Finite Group Databasebeta

← #13161046

Subgroups — #13161046

Order 1728.

1096 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723979C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3#11723449C211yesmaximal, characteristic, complemented
4C2×#88788C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722040C211yesmaximal, characteristic, complemented
8#10592479—31nomaximal
9#10591464—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14#88864—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×#3754—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21#88804—21no—
22#88788—22no—
23#64147—31no—
24#64147—34no—
25#64147—64no—
26C2×#3754D311yescharacteristic, complemented
27C2×#3743—31no—
28#64106—31no—
29C2×C3×#214—34no—
30#63871—34no—
31C2×#3712—34no—
32C32×#102—31no—
33#63643—31no—
34#63634—31no—
35C3×#832D314yescomplemented
36#63575—34no—
37#63566—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43#6312—42no—
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C4×C3⋊(C3⋊D3)—21no—
47C4×C3⋊(C3⋊D3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634—21no—
51C2×C32×Dic3—21no—
52C3×C3⋊(C3⋊C8)D412yescomplemented
53C32×C3⋊C8—21no—
54#5174—91no—
55#4780—94no—
56C2×C32×D4—24no—
57C2×C32×D4C2×D314yes—
58C2×C32×D4C2×D311yescharacteristic
59C2×#638—61no—
60C2×#638—64no—
61C2×#638—68no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—34no—
64C2×C4×C3⋊D3—64no—
65C2×#634C2×D311yescharacteristic
66C2×C3×Dic6—34no—
67C2×C32×C8—31no—
68C32×C4⋊C4—31no—
69#3757—61no—
70#3757—64no—
71#3757—68no—
72#3754—22no—
73#3752—61no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—34no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—64no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79#3743—62no—
80C3×#214—68no—
81C3×C4⋊Dic3C2×D314yes—
82C2×C3×C3⋊C8—34no—
83C2×C3×C3⋊C8C2×D314yes—
84Dic3⋊Dic3—64no—
85#3722—64no—
86#3712—68no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C2×C3⋊(C3⋊D3)—21no—
92C2×C3⋊(C3⋊D3)—21no—
93C2×C3⋊(C3⋊D3)—21no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3⋊(C3⋊Dic3)—21no—
97C3⋊(C3⋊Dic3)—41no—
98C3×C3⋊Dic3—41no—
99C32×Dic3—41no—
100C2×C3×SD16—91no—
101#946—91no—
102#938—94no—
103#938—91no—
104#938—184no—
105C2×#214—34no—
106C2×ΓL(1,𝔽25)—94no—
107C2×ΓL(1,𝔽25)—91no—
108#897—94no—
109C3×#102—34no—
110#835—91no—
111#832C3⋊D311yescharacteristic, complemented
112#818—91no—
113#816—94no—
114#807—94no—
115C23×C32—28no—
116C23×C32—24no—
117C23×C32—21no—
118C22×C3⋊D3—31no—
119C22×C3⋊D3—34no—
120C22×C3⋊D3—64no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—24no—
125C32×D4—44no—
126C32×D4—44no—
127C2×C32×C4—24no—
128C2×C32×C4—61no—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131#638—1216no—
132#638—128no—
133#638—122no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—31no—
136C2×C3⋊Dic3—34no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—68no—
