Finite Group Databasebeta

← #13139189

Subgroups — #13139189

Order 1728.

415 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 73 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#64116⋊C3C211yesmaximal, characteristic, complemented
2#11721073C211yesmaximal, characteristic
3#11721064C211yesmaximal, characteristic, complemented
4#11720562C211yesmaximal, characteristic, complemented
5#63649⋊C3C211yesmaximal, characteristic, complemented
6#11720305C211yesmaximal, characteristic, complemented
7#11720302C211yesmaximal, characteristic, complemented
8#10590977—31nomaximal
9#10590975—31nomaximal
10#88740C2211yescharacteristic
11#3828⋊C3C2211yescharacteristic
12M16×H(3,𝔽3)C2211yescharacteristic
13C4×#1084C2211yescharacteristic
14#3748⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#3743⋊C3—21no—
18C8×(C3⋊D3)⋊C3—21no—
19#88439C411yescharacteristic, complemented
20#88438C411yescharacteristic, complemented
21#88437C411yescharacteristic, complemented
22#88436C411yescharacteristic, complemented
23#88435—21no—
24#88434—21no—
25#64116D311yescharacteristic, complemented
26C3×#907—31no—
27#63829—31no—
28#63822—31no—
29#63815—31no—
30#63814—31no—
31#63649D311yescharacteristic, complemented
32C3×#825—31no—
33C3×#825—31no—
34C3×#825—61no—
35C3×#822—31no—
36#63570—31no—
37#63569—31no—
38Dic3×C3⋊C8—31no—
39Dic3×C3⋊C8—31no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#6235—21no—
42C2×#1084—21no—
43C4⋊#272C2×C411yescharacteristic
44C4×#272—21no—
45#6229C2×C411yescharacteristic
46#638⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49#634⋊C3C2×C411yescharacteristic
50C8×H(3,𝔽3)—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#5144—91no—
57#4895—91no—
58#4824—91no—
59#3828C2×D311yescharacteristic
60C3×#235—31no—
61C3×#235—31no—
62C3×#235—31no—
63C3×#235—61no—
64C32×M16—21no—
65C32×M16C2×D311yescharacteristic
66C32×M16C2×D311yescharacteristic
67#3748C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70#3743—21no—
71C8×C3⋊D3—21no—
72C3×C4×Dic3—31no—
73C3×C4×Dic3—31no—
74C3×C4×Dic3—61no—
75C3×#208—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C3×ΓL(1,𝔽25)—61no—
79C3×D3×C8—61no—
80#3719—31no—
81#3718—31no—
82#3717—31no—
83#3716—31no—
84#3715—31no—
85#3715—31no—
86#3714—31no—
87#3713—31no—
88#3711—61no—
89#3711—61no—
90D3×C3⋊C8—61no—
91D3×C3⋊C8—61no—
92C22×H(3,𝔽3)#4111yescharacteristic
93C2×#272—21no—
94C2×(C3⋊D3)⋊C3—21no—
95C4×H(3,𝔽3)C22×C411yescharacteristic
96C4×H(3,𝔽3)C2×D411yescharacteristic
97#1084—22no—
98#1084—21no—
99(C3⋊Dic3)⋊C3—21no—
100C3×#131—91no—
101#950—91no—
102#950—91no—
103#948—91no—
104#907—31no—
105#907—91no—
106#907—91no—
107#904—91no—
108#901—91no—
109C3×#108—91no—
110C3×#104—91no—
111#837—91no—
112#825—31no—
113#825—31no—
114#825—61no—
115#825—91no—
116#822—31no—
117#822—91no—
118C8×Dic3—91no—
119C8×Dic3—91no—
120#805—91no—
121C4×C3⋊C8—91no—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125#638—21no—
126C4⋊(C3⋊D3)C4×D311yescharacteristic
127C4×C3⋊D3—21no—
128#634C4×D311yescharacteristic
129C3×#67—61no—
130C3×#67—61no—
131C3×#67—61no—
132C3×#67—121no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—121no—
136C3×D12—31no—
137C3×D12—31no—
138C3×D12—31no—
139C3×D12—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—121no—
144C3×Dic6—31no—
145C3×Dic6—31no—
146C3×Dic6—31no—
147C3×Dic6—61no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—41no—
151C3⋊(C3⋊C8)—21no—
152C3⋊(C3⋊C8)C4×D311yescharacteristic
153C3⋊(C3⋊C8)C4×D311yescharacteristic
154C3⋊(C3⋊C8)#6711yescharacteristic
155C3⋊(C3⋊C8)#6711yescharacteristic
156C3×C3⋊C8—31no—
157C3×C3⋊C8—31no—
158C3×C3⋊C8—31no—
159C3×C3⋊C8—31no—
160C3×C3⋊C8—61no—
161#443—271noSylow
162C2×H(3,𝔽3)—21no—
163C2×H(3,𝔽3)C4×D411yescharacteristic
164#272—41no—
165(C3⋊D3)⋊C3—41no—
166C3×#41—91no—
167C3×#41—91no—
168#235—31no—
169#235—31no—
170#235—31no—
171#235—61no—
172#235—91no—
173C3×M16—31no—
174C3×M16—31no—
175C3×M16—61no—
176C3×M16—91no—
177C3×M16—181no—
178C3×M16D3211yescharacteristic
179C2×C3×C8—91no—
180C2×C3×C8—91no—
181C2×C3×C8—181no—
182C3×C42—91no—
183#216—91no—
184#216—91no—
185#215—91no—
186#215—91no—
187#214—91no—
188#214—91no—
189#213—91no—
190#213—91no—
191C4×Dic3—31no—
192C4×Dic3—31no—
193C4×Dic3—61no—
194#208—31no—
195#208—91no—
196#208—181no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—91no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—181no—
