Finite Group Databasebeta

← #13141082

Subgroups — #13141082

Order 1728.

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

normal subgroups only

This group has 410 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
2#64160⋊C3C211yesmaximal, characteristic
3#11721689C211yesmaximal, characteristic
4C2×#88439C211yesmaximal, characteristic, complemented
5#11720327C211yesmaximal, characteristic, complemented
6#11720321C211yesmaximal, characteristic, complemented
7#11720319C211yesmaximal, characteristic, complemented
8#10591683—31nomaximal
9#10591682—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13#88550—21no—
14#88546—21no—
15#88538C2211yescharacteristic
16#3756⋊C3—21no—
17#3754⋊C3—21no—
18#3748⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20#3748⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88439—21no—
24#88439—21no—
25#64160D311yescharacteristic
26#64157D311yescharacteristic
27C3×#951—31no—
28C3×#951—31no—
29C3×#951—31no—
30C3×#951—61no—
31C3×#948—31no—
32C2×#3719—31no—
33C2×#3718—31no—
34#63578—31no—
35#63577—31no—
36#63561—31no—
37#63560—31no—
38#63560—31no—
39#63560—31no—
40Q8×H(3,𝔽3)D411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084—21no—
46#6229—21no—
47#6229—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊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#5204—91no—
57#5204—91no—
58#5143—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C2×D311yescharacteristic
62C2×#634C2×D311yescharacteristic
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67#3756—21no—
68#3754—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×#216—61no—
78C3×#214—62no—
79C3×#214—61no—
80C3×#214—61no—
81C3×#214—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#3719—61no—
91#3719—61no—
92#3718—61no—
93#3718—61no—
94C22×H(3,𝔽3)—21no—
95C22×H(3,𝔽3)#4111yescharacteristic
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)C2×D411yescharacteristic
98C4×H(3,𝔽3)C2×D411yescharacteristic
99#1084—41no—
100(C3⋊Dic3)⋊C3—41no—
101C3×#137—91no—
102C3×#137—91no—
103C3×#131—91no—
104#951—31no—
105#951—31no—
106#951—31no—
107#951—61no—
108#951—91no—
109#948—31no—
110#948—91no—
111C2×#216—91no—
112C2×#216—91no—
113C2×Dic12—91no—
114#836—91no—
115#836—91no—
116#824—91no—
117#824—91no—
118#824—91no—
119#824—91no—
120#822—91no—
121#822—91no—
122#819—91no—
123C32×Q8—21no—
124C32×Q8D1211yescharacteristic
125C32×Q8#6711yescharacteristic
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—41no—
130C32×D4D1211yescharacteristic
131C32×D4#6711yescharacteristic
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—41no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3—21no—
139#634—21no—
140#634—21no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—121no—
145C3×Dic6—62no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—121no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)#6711yescharacteristic
156C3⋊(C3⋊C8)#6711yescharacteristic
157C3×C3⋊C8—31no—
158C3×C3⋊C8—31no—
159C3×C3⋊C8—62no—
160C3×C3⋊C8—61no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164#473—271noSylow
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×#41D3211yescharacteristic
172C2×C3×Q8—91no—
173C2×C3×Q8—91no—
174C2×Dic6—31no—
175C2×Dic6—31no—
176C2×Dic6—31no—
177C2×Dic6—61no—
178C2×Dic6—91no—
179C3×Q16—181no—
180C3×Q16—181no—
181C3×SD16—181no—
182C3×SD16—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—61no—
191#216—62no—
192#216—61no—
193#216—181no—
194#216—181no—
195#216—181no—
196#216—181no—
197#216—181no—
198#214—61no—
199#214—61no—
200#214—62no—
201#214—61no—
202#214—181no—
203#208—31no—
204#208—31no—
205#208—31no—
206#208—31no—
207#208—61no—
208#208—61no—
209#208—91no—
210#208—91no—
211#208—181no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—61no—
214C2×C3⋊C8—91no—
215C2×C3⋊C8—181no—
216Dic12—181no—
217Dic12—181no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—41no—
222C22×C32#23511yescharacteristic
223C22×C32#23711yescharacteristic
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—41no—
229C32×C4C2×D1211yescharacteristic
230C32×C4D3×D411yescharacteristic
231C32×C4C2×#6711yescharacteristic
232C32×C4C2×#6711yescharacteristic
233C3⋊Dic3—41no—
234C3×Dic3—122no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238#137—91no—
239#137—91no—
240C2×Q16—271no—
241#131—91no—
242#108—271no—
243#101—271no—
244#101—271no—
245H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
246C3×Q8—31no—
247C3×Q8—31no—
248C3×Q8—61no—
249C3×Q8—181no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×Q8—181no—
253C3×Q8#62611yescharacteristic
254C3×D4—21no—
255C3×D4—31no—
256C3×D4—31no—
257C3×D4—61no—
258C3×D4—61no—
259C3×D4—61no—
260C3×D4—121no—
261C3×D4#62611yescharacteristic
262C2×C3×C4—21no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—121no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C2×D3211yescharacteristic
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—121no—
276C2×Dic3—181no—
277Dic6—61no—
278Dic6—61no—
279Dic6—61no—
280Dic6—61no—
281Dic6—62no—
282Dic6—61no—
283Dic6—61no—
284Dic6—121no—
285Dic6—181no—
286Dic6—181no—
287C3×C8—91no—
288C3×C8—91no—
289C3×C8—181no—
290C3×C8—181no—
291C3×C8—181no—
292C3⋊C8—31no—
293C3⋊C8—31no—
294C3⋊C8—61no—
295C3⋊C8—61no—
296C3⋊C8—62no—
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#93011yescharacteristic
311C2×C32#93911yescharacteristic
312#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
313C2×Q8—91no—
314C2×Q8—91no—
315Q16—181no—
316Q16—181no—
317Q16—541no—
318Q16—541no—
319SD16—181no—
320SD16—181no—
321M16—91no—
322M16—91no—
323M16—91no—
324M16—181no—
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×C3#380411yescharacteristic
335C3×C4—21no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—121no—
345C3×C4—361no—
346C3×C4—361no—
347C3×C4C2×#62611yescharacteristic
348C3×C4D3×#6711yescharacteristic
349Dic3—121no—
350Dic3—121no—
351Dic3—122no—
352Dic3—121no—
353Dic3—361no—
354C32—21no—
355C32#514311yescharacteristic, complemented
356C32#520411yescharacteristic, complemented
357Q8—181no—
358Q8—181no—
359Q8—181no—
360Q8—181no—
361Q8#620011yescharacteristic
362D4—21no—
363D4#620011yescharacteristic
364C2×C4—21no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C8—91no—
369C8—91no—
370C8—181no—
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×C3#6397811yescharacteristic
385C22—21no—
386C22#8867111yescharacteristic
387C4—21no—
388C4—361no—
389C4—361no—
390C4C2×#620011yescharacteristic
391C4#8867611yescharacteristic
392C3—31no—
393C3—31no—
394C3—61no—
395C3#1059168211yescharacteristic
396C2—21no—
397C2—41no—
398C2#1172132211yescharacteristic
subgroupG/H
centerC2#11721322
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radical#41×H(3,𝔽3)C22
socleC2×C3#63978
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#473
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
4. C2×#88439C2semidirect
5. #11720327C2semidirect
6. #11720321C2semidirect
7. #11720319C2semidirect
245. H(3,𝔽3)#473semidirect
355. C32#5143semidirect
356. C32#5204semidirect

Maximal subgroups

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

subgroupindex
#117218722
#64160⋊C32
#117216892
C2×#884392
#117203272
#117203212
#117203192
#105916833
#105916823