Finite Group Databasebeta

← #13160054

Subgroups — #13160054

Order 1728.

1116 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 170 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×D3×#634C211yesmaximal, characteristic, complemented
2C2×C3×#3744C211yesmaximal, characteristic, complemented
3#11723515C211yesmaximal, characteristic
4C2×#88796C211yesmaximal, characteristic, complemented
5#11722186C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11722040C211yesmaximal, characteristic, complemented
8#10592437—31nomaximal
9#10590822—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6293—21no—
15C2×#6290—21no—
16C2×D3×C3⋊Dic3—21no—
17D3×#634—22no—
18D3×#634—22no—
19C2×C33×C8C2211yescharacteristic
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C3×#3744—22no—
22C2×C32×C3⋊C8C2211yescharacteristic
23#88803—21no—
24#88796—22no—
25C22×#634—31no—
26C2×D3×Dic6—34no—
27#64139—31no—
28C2×#3744—31no—
29C2×#3744D311yescharacteristic, complemented
30C2×C3×#204—34no—
31#63908—34no—
32C2×#3716—34no—
33C32×C22⋊C8—31no—
34#63643—31no—
35#63643—31no—
36#63643—34no—
37#63643—64no—
38C3×(C2×D3)⋊C8D314yescomplemented
39#63566—34no—
40C22×C3×C3⋊D3—21no—
41C22×C32×D3D411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)—21no—
45#6308—21no—
46#6308D412yescomplemented
47C2×C3×C3⋊Dic3—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×#634—21no—
51C3×#634D412yescomplemented
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55#6293—24no—
56#6290—42no—
57D3×C3⋊Dic3—24no—
58C33×C8—21no—
59C32×C3⋊C8—21no—
60#5170—91no—
61#4738—94no—
62C22×C32×C4—31no—
63C2×#638—61no—
64C22×C3⋊Dic3—61no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×#634—31no—
67C2×#634—34no—
68C2×#634—62no—
69C2×#634—61no—
70C2×#634—62no—
71C2×#634—64no—
72C2×#634C2×D311yescharacteristic
73C2×C3×D12—34no—
74C2×C3×C4×D3C2×D314yes—
75C2×C3×Dic6—34no—
76C2×#627—64no—
77C2×#625—64no—
78C2×D3×Dic3—64no—
79D3×Dic6—68no—
80D3×Dic6—68no—
81C2×C32×C8—24no—
82C2×C32×C8—31no—
83C2×C32×C8C2×D314yes—
84C2×C32×C8C2×D311yescharacteristic
85#3752—61no—
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—34no—
88C4⋊(C3⋊Dic3)—64no—
89#3744—22no—
90#3744—62no—
91C2×C3×C3⋊C8C2×D314yes—
92C3×#204—68no—
93(C2×D3)⋊Dic3—64no—
94#3716—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3—41no—
98C2×C3×C3⋊D3—41no—
99C2×C32×D3—21no—
100C2×C32×D3—21no—
101C2×C32×D3SD1612yescomplemented
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—22no—
105C3⋊(C3⋊Dic3)—41no—
106C3×C3⋊Dic3—22no—
107C32×Dic3—21no—
108C2×D3×Q8—91no—
109C22×Dic6—94no—
110C2×C3×SD16—91no—
111#938—91no—
112C2×#214—91no—
113#930—94no—
114C2×#204—34no—
115C2×#204—94no—
116C3×C22⋊C8—34no—
117#835—91no—
118(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
119#816—94no—
120#816—94no—
121#816—91no—
122#816—184no—
123C23×C32—31no—
124C22×C3⋊D3—21no—
125C22×C3×D3—64no—
126C22×C3×D3D1214yes—
127C2×C32×C4—24no—
128C2×C32×C4—31no—
129C2×C32×C4—61no—
130C2×C32×C4—61no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133#638—122no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—61no—
