Finite Group Databasebeta

← #13159980

Subgroups — #13159980

Order 1728.

1048 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1048 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
1C2×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2#11723531C211yesmaximal, characteristic
3#11723449C211yesmaximal, characteristic, complemented
4C2×#88792C211yesmaximal, characteristic, complemented
5C3×#63643C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722053C211yesmaximal, characteristic, complemented
8#10592477—31nomaximal
9#10590841—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6291—21no—
14C2×C33×C8C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C8×C3⋊(C3⋊D3)—22no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20#88804—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88792—22no—
23#64147—31no—
24C2×#3754—31no—
25C2×C8×C3⋊D3—31no—
26C2×C8×C3⋊D3—34no—
27C2×C8×C3⋊D3—64no—
28#64106—31no—
29#63916—34no—
30#63871—34no—
31C2×#3715—34no—
32C32×#102—31no—
33#63643D311yescharacteristic, complemented
34#63635—31no—
35C3×#821D314yescomplemented
36C3×#816—34no—
37#63574—34no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C22×C32×D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C4×C3⋊(C3⋊D3)—21no—
43C4×C3⋊(C3⋊D3)—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634—21no—
48C2×C32×Dic3—21no—
49C32×D12—21no—
50C32×D12—21no—
51#6291—42no—
52C33×C8—21no—
53C3⋊(C3⋊(C3⋊C8))D412yescomplemented
54#4807—91no—
55#4778—94no—
56C2×C32×D4—31no—
57C2×#638—61no—
58C2×C4×C3⋊D3—31no—
59C2×C4×C3⋊D3—34no—
60C2×C4×C3⋊D3—64no—
61C2×#634C2×D311yescharacteristic
62C2×C3×D12C2×D314yes—
63C2×C3×Dic6—34no—
64C2×#626—64no—
65C2×C32×C8—24no—
66C2×C32×C8C2×D314yes—
67C2×C32×C8C2×D311yescharacteristic
68C32×C4⋊C4—31no—
69#3757—61no—
70#3754—62no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—64no—
77C8×C3⋊D3—62no—
78C8×C3⋊D3—68no—
79C8×C3⋊D3—616no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3C2×D314yes—
82Dic3⋊Dic3—64no—
83#3722—64no—
84(C2×D3)⋊Dic3—64no—
85#3715—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)SD1612yescomplemented
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—21no—
96C3×C3⋊Dic3—41no—
97C3×C3⋊Dic3—41no—
98C32×Dic3—41no—
99#946—91no—
100C2×#215—91no—
101#938—94no—
102C2×#214—94no—
103C2×D3×C8—94no—
104C2×D3×C8—91no—
105C2×D3×C8—184no—
106#897—94no—
107#895—91no—
108C3×#103—91no—
109C3×#102—34no—
110#821C3⋊D311yescharacteristic, complemented
111#816—34no—
112#808—94no—
113#808—91no—
114C23×C32—61no—
115C22×C3⋊D3—31no—
116C22×C3⋊D3—34no—
117C22×C3⋊D3—64no—
118C22×C3×D3—24no—
119C32×D4—61no—
120C32×D4—61no—
121C2×C32×C4—24no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125#638—122no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—34no—
130C2×C3⋊Dic3—64no—
131C4×C3⋊D3—68no—
132C4×C3⋊D3—68no—
133C4×C3⋊D3—64no—
134C4×C3⋊D3—61no—
135C4×C3⋊D3—64no—
136C4×C3⋊D3—61no—
137#634—21no—
138#634—21no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—64no—
142C3×D12—24no—
143C3×D12—24no—
144C3×Dic6—64no—
145C3×Dic6—64no—
146#626—128no—
147C32×C8—28no—
148C32×C8—24no—
149C32×C8—21no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153#465—271noSylow
154C2×C33#12111yescharacteristic
155C2×C33C2×SD1611yescharacteristic
156C2×C33#13511yescharacteristic
157C3⋊(C3⋊D3)—22no—
158C32×D3—81no—
159C2×C3×Q8—91no—
160C2×C3×D4—34no—
161C2×#67—184no—
162C2×D12—181no—
163C2×D12C2×C3⋊D311yescharacteristic
164C2×C4×D3—94no—
165C2×C4×D3—91no—
166C2×C4×D3—184no—
167C2×Dic6—34no—
168C2×C3×C8—24no—
