711yescharacteristic
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—41no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—21no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—121no—
146C3×D12—61no—
147C3×D12—61no—
148C3×Dic6—62no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—121no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)#6711yescharacteristic
157C3⋊(C3⋊C8)#6711yescharacteristic
158C3×C3⋊C8—31no—
159C3×C3⋊C8—31no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165#471—271noSylow
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×#41D3211yescharacteristic
174C2×C3×Q8—91no—
175C2×C3×D4—91no—
176C2×D12—31no—
177C2×D12—91no—
178C2×Dic6—31no—
179C2×Dic6—31no—
180C2×Dic6—61no—
181C3×Q16—181no—
182C3×SD16—181no—
183C3×SD16—181no—
184C3×D8—181no—
185C3×M16—91no—
186C3×M16—91no—
187C3×M16—91no—
188C3×M16—181no—
189C2×C3×C8—91no—
190C2×C3×C8—181no—
191#216—61no—
192#216—61no—
193#216—62no—
194#215—61no—
195#215—181no—
196#215—181no—
197#215—181no—
198#214—61no—
199#214—61no—
200#214—62no—
201#214—181no—
202#214—181no—
203#213—61no—
204#213—181no—
205#208—31no—
206#208—31no—
207#208—31no—
208#208—31no—
209#208—61no—
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—
218#204—181no—
219#204—181no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—41no—
224C22×C32#23511yescharacteristic
225C22×C32#23711yescharacteristic
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×C4C2×D1211yescharacteristic
236C32×C4D3×D411yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C32×C4C2×#6711yescharacteristic
239C3×Dic3—122no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242#137—91no—
243AGL(1,ℤ/8)—91no—
244C2×SD16—271no—
245#131—91no—
246#108—271no—
247#101—271no—
248#100—271no—
249H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
250C23×C3—181no—
251C22×D3—61no—
252C22×D3—181no—
253C3×Q8—31no—
254C3×Q8—31no—
255C3×Q8—61no—
256C3×Q8—181no—
257C3×Q8—181no—
258C3×Q8#62611yescharacteristic
259C3×D4—21no—
260C3×D4—31no—
261C3×D4—31no—
262C3×D4—61no—
263C3×D4—61no—
264C3×D4—61no—
265C3×D4—121no—
266C3×D4—181no—
267C3×D4—181no—
268C3×D4#62611yescharacteristic
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—121no—
276C2×C3×C4—181no—
277C2×C3×C4C2×D3211yescharacteristic
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—121no—
281D12—61no—
282D12—61no—
283D12—181no—
284D12—181no—
285Dic6—61no—
286Dic6—61no—
287Dic6—61no—
288Dic6—61no—
289Dic6—62no—
290Dic6—121no—
291C3×C8—91no—
292C3×C8—91no—
293C3×C8—181no—
294C3×C8—181no—
295C3×C8—181no—
296C3⋊C8—31no—
297C3⋊C8—31no—
298C3⋊C8—61no—
299C3⋊C8—61no—
300C3⋊C8—62no—
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#93011yescharacteristic
315C2×C32#93911yescharacteristic
316C3⋊D3—81no—
317C3×D3—241no—
318#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
319C2×Q8—91no—
320C2×D4—91no—
321Q16—181no—
322SD16—181no—
323SD16—181no—
324SD16—541no—
325SD16—541no—
326D8—181no—
327M16—91no—
328M16—91no—
329M16—91no—
330M16—181no—
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×C3#380411yescharacteristic
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—61no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—61no—
356C3×C4—121no—
357C3×C4—361no—
358C3×C4C2×#62611yescharacteristic
359C3×C4D3×#6711yescharacteristic
360Dic3—121no—
361Dic3—121no—
362Dic3—122no—
363C32—21no—
364C32#514111yescharacteristic, complemented
365C32#520011yescharacteristic, complemented
366C23—181no—
367Q8—181no—
368Q8—181no—
369Q8#620011yescharacteristic
370D4—21no—
371D4—181no—
372D4—181no—
373D4#620011yescharacteristic
374C2×C4—21no—
375C2×C4—181no—
376C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C8—91no—
378C8—91no—
379C8—181no—
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×C3#6397811yescharacteristic
395D3—241no—
396D3—721no—
397C22—21no—
398C22—361no—
399C22—721no—
400C22#8867111yescharacteristic
401C4—21no—
402C4—361no—
403C4C2×#620011yescharacteristic
404C4#8867611yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059167911yescharacteristic
409C2—21no—
410C2—41no—
411C2—721no—
412C2#1172132211yescharacteristic
subgroupG/H
centerC2#11721322
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3#63978
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. #11721872C2semidirect
3. #11721689C2semidirect
4. C2×#88438C2semidirect
5. #11720327C2semidirect
6. #11720320C2semidirect
7. #11720319C2semidirect
26. #64157D3semidirect
249. H(3,𝔽3)#471semidirect
364. C32#5141semidirect
365. C32#5200semidirect

Maximal subgroups

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

subgroupindex
#117218722
#64158⋊C32
#117216892
C2×#884382
#117203272
#117203202
#117203192
#105916813
#105916803