Finite Group Databasebeta

← #13160986

Subgroups — #13160986

Order 1728.

1272 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1272 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C32×#1004C211yesmaximal, characteristic
2#11723976C211yesmaximal, characteristic, complemented
3#11723603C212yesmaximal, complemented
4C3×#63661C211yesmaximal, characteristic, complemented
5#11722062C212yesmaximal, complemented
6#10592942—31nomaximal
7#10591534—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
10C2×C32×#67—21no—
11C2×C32×#67C2212yes—
12C32×#237—22no—
13C32×D3×D4—22no—
14C32×#235—21no—
15#88864—21no—
16#88864C2211yescharacteristic, complemented
17#88857—21no—
18C3×#3757C2212yescomplemented
19#88807—22no—
20(C2×D3)⋊(C3⋊Dic3)—22no—
21C32×H(5,𝔽2)—31no—
22C3×#1004D314yes—
23#64144—31no—
24#64144—32no—
25#64144—34no—
26#64144—64no—
27#63962—38no—
28#63661—32no—
29#63661D311yescharacteristic, complemented
30C3×#834—34no—
31#63580—38no—
32C23×C33D411yescharacteristic
33C23×C33C2311yescharacteristic
34C22×C32×D3—21no—
35C22×C32×D3D412yes—
36C33×D4—22no—
37C2×C33×C4D411yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)—22no—
41C2×C3×C3⋊Dic3—22no—
42C32×#67—22no—
43C32×#67—22no—
44C32×#67—22no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3D412yes—
47C32×D12—21no—
48C32×C4×D3—22no—
49C32×Dic6—21no—
50#5243—94no—
51#4742—91no—
52C32×#41—62no—
53C32×#41—61no—
54C2×C32×D4—24no—
55C2×C32×D4—32no—
56C2×C32×D4—62no—
57C2×C32×D4—61no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D314yes—
60C22×C3⋊Dic3—31no—
61C22×C3⋊Dic3—34no—
62C22×C3⋊Dic3—64no—
63C2×C3×#67—24no—
64C2×C3×#67C2×D318yes—
65C3×#237—28no—
66C3×D3×D4—28no—
67C3×#235—24no—
68#3757—31no—
69#3757—34no—
70#3757—62no—
71#3757—64no—
72#3757—68no—
73#3757—61no—
74#3757—64no—
75#3757C2×D312yescomplemented
76#3750—62no—
77#3750—68no—
78#3750—61no—
79#3750—64no—
80C3×#217—38no—
81Dic3⋊Dic3—68no—
82(C2×D3)⋊Dic3—68no—
83C22×C33—21no—
84C22×C33—41no—
85C22×C33—41no—
86C22×C33C2×D411yescharacteristic
87C22×C33C2×D412yes—
88C2×C32×D3—22no—
89C2×C32×D3—42no—
90C2×C32×D3—81no—
91C33×C4—21no—
92C3⋊(C3⋊Dic3)—41no—
93C3⋊(C3⋊Dic3)—41no—
94C3×C3⋊Dic3—42no—
95C32×Dic3—22no—
96C3×H(5,𝔽2)—34no—
97#1004C3⋊D311yescharacteristic
98#935—98no—
99#935—94no—
100#935—91no—
101#935—184no—
102#886—92no—
103C3×#99—91no—
104#834—34no—
105#834—98no—
106#806—92no—
107C23×C32—24no—
108C23×C32—24no—
109C23×C32—32no—
110C23×C32—61no—
111C23×C32D1211yescharacteristic
112C23×C32#6714yes—
113C23×C32C22×D314yes—
114C23×C32C22×D311yescharacteristic
115C22×C3×D3—24no—
116C22×C3×D3#6718yes—
117C32×Q8—61no—
118C32×D4—22no—
119C32×D4—28no—
120C32×D4—48no—
121C32×D4—61no—
122C32×D4—62no—
123C32×D4—62no—
124C32×D4—62no—
125C2×C32×C4—24no—
126C2×C32×C4—32no—
127C2×C32×C4—62no—