137C2×C3⋊Dic3—61no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—64no—
140C2×C3⋊Dic3—124no—
141C2×C3⋊Dic3—124no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144#634—21no—
145#634—32no—
146#634—38no—
147#634—68no—
148#634—64no—
149#634—61no—
150#634—64no—
151#634—124no—
152#634#6712yescomplemented
153C2×C3×Dic3—64no—
154C2×C3×Dic3D1214yes—
155C3×D12—64no—
156C3×D12—64no—
157C3×C4×D3—24no—
158C3×C4×D3—24no—
159C3×Dic6—38no—
160C3×Dic6—64no—
161#627—616no—
162#625—128no—
163D3×Dic3—616no—
164C32×C8—21no—
165C32×C8—24no—
166C32×C8—44no—
167C32×C8—61no—
168C3×C3⋊C8—24no—
169#448—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33C2×SD1611yescharacteristic
172C2×C33#13711yescharacteristic
173C3×C3⋊D3—81no—
174C32×D3—22no—
175C2×C3×Q8—91no—
176C2×C3×D4—91no—
177C22×C3×C4—34no—
178C2×#67—184no—
179C2×#67—181no—
180C22×Dic3—184no—
181D3×Q8—182no—
182D3×Q8—182no—
183C2×D12—34no—
184C2×C4×D3—181no—
185C2×C4×D3C2×C3⋊D311yescharacteristic
186C2×Dic6—34no—
187C2×Dic6—94no—
188C2×Dic6—91no—
189C2×Dic6—184no—
190C2×Dic6—188no—
191C2×Dic6—188no—
192C2×Dic6—184no—
193C2×Dic6—181no—
194C3×SD16—182no—
195C2×C3×C8—24no—
196C2×C3×C8—34no—
197C2×C3×C8D3214yes—
198C2×C3×C8C2×C3⋊D311yescharacteristic
199#217—181no—
200#214—182no—
201(C2×D3)⋊C4—184no—
202C4⋊Dic3—94no—
203C4⋊Dic3—91no—
204C4⋊Dic3—184no—
205C2×C3⋊C8C2×C3⋊D311yescharacteristic
206#204—68no—
207#204—188no—
208C22×C32—24no—
209C22×C32—32no—
210C22×C32—61no—
211C22×C32—61no—
212C22×C32C2×D1214yes—
213C22×C32D3×D411yescharacteristic
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3×D3—24no—
218C2×C3×D3—24no—
219C2×C3×D3—124no—
220C2×C3×D3—124no—
221C2×C3×D3—124no—
222C2×C3×D3#20418yescomplemented
223C32×C4—24no—
224C32×C4—24no—
225C32×C4—61no—
226C32×C4D3×D414yes—
227C32×C4D3×D414yes—
228C32×C4D3×D411yescharacteristic
229C32×C4C2×#6711yescharacteristic
230C3⋊Dic3—22no—
231C3⋊Dic3—62no—
232C3⋊Dic3—68no—
233C3⋊Dic3—128no—
234C3⋊Dic3—121no—
235C3⋊Dic3—124no—
236C3⋊Dic3—244no—
237C3×Dic3—24no—
238C3×Dic3—68no—
239C22×Q8—271no—
240C2×SD16—91no—
241C2×SD16—271no—
242#121—271no—
243#103—271no—
244#103—271no—
245C22⋊C8—31no—
246C33#44811yescharacteristic, complemented, Sylow
247C23×C3—34no—
248C23×C3—181no—
249C22×D3—64no—
250C22×D3C4⋊(C3⋊D3)11yescharacteristic
251C3×Q8—92no—
252C3×Q8—181no—
253C3×D4—181no—
254C3×D4—181no—
255C2×C3×C4—24no—
256C2×C3×C4—34no—
257C2×C3×C4—64no—
258C2×C3×C4—64no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262#67—368no—
263#67—362no—
264C2×Dic3—64no—
265C2×Dic3—1816no—
266C2×Dic3—184no—
267C2×Dic3—181no—
268C2×Dic3—184no—
269C2×Dic3—181no—
270C2×Dic3—364no—
271C2×Dic3—364no—
272C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
273D12—64no—
274D12—64no—
275C4×D3—21no—
276C4×D3—21no—
277C4×D3—184no—
278Dic6—38no—
279Dic6—64no—
280Dic6—98no—
281Dic6—92no—
282Dic6—188no—
283Dic6—184no—
284Dic6—1816no—
285Dic6—184no—
286Dic6—181no—
287Dic6—364no—
288C3×C8—24no—
289C3×C8—21no—
290C3×C8—44no—
291C3×C8—64no—