139C2×C3⋊Dic3—64no—
140C2×C3⋊Dic3—61no—
141C4×C3⋊D3—61no—
142C4×C3⋊D3—61no—
143C4×C3⋊D3—64no—
144C4×C3⋊D3—64no—
145C4×C3⋊D3—124no—
146C4×C3⋊D3—124no—
147#634—21no—
148#634—21no—
149C2×C3×Dic3—24no—
150C2×C3×Dic3—64no—
151C3×Dic6—64no—
152C3×Dic6—64no—
153C32×C8—61no—
154C3⋊(C3⋊C8)#6712yescomplemented
155C3×C3⋊C8—24no—
156C3×C3⋊C8—38no—
157#468—271noSylow
158C2×C33—81no—
159C2×C33#12111yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C2×C33#13711yescharacteristic
162C3⋊(C3⋊D3)—41no—
163C2×C3×Q8—91no—
164C2×C3×D4—24no—
165C2×C3×D4D3214yes—
166C2×C3×D4C2×C3⋊D311yescharacteristic
167C2×#67—184no—
168C2×#67—181no—
169C2×#67—188no—
170C2×C4×D3—94no—
171C2×C4×D3—91no—
172C2×C4×D3—184no—
173C2×Dic6—34no—
174C3×SD16—182no—
175C2×C3×C8—34no—
176C2×C3×C8—91no—
177C3×C4⋊C4—34no—
178#217—184no—
179#217—181no—
180#217—188no—
181#214—68no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—184no—
184C4⋊Dic3—94no—
185C4⋊Dic3—91no—
186C4⋊Dic3—184no—
187C4⋊Dic3—184no—
188C4⋊Dic3C2×C3⋊D311yescharacteristic
189Dic3⋊C4—181no—
190C2×C3⋊C8—34no—
191C2×C3⋊C8C2×C3⋊D311yescharacteristic
192ΓL(1,𝔽25)—182no—
193ΓL(1,𝔽25)—188no—
194C22×C32—24no—
195C22×C32—48no—
196C22×C32—48no—
197C22×C32—48no—
198C22×C32—44no—
199C22×C32—44no—
200C22×C32—41no—
201C22×C32—44no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32D3×D414yes—
205C22×C32D3×D411yescharacteristic
206C2×C3⋊D3—68no—
207C2×C3⋊D3—64no—
208C2×C3⋊D3—61no—
209C2×C3⋊D3—61no—
210C2×C3⋊D3—61no—
211C2×C3⋊D3—64no—
212C2×C3⋊D3—64no—
213C2×C3⋊D3—124no—
214C2×C3⋊D3—124no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—121no—
218C32×C4#23514yes—
219C32×C4D3×D414yes—
220C32×C4#23711yescharacteristic
221C32×C4C2×#6711yescharacteristic
222C3⋊Dic3—41no—
223C3⋊Dic3—61no—
224C3⋊Dic3—64no—
225C3⋊Dic3—124no—
226C3⋊Dic3—128no—
227C3⋊Dic3—124no—
228C3⋊Dic3—121no—
229C3×Dic3—44no—
230C3×Dic3—124no—
231C2×SD16—91no—
232C2×M16—271no—
233C22⋊Q8—271no—
234#121—271no—
235#107—271no—
236#103—271no—
237#102—31no—
238C33#46811yescharacteristic, complemented, Sylow
239C23×C3—28no—
240C23×C3—24no—
241C23×C3—21no—
242C22×D3—94no—
243C22×D3—91no—
244C22×D3—184no—
245C3×Q8—181no—
246C3×Q8—181no—
247C3×D4—24no—
248C3×D4—24no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—44no—
252C3×D4—44no—
253C2×C3×C4—24no—
254C2×C3×C4—64no—
255C2×C3×C4—181no—
256C2×C3×C4C2×D3214yes—
257C2×C3×C4C22×C3⋊D311yescharacteristic
258#67—3616no—
259#67—362no—
260#67—368no—
261C2×Dic3—21no—
262C2×Dic3—64no—
263C2×Dic3—94no—
264C2×Dic3—91no—
265C2×Dic3—184no—
266C2×Dic3—188no—
267C2×Dic3—184no—
268C2×Dic3—181no—
269C4×D3—184no—
270C4×D3—184no—
271C4×D3—181no—
272C4×D3—181no—
273C4×D3—364no—
274C4×D3—364no—
275Dic6—64no—
276Dic6—64no—
277C3×C8—64no—
278C3×C8—92no—
279C3⋊C8—21no—
280C3⋊C8—38no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—88no—
285C2×C32—81no—
286C2×C32—84no—
287C2×C32#88414yes—