202Dic12—91no—
203D24—91no—
204#204—91no—
205#204—91no—
206ΓL(1,𝔽25)—61no—
207ΓL(1,𝔽25)—181no—
208ΓL(1,𝔽25)—181no—
209ΓL(1,𝔽25)—181no—
210D3×C8—61no—
211D3×C8—181no—
212D3×C8—181no—
213D3×C8—181no—
214C22×C32—21no—
215C22×C32#23511yescharacteristic
216C22×C32#23911yescharacteristic
217C2×C3⋊D3—21no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C2×C3×D3—121no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4C2×C4×D311yescharacteristic
225C32×C4C2×C4×D311yescharacteristic
226C32×C4D3×D411yescharacteristic
227C32×C4C2×#6711yescharacteristic
228C3⋊Dic3—21no—
229C3×Dic3—62no—
230C3×Dic3—62no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—121no—
235C3×Dic3—122no—
236#135—271no—
237#131—91no—
238#131—271no—
239#108—91no—
240#104—91no—
241#104—271no—
242C4×C8—271no—
243H(3,𝔽3)#44311yescharacteristic, complemented, Sylow
244C3×Q8—91no—
245C3×Q8—91no—
246C3×D4—91no—
247C3×D4—91no—
248C3×D4—181no—
249C3×D4—181no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—61no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4C2×D3211yescharacteristic
257#67—61no—
258#67—61no—
259#67—61no—
260#67—121no—
261#67—181no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—121no—
265D12—31no—
266D12—31no—
267D12—31no—
268D12—61no—
269D12—91no—
270C4×D3—61no—
271C4×D3—61no—
272C4×D3—61no—
273C4×D3—121no—
274C4×D3—181no—
275Dic6—31no—
276Dic6—31no—
277Dic6—31no—
278Dic6—61no—
279Dic6—91no—
280C3×C8—21no—
281C3×C8—61no—
282C3×C8—61no—
283C3×C8—91no—
284C3×C8—91no—
285C3×C8—91no—
286C3×C8—91no—
287C3×C8—121no—
288C3×C8—181no—
289C3⋊C8—31no—
290C3⋊C8—31no—
291C3⋊C8—31no—
292C3⋊C8—31no—
293C3⋊C8—61no—
294C3⋊C8—91no—
295C3⋊C8—91no—
296C3⋊C8—91no—
297C3⋊C8—91no—
298C3⋊C8—181no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—41no—
303C2×C32#89311yescharacteristic
304C2×C32C4×#6711yescharacteristic
305C3⋊D3—41no—
306C3×D3—121no—
307C3×D3—121no—
308C3×D3—121no—
309C3×D3—241no—
310#41—91no—
311#41—91no—
312Q16—271no—
313SD16—271no—
314SD16—271no—
315D8—271no—
316M16—91no—
317M16—181no—
318M16—541no—
319M16((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C2×C8—91no—
321C2×C8—91no—
322C2×C8—181no—
323C2×C8—541no—
324C42—91no—
325C22×C3—31no—
326C22×C3—31no—
327C22×C3—61no—
328C22×C3—181no—
329C22×C3—181no—
330C22×C3#379911yescharacteristic
331C2×D3—61no—
332C2×D3—61no—
333C2×D3—61no—
334C2×D3—121no—
335C2×D3—181no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—182no—
343C3×C4—181no—
344C3×C4—181no—
345C3×C4C4×D3211yescharacteristic
346C3×C4D3×#6711yescharacteristic
347Dic3—61no—
348Dic3—61no—
349Dic3—62no—
350Dic3—62no—
351Dic3—61no—
352Dic3—121no—
353Dic3—122no—
354Dic3—181no—
355C32—21no—
356C32#489511yescharacteristic, complemented
357C32#514411yescharacteristic, complemented
358Q8—91no—
359Q8—91no—
360D4—91no—
361D4—91no—
362D4—181no—
363D4—181no—
364C2×C4—181no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C8—21no—
369C8—91no—
370C8—91no—
371C8—91no—
372C8—91no—
373C8—181no—
374C2×C3—21no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—121no—
381C2×C3—361no—
382C2×C3—361no—
383C2×C3#6390411yescharacteristic
384D3—121no—
385D3—121no—
386D3—121no—
387D3—241no—
388D3—361no—
389C22—181no—
390C22—181no—
391C22#8865211yescharacteristic
392C4—182no—
393C4—181no—
394C4—181no—
395C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
396C4#8867611yescharacteristic
397C3—31no—
398C3—31no—
399C3—61no—
400C3#1059099511yescharacteristic
401C2—21no—
402C2—361no—
403C2—361no—
404C2#1172115411yescharacteristic
subgroupG/H
centerC4#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingM16×H(3,𝔽3)C22
socleC2×C3#63904
prime radical#11721073C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterM16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#443
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #64116⋊C3C2semidirect
3. #11721064C2semidirect
4. #11720562C2semidirect
5. #63649⋊C3C2semidirect
6. #11720305C2semidirect
7. #11720302C2semidirect
19. #88439C4semidirect
20. #88438C4semidirect
21. #88437C4semidirect
22. #88436C4semidirect
25. #64116D3semidirect
31. #63649D3semidirect
243. H(3,𝔽3)#443semidirect
356. C32#4895semidirect
357. C32#5144semidirect

Maximal subgroups

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

subgroupindex
#64116⋊C32
#117210732
#117210642
#117205622
#63649⋊C32
#117203052
#117203022
#105909773
#105909753