Finite Group Databasebeta

← #13160056

Subgroups — #13160056

Order 1728.

1148 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 160 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
1C2×#89019C211yesmaximal, characteristic, complemented
2C2×C3×#3744C211yesmaximal, characteristic, complemented
3#11723532C211yesmaximal, characteristic
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722039C211yesmaximal, characteristic, complemented
8#10592505—31nomaximal
9#10590824—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293—21no—
15C2×#6292—21no—
16C2×#6289—21no—
17#89019—22no—
18#89019—22no—
19C2×C33×C8C2211yescharacteristic
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×#3744—22no—
22C32×C4⋊Dic3C2211yescharacteristic
23#88803—21no—
24#88793—22no—
25C2×Q8×C3⋊D3—31no—
26C2×#3797—34no—
27C2×#3755—31no—
28C2×#3744D311yescharacteristic, complemented
29#64104—31no—
30C2×C3×#204—34no—
31#63916—34no—
32C2×#3715—34no—
33C32×#103—31no—
34#63647—31no—
35#63647—34no—
36#63647—64no—
37#63637—31no—
38C3×#816D314yescomplemented
39#63565—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C22×C3×C3⋊D3—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C4×C3⋊(C3⋊D3)—21no—
45C4×C3⋊(C3⋊D3)—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×#634—21no—
50C3×#634D412yescomplemented
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6D412yescomplemented
55#6293—24no—
56#6292—42no—
57#6289—24no—
58C33×C8—21no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60#5170—91no—
61#4806—94no—
62C2×C32×Q8—31no—
63Q8×C3⋊D3—62no—
64Q8×C3⋊D3—62no—
65C2×C4⋊(C3⋊D3)—61no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—34no—
69C2×C4×C3⋊D3—61no—
70C2×C4×C3⋊D3—64no—
71C2×#634—61no—
72C2×#634C2×D311yescharacteristic
73C2×C3×D12—34no—
74C2×C3×Dic6—34no—
75C2×C3×Dic6C2×D314yes—
76C2×#627—64no—
77C2×#626—64no—
78C2×#624—64no—
79#3797—68no—
80#3797—68no—
81C2×C32×C8—24no—
82C2×C32×C8C2×D314yes—
83C2×C32×C8C2×D311yescharacteristic
84C32×C4⋊C4—31no—
85#3755—62no—
86#3752—61no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—34no—
89C2×C3⋊(C3⋊C8)—64no—
90#3744—22no—
91C3×C4⋊Dic3C2×D314yes—
92C3×#204—68no—
93(C2×D3)⋊Dic3—64no—
94#3715—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3⋊(C3⋊D3)—21no—
97C2×C3⋊(C3⋊D3)—21no—
98C2×C3⋊(C3⋊D3)SD1612yescomplemented
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—21no—
105C3×C3⋊Dic3—22no—
106C32×Dic3—22no—
107C32×Dic3—41no—
108C2×D3×Q8—94no—
109C2×D3×Q8—91no—
110C2×C3×SD16—91no—
111C2×#215—94no—
112#938—91no—
113C2×#214—91no—
114C2×#204—34no—
115#895—94no—
116C3×#103—34no—
117#835—91no—
118(C2×D3)⋊C8—94no—
119(C2×D3)⋊C8—91no—
120(C2×D3)⋊C8—184no—
121#816C3⋊D311yescharacteristic, complemented
122#810—94no—
123C22×C3⋊D3—21no—
124C22×C3⋊D3—31no—
125C22×C3⋊D3—34no—
126C22×C3⋊D3—64no—
127C22×C3×D3—64no—
128C32×Q8—32no—
129C32×Q8—61no—
130C2×C32×C4—24no—
131C2×C32×C4—61no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—31no—
137C2×C3⋊Dic3—34no—
138C2×C3⋊Dic3—64no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—122no—
142C4×C3⋊D3—64no—