128C2×C32×C4—61no—
129C2×C32×C4D1211yescharacteristic
130C2×C32×C4#6714yes—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—68no—
133C2×C3⋊Dic3—68no—
134C2×C3⋊Dic3—61no—
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—62no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—68no—
140C2×C3⋊Dic3—128no—
141C3×#67—28no—
142C3×#67—28no—
143C3×#67—28no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—68no—
146C2×C3×Dic3#6718yes—
147C3×D12—24no—
148C3×C4×D3—28no—
149C3×Dic6—24no—
150#458—271noSylow
151C2×C33—21no—
152C2×C33—22no—
153C2×C33—41no—
154C2×C33#12011yescharacteristic
155C32×D3—42no—
156C3×#41—68no—
157C3×#41—64no—
158C2×C3×D4—24no—
159C2×C3×D4—38no—
160C2×C3×D4—68no—
161C2×C3×D4—64no—
162C2×C3×D4D3214yes—
163C2×C3×D4C2×C3⋊D311yescharacteristic
164C2×#67—21no—
165C2×#67C2×C3⋊D312yes—
166C22×Dic3—94no—
167C22×Dic3—91no—
168C22×Dic3—184no—
169#237—22no—
170D3×D4—22no—
171#235—21no—
172C3×C22⋊C4—92no—
173#217—38no—
174#217—94no—
175#217—91no—
176#217—184no—
177#217—188no—
178#217—188no—
179#217—184no—
180#217—181no—
181(C2×D3)⋊C4—182no—
182C4⋊Dic3—182no—
183Dic3⋊C4—188no—
184Dic3⋊C4—188no—
185Dic3⋊C4—184no—
186Dic3⋊C4—181no—
187C22×C32—21no—
188C22×C32—24no—
189C22×C32—28no—
190C22×C32—24no—
191C22×C32—44no—
192C22×C32—41no—
193C22×C32—44no—
194C22×C32—41no—
195C22×C32—44no—
196C22×C32—48no—
197C22×C32—48no—
198C22×C32—62no—
199C22×C32—122no—
200C22×C32—241no—
201C22×C32C2×D1211yescharacteristic
202C22×C32D3×D412yes—
203C22×C32C2×#6714yes—
204C22×C32C2×#6718yes—
205C2×C3×D3—28no—
206C2×C3×D3—48no—
207C2×C3×D3—84no—
208C32×C4—21no—
209C32×C4—24no—
210C32×C4—44no—
211C32×C4—62no—
212C3⋊Dic3—42no—
213C3⋊Dic3—128no—
214C3⋊Dic3—124no—
215C3⋊Dic3—121no—
216C3⋊Dic3—128no—
217C3⋊Dic3—121no—
218C3⋊Dic3—124no—
219C3×Dic3—28no—
220C3×Dic3—128no—
221H(5,𝔽2)—31no—
222#123—272no—
223#123—271no—
224#99—91no—
225#99—272no—
226C33#45811yescharacteristic, complemented, Sylow
227C23×C3—24no—
228C23×C3—24no—
229C23×C3—38no—
230C23×C3—64no—
231C23×C3#62614yes—
232C23×C3#63811yescharacteristic
233C23×C3C2×D3214yes—
234C23×C3C22×C3⋊D311yescharacteristic
235C22×D3—21no—
236C22×D3#63812yes—
237C3×Q8—64no—
238C3×D4—28no—
239C3×D4—22no—
240C3×D4—48no—
241C3×D4—64no—
242C3×D4—68no—
243C3×D4—68no—
244C3×D4—68no—
245C2×C3×C4—24no—
246C2×C3×C4—38no—
247C2×C3×C4—68no—
248C2×C3×C4—64no—
249C2×C3×C4—182no—
250C2×C3×C4#62614yes—
251C2×C3×C4#63811yescharacteristic
252#67—22no—
253#67—22no—
254#67—22no—
255C2×Dic3—21no—
256C2×Dic3—68no—
257C2×Dic3—188no—
258C2×Dic3—188no—
259C2×Dic3—184no—
260C2×Dic3—184no—
261C2×Dic3—188no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—182no—
265C2×Dic3—368no—
266C2×Dic3#63812yes—
267D12—21no—
268C4×D3—22no—
269Dic6—21no—
270C2×C32—28no—
271C2×C32—24no—
272C2×C32—22no—
