Finite Group Databasebeta

← #13141066

Subgroups — #13141066

Order 1728.

415 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 85 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#64159⋊C3C211yesmaximal, characteristic
2#11721689C211yesmaximal, characteristic, complemented
3#11721054C211yesmaximal, characteristic, complemented
4#11720586C211yesmaximal, characteristic, complemented
5#11720319C211yesmaximal, characteristic, complemented
6#11720305C211yesmaximal, characteristic, complemented
7#11720302C211yesmaximal, characteristic, complemented
8#10591675—31nomaximal
9#10591662—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13#88538C2211yescharacteristic
14#3755⋊C3C411yescharacteristic, complemented
15#3756⋊C3C411yescharacteristic, complemented
16#3753⋊C3C411yescharacteristic, complemented
17#3754⋊C3C411yescharacteristic, complemented
18#3748⋊C3—21no—
19#3748⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88430—21no—
24((C3⋊D3)⋊C3)⋊C8—21no—
25#64159D311yescharacteristic
26#64157D311yescharacteristic, complemented
27C3×#950—31no—
28C3×#948—31no—
29#63817—31no—
30#63815—31no—
31C3×#837—31no—
32C3×#837—31no—
33C3×#837—61no—
34#63578—31no—
35#63577—31no—
36#63570—31no—
37#63569—31no—
38Dic3×C3⋊C8—31no—
39Dic3×C3⋊C8—31no—
40Q8×H(3,𝔽3)C2×C411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)C2×C411yescharacteristic
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084—21no—
46#638⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49#634⋊C3C2×C411yescharacteristic
50#6180—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
53(C3⋊(C3⋊C8))⋊C3C2×C411yescharacteristic
54(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
55(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
56#5196—91no—
57#5144—91no—
58#4824—91no—
59C32×#41—21no—
60C32×#41C2×D311yescharacteristic
61C32×#41C2×D311yescharacteristic
62#3828C2×D311yescharacteristic
63C3×#235—31no—
64#3756Dic311yescharacteristic, complemented
65#3755Dic311yescharacteristic, complemented
66#3754Dic311yescharacteristic, complemented
67#3753Dic311yescharacteristic, complemented
68#3748—21no—
69#3748C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)—21no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×#216—31no—
74C3×#215—31no—
75C3×#214—31no—
76C3×#213—31no—
77C3×C4×Dic3—31no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—61no—
80C3×#208—31no—
81C3×#208—31no—
82C3×#208—31no—
83C3×#208—61no—
84C3×#208—61no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—61no—
88#3712—61no—
89#3711—61no—
90#3710—61no—
91D3×C3⋊C8—61no—
92C22×H(3,𝔽3)—21no—
93C22×H(3,𝔽3)#4111yescharacteristic
94C2×(C3⋊D3)⋊C3—21no—
95C4×H(3,𝔽3)—21no—
96C4×H(3,𝔽3)C22×C411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98#1084—22no—
99(C3⋊Dic3)⋊C3—21no—
100C3×#135—91no—
101C3×#131—91no—
102#950—31no—
103#950—91no—
104#948—31no—
105#948—91no—
106#948—91no—
107#907—91no—
108#901—91no—
109C3×#104—91no—
110#837—31no—
111#837—31no—
112#837—61no—
113#837—91no—
114#825—91no—
115#822—91no—
116#822—91no—
117#819—91no—
118C8×Dic3—91no—
119C8×Dic3—91no—
120#805—91no—
121C4×C3⋊C8—91no—
122C32×Q8—21no—
123C32×Q8C4×D311yescharacteristic
124C32×Q8C2×Dic311yescharacteristic
125C32×D4—21no—
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—41no—
129C32×D4C4×D311yescharacteristic
130C32×D4C2×Dic311yescharacteristic
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—41no—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137#638—21no—
138C4⋊(C3⋊D3)C2×Dic311yescharacteristic
139C4×C3⋊D3—21no—
140#634C2×Dic311yescharacteristic
141C3×#67—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—121no—
145C3×D12—31no—
146C3×C4×D3—61no—
147C3×Dic6—31no—
148C3⋊(C3⋊C8)—21no—
149C3⋊(C3⋊C8)C2×Dic311yescharacteristic
150C3⋊(C3⋊C8)C2×Dic311yescharacteristic
151C3⋊(C3⋊C8)#6711yescharacteristic
152C3⋊(C3⋊C8)#6711yescharacteristic
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—61no—
160C3×C3⋊C8—121no—
161#443—271noSylow
162C2×H(3,𝔽3)—21no—
163C2×H(3,𝔽3)—41no—
164C2×H(3,𝔽3)C4×D411yescharacteristic
165(C3⋊D3)⋊C3—41no—
166C3×#41—31no—
167C3×#41—31no—
168C3×#41—61no—
169C3×#41—91no—
170C3×#41D3211yescharacteristic
171#235—31no—
172#235—91no—
173C3×Q16—91no—
174C3×SD16—91no—
175C3×SD16—91no—
176C3×D8—91no—
177C3×M16—91no—
178C3×M16—91no—
179C3×M16—181no—
180C2×C3×C8—91no—
181C2×C3×C8—91no—
182C2×C3×C8—181no—
183C3×C42—91no—
184#216—31no—
185#216—91no—
186#215—31no—
187#215—91no—
188#214—31no—
189#214—91no—
190#213—31no—
191#213—91no—
192C4×Dic3—31no—
193C4×Dic3—31no—
194C4×Dic3—61no—
195#208—31no—