143C4×C3⋊D3—61no—
144C4×C3⋊D3—61no—
145C4×C3⋊D3—64no—
146C4×C3⋊D3—64no—
147C4×C3⋊D3—124no—
148C4×C3⋊D3—124no—
149#634—21no—
150#634—64no—
151#634#6712yescomplemented
152C2×C3×Dic3—24no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—64no—
155C3×D12—64no—
156C3×D12—64no—
157C3×Dic6—24no—
158C3×Dic6—38no—
159C3×Dic6—64no—
160C3×Dic6D1218yescomplemented
161#627—616no—
162#626—128no—
163#624—616no—
164C32×C8—21no—
165C32×C8—24no—
166C32×C8—44no—
167C3⋊(C3⋊C8)—61no—
168C3⋊(C3⋊C8)—64no—
169C3⋊(C3⋊C8)—124no—
170#448—271noSylow
171C2×C33#12011yescharacteristic
172C2×C33C2×SD1611yescharacteristic
173C2×C33#13711yescharacteristic
174C3⋊(C3⋊D3)—22no—
175C3×C3⋊D3—81no—
176C2×C3×Q8—34no—
177C2×C3×Q8—91no—
178C2×C3×D4—91no—
179C2×#67—181no—
180D3×Q8—182no—
181D3×Q8—188no—
182D3×Q8—188no—
183D3×Q8—182no—
184C2×D12—34no—
185C2×D12—184no—
186C2×C4×D3—94no—
187C2×C4×D3—91no—
188C2×C4×D3—184no—
189C2×C4×D3—181no—
190C2×C4×D3—184no—
191C2×Dic6—34no—
192C2×Dic6—181no—
193C2×Dic6—184no—
194C2×Dic6C2×C3⋊D311yescharacteristic
195C3×SD16—182no—
196C2×C3×C8—24no—
197C2×C3×C8D3214yes—
198C2×C3×C8C2×C3⋊D311yescharacteristic
199C3×C4⋊C4—34no—
200#217—181no—
201#215—188no—
202#214—182no—
203(C2×D3)⋊C4—184no—
204C4⋊Dic3C2×C3⋊D311yescharacteristic
205C2×C3⋊C8—94no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—184no—
208#204—68no—
209C22×C32—24no—
210C22×C32D3×D414yes—
211C22×C32D3×D411yescharacteristic
212C2×C3⋊D3—32no—
213C2×C3⋊D3—38no—
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—68no—
218C2×C3⋊D3—61no—
219C2×C3⋊D3—61no—
220C2×C3⋊D3—64no—
221C2×C3⋊D3—64no—
222C2×C3⋊D3—124no—
223C2×C3⋊D3—124no—
224C2×C3×D3—124no—
225C2×C3×D3—124no—
226C2×C3×D3—124no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—62no—
230C32×C4—121no—
231C32×C4C2×D1214yes—
232C32×C4D3×D414yes—
233C32×C4D3×D411yescharacteristic
234C32×C4C2×#6711yescharacteristic
235C3⋊Dic3—22no—
236C3⋊Dic3—61no—
237C3⋊Dic3—64no—
238C3⋊Dic3—124no—
239C3×Dic3—28no—
240C3×Dic3—44no—
241C3×Dic3—68no—
242C22×Q8—271no—
243C2×SD16—91no—
244C2×SD16—271no—
245#121—271no—
246#103—31no—
247#103—271no—
248C22⋊C8—271no—
249C33#44811yescharacteristic, complemented, Sylow
250C23×C3—181no—
251C22×D3—64no—
252C22×D3—94no—
253C22×D3—91no—
254C22×D3—184no—
255C3×Q8—38no—
256C3×Q8—64no—
257C3×Q8—92no—
258C3×Q8—181no—
259C3×D4—181no—
260C3×D4—181no—
261C2×C3×C4—24no—
262C2×C3×C4—64no—
263C2×C3×C4—64no—
264C2×C3×C4—181no—
265C2×C3×C4C2×D3214yes—
266C2×C3×C4C22×C3⋊D311yescharacteristic
267#67—362no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—64no—
271C2×Dic3—94no—
272C2×Dic3—91no—
273C2×Dic3—184no—
274D12—64no—
275D12—64no—
276D12—368no—
277C4×D3—184no—
278C4×D3—1816no—
279C4×D3—184no—
280C4×D3—184no—
281C4×D3—181no—
282C4×D3—181no—
283C4×D3—364no—
284C4×D3—364no—
285Dic6—21no—
286Dic6—38no—
287Dic6—64no—
288Dic6—184no—
289Dic6—1816no—
290Dic6C4⋊(C3⋊D3)12yescomplemented
291C3×C8—24no—
292C3×C8—21no—
293C3×C8—44no—
294C3⋊C8—184no—
295C3⋊C8—181no—
296C3⋊C8—364no—
297C2×C32—24no—
298C2×C32—24no—
299C2×C32—24no—
300C2×C32#88214yes—