273C2×C32—21no—
274C2×C32—24no—
275C2×C32—28no—
276C2×C32—48no—
277C2×C32—41no—
278C2×C32—44no—
279C2×C32—84no—
280C2×C32—122no—
281C2×C32#88211yescharacteristic
282C2×C32#95314yes—
283C3×D3—48no—
284#41—62no—
285#41—61no—
286C2×D4—32no—
287C2×D4—62no—
288C2×D4—61no—
289C2×D4D3×C3⋊D311yescharacteristic
290C22×C4—271no—
291C4⋊C4—542no—
292C4⋊C4—541no—
293C22⋊C4—92no—
294C22⋊C4—271no—
295C22⋊C4—542no—
296C22⋊C4—541no—
297C22×C3—21no—
298C22×C3—24no—
299C22×C3—24no—
300C22×C3—28no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—41no—
304C22×C3—44no—
305C22×C3—41no—
306C22×C3—48no—
307C22×C3—48no—
308C22×C3—68no—
309C22×C3—128no—
310C22×C3—244no—
311C22×C3C2×#62614yes—
312C22×C3D3×#6718yes—
313C22×C3C2×#63811yescharacteristic
314C22×C3C2×#63812yes—
315C2×D3—22no—
316C2×D3—42no—
317C2×D3—81no—
318C3×C4—21no—
319C3×C4—24no—
320C3×C4—44no—
321C3×C4—68no—
322C3×C4—362no—
323Dic3—22no—
324Dic3—128no—
325Dic3—368no—
326Dic3—368no—
327Dic3—364no—
328Dic3—361no—
329Dic3—361no—
330Dic3—364no—
331C32—24no—
332C32#474211yescharacteristic, complemented
333C32#524314yescomplemented
334C23—32no—
335C23—61no—
336C23#629111yescharacteristic
337C23C2×D3×C3⋊D311yescharacteristic
338Q8—61no—
339D4—22no—
340D4—61no—
341D4—62no—
342D4—62no—
343D4—62no—
344C2×C4—32no—
345C2×C4—62no—
346C2×C4—61no—
347C2×C4—182no—
348C2×C4—541no—
349C2×C4—541no—
350C2×C4—542no—
351C2×C4#629111yescharacteristic
352C2×C3—28no—
353C2×C3—28no—
354C2×C3—24no—
355C2×C3—21no—
356C2×C3—22no—
357C2×C3—24no—
358C2×C3—48no—
359C2×C3—44no—
360C2×C3—41no—
361C2×C3—84no—
362C2×C3—128no—
363C2×C3#6397714yes—
364C2×C3#6416211yescharacteristic
365D3—42no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—62no—
370C22—122no—
371C22—241no—
372C22C2×#629111yescharacteristic
373C22D3×#63812yes—
374C4—21no—
375C4—62no—
376C4—362no—
377C4—1081no—
378C4—1081no—
379C3—24no—
380C3#1059153414yescomplemented
381C3#1059294211yescharacteristic, complemented
382C2—22no—
383C2—21no—
384C2—41no—
385C2—122no—
386C2#1172363011yescharacteristic
subgroupG/H
centerC2#11723630
commutator, cosocleC23×C33C23
FrattiniC23C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#120
prime radicalC32×#1004C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#458
LCSG ▷ C23×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723976C2semidirect
3. #11723603C2semidirect
4. C3×#63661C2semidirect
5. #11722062C2semidirect
9. C22×C3⋊(C3⋊Dic3)C22semidirect
16. #88864C22semidirect
18. C3×#3757C22semidirect
29. #63661D3semidirect
75. #3757C2×D3semidirect
226. C33#458semidirect
332. C32#4742semidirect
333. C32#5243semidirect
380. C3#10591534semidirect
381. C3#10592942semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
#117239762
#117236032
C3×#636612
#117220622
#105929423
#105915343