292C3⋊C8—21no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—62no—
297C2×C32#88214yes—
298C2×C32C2×#20414yes—
299C2×C32#90914yes—
300C2×C32D3×SD1611yescharacteristic
301C2×C32#91511yescharacteristic
302C2×C32#93711yescharacteristic
303C3⋊D3—81no—
304C3×D3—28no—
305C3×D3—244no—
306C2×Q8—91no—
307C2×Q8—271no—
308C2×Q8—541no—
309C2×Q8—542no—
310C2×Q8—542no—
311C2×D4—91no—
312C2×D4—541no—
313C22×C4—31no—
314C22×C4—541no—
315SD16—182no—
316SD16—542no—
317C2×C8—31no—
318C2×C8D3×C3⋊D311yescharacteristic
319C4⋊C4—271no—
320C22⋊C4—541no—
321C22×C3—24no—
322C22×C3—38no—
323C22×C3—64no—
324C22×C3—64no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3D3×D1214yes—
329C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
330C2×D3—21no—
331C2×D3—21no—
332C2×D3—124no—
333C2×D3—124no—
334C2×D3—124no—
335C2×D3#374412yescomplemented
336C3×C4—24no—
337C3×C4—24no—
338C3×C4—64no—
339C3×C4—182no—
340C3×C4#380214yes—
341C3×C4D3×#6714yes—
342C3×C4D4×C3⋊D311yescharacteristic
343C3×C4D4×C3⋊D311yescharacteristic
344Dic3—21no—
345Dic3—68no—
346Dic3—182no—
347Dic3—188no—
348Dic3—368no—
349Dic3—364no—
350Dic3—361no—
351Dic3—724no—
352C32—24no—
353C32#473814yescomplemented
354C32#517011yescharacteristic, complemented
355C23—31no—
356C23—181no—
357Q8—92no—
358Q8—181no—
359Q8—272no—
360Q8—544no—
361Q8—541no—
362D4—181no—
363D4—181no—
364D4—1082no—
365C2×C4—31no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—181no—
369C2×C4—544no—
370C2×C4—541no—
371C2×C4—541no—
372C2×C4C2×D3×C3⋊D311yescharacteristic
373C8—21no—
374C8—61no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—68no—
379C2×C3—721no—
380C2×C3D3×#20414yes—
381C2×C3#6380014yes—
382C2×C3#6392214yes—
383C2×C3#6409111yescharacteristic
384C2×C3C2×#374411yescharacteristic
385C2×C3#6411811yescharacteristic
386D3—22no—
387D3—244no—
388C22—32no—
389C22—61no—
390C22—61no—
391C22—361no—
392C22—361no—
393C22—361no—
394C22D3×C4⋊(C3⋊D3)11yescharacteristic
395C4—61no—
396C4—182no—
397C4—542no—
398C4—1081no—
399C4#8902611yescharacteristic
400C4C3⋊D3×#6711yescharacteristic
401C3—24no—
402C3#1059082214yescomplemented
403C3#1059243711yescharacteristic, complemented
404C2—62no—
405C2—721no—
406C2D3×#374411yescharacteristic
407C2#1172332611yescharacteristic
408C2#1172354111yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723515C2
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×D3×#634C2semidirect
2. C2×C3×#3744C2semidirect
4. C2×#88796C2semidirect
5. #11722186C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. #11722040C2semidirect
29. C2×#3744D3semidirect
38. C3×(C2×D3)⋊C8D3semidirect
46. #6308D4semidirect
51. C3×#634D4semidirect
101. C2×C32×D3SD16semidirect
118. (C2×D3)⋊C8C3⋊D3semidirect
152. #634#67semidirect
222. C2×C3×D3#204semidirect
246. C33#448semidirect
335. C2×D3#3744semidirect
353. C32#4738semidirect
354. C32#5170semidirect
402. C3#10590822semidirect
403. C3#10592437semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
C2×C3×#37442
#117235152
C2×#887962
#117221862
C32×(C2×D3)⋊C82
#117220402
#105924373
#105908223