301C2×C32D3×SD1614yes—
302C2×C32D3×SD1611yescharacteristic
303C2×C32#91511yescharacteristic
304C2×C32#91814yes—
305C2×C32#93711yescharacteristic
306C3⋊D3—62no—
307C3⋊D3—68no—
308C3⋊D3—81no—
309C3⋊D3—128no—
310C3×D3—244no—
311C2×Q8—31no—
312C2×Q8—91no—
313C2×Q8—541no—
314C2×Q8—542no—
315C2×Q8—542no—
316C2×D4—91no—
317C2×D4—541no—
318C22×C4—271no—
319C22×C4—541no—
320SD16—182no—
321SD16—542no—
322C2×C8—271no—
323C2×C8D3×C3⋊D311yescharacteristic
324C4⋊C4—31no—
325C22⋊C4—541no—
326C22×C3—24no—
327C22×C3—361no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3#380214yes—
331C22×C3D4×C3⋊D311yescharacteristic
332C2×D3—98no—
333C2×D3—92no—
334C2×D3—124no—
335C2×D3—124no—
336C2×D3—124no—
337C2×D3—188no—
338C2×D3—184no—
339C2×D3—181no—
340C2×D3—181no—
341C2×D3—184no—
342C2×D3—364no—
343C2×D3—364no—
344C3×C4—24no—
345C3×C4—24no—
346C3×C4—68no—
347C3×C4—124no—
348C3×C4—182no—
349C3×C4D3×D1214yes—
350C3×C4D3×#6714yes—
351C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
352C3×C4D4×C3⋊D311yescharacteristic
353Dic3—22no—
354Dic3—41no—
355Dic3—68no—
356Dic3—181no—
357Dic3—184no—
358Dic3—364no—
359C32—24no—
360C32#480614yescomplemented
361C32#517011yescharacteristic, complemented
362C23—181no—
363C23—271no—
364Q8—32no—
365Q8—61no—
366Q8—92no—
367Q8—181no—
368Q8—544no—
369D4—181no—
370D4—181no—
371D4—1082no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—181no—
375C2×C4—271no—
376C2×C4—544no—
377C2×C4—541no—
378C2×C4—541no—
379C2×C4C2×D3×C3⋊D311yescharacteristic
380C8—21no—
381C8—541no—
382C2×C3—24no—
383C2×C3—24no—
384C2×C3—24no—
385C2×C3—721no—
386C2×C3#6379614yes—
387C2×C3#6381114yes—
388C2×C3#6392214yes—
389C2×C3#6409111yescharacteristic
390C2×C3SD16×C3⋊D311yescharacteristic
391C2×C3#6412711yescharacteristic
392D3—188no—
393D3—182no—
394D3—244no—
395D3—368no—
396C22—272no—
397C22—361no—
398C22—361no—
399C22—361no—
400C22—541no—
401C22—541no—
402C22#8902611yescharacteristic
403C4—62no—
404C4—121no—
405C4—182no—
406C4—541no—
407C4D3×C4⋊(C3⋊D3)11yescharacteristic
408C4C3⋊D3×#6711yescharacteristic
409C3—24no—
410C3#1059082414yescomplemented
411C3#1059250511yescharacteristic, complemented
412C2—542no—
413C2—721no—
414C2#1172332111yescharacteristic
415C2#1172334611yescharacteristic
416C2#1172354111yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723532C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#448
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#89019C2semidirect
2. C2×C3×#3744C2semidirect
4. C2×#88793C2semidirect
5. #11722190C2semidirect
6. C32×#816C2semidirect
7. #11722039C2semidirect
28. C2×#3744D3semidirect
38. C3×#816D3semidirect
50. C3×#634D4semidirect
54. C32×Dic6D4semidirect
98. C2×C3⋊(C3⋊D3)SD16semidirect
121. #816C3⋊D3semidirect
151. #634#67semidirect
160. C3×Dic6D12semidirect
249. C33#448semidirect
290. Dic6C4⋊(C3⋊D3)semidirect
360. C32#4806semidirect
361. C32#5170semidirect
410. C3#10590824semidirect
411. C3#10592505semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
C2×C3×#37442
#117235322
C2×#887932
#117221902
C32×#8162
#117220392
#105925053
#105908243