196#208—31no—
197#208—31no—
198#208—61no—
199#208—61no—
200#208—91no—
201#208—181no—
202#208—181no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—61no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—91no—
208C2×C3⋊C8—181no—
209C2×C3⋊C8—181no—
210ΓL(1,𝔽25)—181no—
211ΓL(1,𝔽25)—181no—
212D3×C8—181no—
213D3×C8—181no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32#23511yescharacteristic
219C22×C32#23711yescharacteristic
220C2×C3⋊D3—21no—
221C2×C3×D3—61no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—41no—
227C32×C4C2×C4×D311yescharacteristic
228C32×C4D3×D411yescharacteristic
229C32×C4C22×Dic311yescharacteristic
230C32×C4C2×#6711yescharacteristic
231C3⋊Dic3—21no—
232C3×Dic3—62no—
233C3×Dic3—62no—
234C3×Dic3—61no—
235C3×Dic3—122no—
236#135—91no—
237#131—91no—
238#131—271no—
239#108—271no—
240#104—91no—
241#104—271no—
242C4×C8—271no—
243H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
244C3×Q8—31no—
245C3×Q8—31no—
246C3×Q8—61no—
247C3×Q8—91no—
248C3×Q8D3×Dic311yescharacteristic
249C3×D4—21no—
250C3×D4—31no—
251C3×D4—31no—
252C3×D4—61no—
253C3×D4—61no—
254C3×D4—61no—
255C3×D4—91no—
256C3×D4—121no—
257C3×D4—181no—
258C3×D4D3×Dic311yescharacteristic
259C2×C3×C4—21no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—121no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4C2×D3211yescharacteristic
269#67—61no—
270#67—181no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—121no—
274D12—31no—
275D12—91no—
276C4×D3—61no—
277C4×D3—181no—
278Dic6—31no—
279Dic6—91no—
280C3×C8—91no—
281C3×C8—91no—
282C3×C8—91no—
283C3×C8—91no—
284C3×C8—181no—
285C3×C8—181no—
286C3⋊C8—31no—
287C3⋊C8—31no—
288C3⋊C8—31no—
289C3⋊C8—31no—
290C3⋊C8—61no—
291C3⋊C8—61no—
292C3⋊C8—61no—
293C3⋊C8—91no—
294C3⋊C8—91no—
295C3⋊C8—91no—
296C3⋊C8—91no—
297C3⋊C8—121no—
298C3⋊C8—181no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—41no—
303C2×C32—41no—
304C2×C32—41no—
305C2×C32—81no—
306C2×C32C4×#6711yescharacteristic
307C2×C32D4×Dic311yescharacteristic
308C3⋊D3—41no—
309C3×D3—121no—
310#41—91no—
311#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
312Q16—91no—
313SD16—91no—
314SD16—91no—
315D8—91no—
316M16—91no—
317M16—91no—
318M16—181no—
319M16—541no—
320C2×C8—91no—
321C2×C8—91no—
322C2×C8—181no—
323C2×C8—541no—
324C42—91no—
325C22×C3—21no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—61no—
330C22×C3—61no—
331C22×C3—121no—
332C22×C3—181no—
333C22×C3#380411yescharacteristic
334C2×D3—61no—
335C2×D3—181no—
336C3×C4—21no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—121no—
346C3×C4—182no—
347C3×C4—181no—
348C3×C4C2×D3×Dic311yescharacteristic
349C3×C4D3×#6711yescharacteristic
350Dic3—62no—
351Dic3—62no—
352Dic3—61no—
353Dic3—122no—
354Dic3—181no—
355C32—21no—
356C32#514411yescharacteristic, complemented
357C32#519611yescharacteristic, complemented
358Q8—91no—
359Q8((C3⋊D3)⋊C3)⋊C411yescharacteristic
360D4—21no—
361D4—91no—
362D4—181no—
363D4((C3⋊D3)⋊C3)⋊C411yescharacteristic
364C2×C4—21no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C8—91no—
369C8—91no—
370C8—91no—
371C8—91no—
372C8—181no—
373C8—181no—
374C2×C3—21no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—41no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—121no—
382C2×C3—121no—
383C2×C3—121no—
384C2×C3—241no—
385C2×C3—361no—
386C2×C3Dic3×#6711yescharacteristic
387D3—121no—
388D3—361no—
389C22—21no—
390C22—181no—
391C22#8867111yescharacteristic
392C4—21no—
393C4—182no—
394C4—181no—
395C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
396C4#8867611yescharacteristic
397C3—31no—
398C3—31no—
399C3—61no—
400C3#1059167311yescharacteristic
401C2—21no—
402C2—41no—
403C2—361no—
404C2#1172131411yescharacteristic
subgroupG/H
centerC4#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3Dic3×#67
prime radical#64159⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#443
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #11721689C2semidirect
3. #11721054C2semidirect
4. #11720586C2semidirect
5. #11720319C2semidirect
6. #11720305C2semidirect
7. #11720302C2semidirect
14. #3755⋊C3C4semidirect
15. #3756⋊C3C4semidirect
16. #3753⋊C3C4semidirect
17. #3754⋊C3C4semidirect
26. #64157D3semidirect
64. #3756Dic3semidirect
65. #3755Dic3semidirect
66. #3754Dic3semidirect
67. #3753Dic3semidirect
243. H(3,𝔽3)#443semidirect
356. C32#5144semidirect
357. C32#5196semidirect

Maximal subgroups

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

subgroupindex
#64159⋊C32
#117216892
#117210542
#117205862
#117203192
#117203052
#117203022
#105916753
#105916623