288C2×C32#91114yes—
289C2×C32#91411yescharacteristic
290C2×C32#91514yes—
291C2×C32#91511yescharacteristic
292C2×C32#93811yescharacteristic
293C3⋊D3—128no—
294C3⋊D3—121no—
295C3⋊D3—124no—
296C2×Q8—91no—
297C2×D4—541no—
298C2×D4D3×C3⋊D311yescharacteristic
299C22×C4—271no—
300SD16—182no—
301M16—542no—
302C2×C8—31no—
303C2×C8—91no—
304C4⋊C4—31no—
305C4⋊C4—271no—
306C4⋊C4—541no—
307C22⋊C4—541no—
308C22⋊C4—541no—
309C22×C3—24no—
310C22×C3—48no—
311C22×C3—48no—
312C22×C3—48no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—44no—
317C22×C3—44no—
318C22×C3—44no—
319C22×C3#381114yes—
320C22×C3D4×C3⋊D311yescharacteristic
321C2×D3—188no—
322C2×D3—181no—
323C2×D3—181no—
324C2×D3—181no—
325C2×D3—184no—
326C2×D3—184no—
327C2×D3—184no—
328C2×D3—364no—
329C2×D3—364no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—124no—
333C3×C4—361no—
334C3×C4#380414yes—
335C3×C4D3×#6714yes—
336C3×C4#382811yescharacteristic
337C3×C4D4×C3⋊D311yescharacteristic
338Dic3—41no—
339Dic3—124no—
340Dic3—181no—
341Dic3—184no—
342Dic3—364no—
343Dic3—368no—
344Dic3—361no—
345Dic3—364no—
346C32—24no—
347C32#478014yescomplemented
348C32#517411yescharacteristic, complemented
349C23—21no—
350C23—271no—
351Q8—181no—
352Q8—181no—
353D4—21no—
354D4—21no—
355D4—1082no—
356C2×C4—61no—
357C2×C4—181no—
358C2×C4—271no—
359C2×C4—541no—
360C2×C4—541no—
361C2×C4—541no—
362C2×C4C2×D3×C3⋊D311yescharacteristic
363C8—61no—
364C8—92no—
365C2×C3—24no—
366C2×C3—24no—
367C2×C3—24no—
368C2×C3—88no—
369C2×C3—81no—
370C2×C3—84no—
371C2×C3#6393114yes—
372C2×C3#6393714yes—
373C2×C3#6395814yes—
374C2×C3#6409311yescharacteristic
375C2×C3#6412011yescharacteristic
376C2×C3#6412411yescharacteristic
377D3—368no—
378D3—361no—
379D3—364no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—541no—
384C22—541no—
385C22—541no—
386C22#8903911yescharacteristic
387C4—121no—
388C4—361no—
389C4—541no—
390C4—1081no—
391C4#8903111yescharacteristic
392C4C3⋊D3×#6711yescharacteristic
393C3—24no—
394C3#1059146414yescomplemented
395C3#1059247911yescharacteristic, complemented
396C2—81no—
397C2—1081no—
398C2#1172355411yescharacteristic
399C2#1172356411yescharacteristic
400C2#1172362311yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radical#11723979C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3754C2semidirect
3. #11723449C2semidirect
4. C2×#88788C2semidirect
5. C32×#832C2semidirect
6. #11722057C2semidirect
7. #11722040C2semidirect
26. C2×#3754D3semidirect
35. C3×#832D3semidirect
52. C3×C3⋊(C3⋊C8)D4semidirect
111. #832C3⋊D3semidirect
154. C3⋊(C3⋊C8)#67semidirect
238. C33#468semidirect
347. C32#4780semidirect
348. C32#5174semidirect
394. C3#10591464semidirect
395. C3#10592479semidirect

Maximal subgroups

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

subgroupindex
#117239792
C2×C3×#37542
#117234492
C2×#887882
C32×#8322
#117220572
#117220402
#105924793
#105914643