169C2×C3×C8D3214yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C3×C4⋊C4—34no—
172C3×C4⋊C4—91no—
173#217—184no—
174#215—182no—
175#214—188no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—184no—
178(C2×D3)⋊C4—181no—
179C4⋊Dic3—34no—
180C4⋊Dic3—184no—
181C4⋊Dic3C2×C3⋊D311yescharacteristic
182Dic3⋊C4—181no—
183C2×C3⋊C8—94no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—184no—
186D3×C8—188no—
187D3×C8—182no—
188D3×C8—1816no—
189C22×C32—24no—
190C22×C32—121no—
191C22×C32—121no—
192C22×C32—121no—
193C22×C32D3×D414yes—
194C22×C32D3×D411yescharacteristic
195C2×C3⋊D3—32no—
196C2×C3⋊D3—38no—
197C2×C3⋊D3—68no—
198C2×C3⋊D3—68no—
199C2×C3⋊D3—68no—
200C2×C3⋊D3—64no—
201C2×C3⋊D3—61no—
202C2×C3⋊D3—64no—
203C2×C3⋊D3—61no—
204C2×C3×D3—44no—
205C2×C3×D3—44no—
206C2×C3×D3—44no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—121no—
210C32×C4#23514yes—
211C32×C4#23511yescharacteristic
212C32×C4D3×D414yes—
213C32×C4D3×D411yescharacteristic
214C3⋊Dic3—41no—
215C3⋊Dic3—41no—
216C3⋊Dic3—68no—
217C3⋊Dic3—64no—
218C3⋊Dic3—61no—
219C3×Dic3—44no—
220C3×Dic3—124no—
221C3×Dic3—124no—
222C2×SD16—271no—
223C22×C8—271no—
224C22⋊Q8—271no—
225#121—271no—
226#106—271no—
227#103—91no—
228#102—31no—
229C33#46511yescharacteristic, complemented, Sylow
230C23×C3—64no—
231C22×D3—21no—
232C22×D3—94no—
233C22×D3—91no—
234C22×D3—184no—
235C3×Q8—181no—
236C3×Q8—181no—
237C3×D4—64no—
238C3×D4—64no—
239C2×C3×C4—24no—
240C2×C3×C4—64no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245#67—368no—
246C2×Dic3—21no—
247C2×Dic3—64no—
248C2×Dic3—64no—
249C2×Dic3—94no—
250C2×Dic3—91no—
251C2×Dic3—184no—
252D12—21no—
253D12—21no—
254D12—362no—
255C4×D3—188no—
256C4×D3—188no—
257C4×D3—184no—
258C4×D3—184no—
259C4×D3—181no—
260C4×D3—181no—
261Dic6—64no—
262Dic6—64no—
263C3×C8—28no—
264C3×C8—24no—
265C3×C8—21no—
266C3⋊C8—98no—
267C3⋊C8—92no—
268C3⋊C8—188no—
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—241no—
273C2×C32#88414yes—
274C2×C32#88411yescharacteristic
275C2×C32#91214yes—
276C2×C32D3×SD1614yes—
277C2×C32D3×SD1611yescharacteristic
278C2×C32#91911yescharacteristic
279C3⋊D3—616no—
280C3⋊D3—68no—
281C3⋊D3—62no—
282C3×D3—84no—
283C2×Q8—91no—
284C2×D4—31no—
285C2×D4—541no—
286C22×C4—271no—
287SD16—542no—
288C2×C8—271no—
289C2×C8—542no—
290C2×C8D3×C3⋊D311yescharacteristic
291C4⋊C4—31no—
292C4⋊C4—91no—
293C4⋊C4—541no—
294C22⋊C4—541no—
295C22⋊C4—541no—
296C22×C3—24no—
297C22×C3—124no—
298C22×C3—124no—
299C22×C3—124no—
300C22×C3#380214yes—
301C22×C3D4×C3⋊D311yescharacteristic
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3—98no—
306C2×D3—92no—
307C2×D3—188no—
308C2×D3—188no—
309C2×D3—188no—
310C2×D3—184no—
311C2×D3—181no—
312C2×D3—181no—
313C2×D3—184no—
314C3×C4—24no—
315C3×C4—24no—
316C3×C4—124no—
317C3×C4—361no—
318C3×C4—361no—
319C3×C4#379814yes—
320C3×C4#381114yes—
321C3×C4#382811yescharacteristic
322C3×C4D4×C3⋊D311yescharacteristic
323Dic3—41no—
324Dic3—124no—
325Dic3—124no—
326Dic3—188no—
327Dic3—181no—
328Dic3—184no—
329C32—24no—
330C32#477814yescomplemented
331C32#480711yescharacteristic, complemented
332C23—61no—
333C23—271no—
334Q8—181no—
335Q8—181no—
336D4—61no—
337D4—61no—
338D4—1082no—
339C2×C4—61no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—271no—
343C2×C4—541no—
344C2×C4—541no—
345C2×C4C2×D3×C3⋊D311yescharacteristic
346C8—21no—
347C8—272no—
348C2×C3—24no—
349C2×C3—24no—
350C2×C3—24no—
351C2×C3—244no—
352C2×C3#6379614yes—
353C2×C3#6381014yes—
354C2×C3#6391214yes—
355C2×C3#6409311yescharacteristic
356C2×C3#6412111yescharacteristic
357C2×C3SD16×C3⋊D311yescharacteristic
358D3—81no—
359D3—1816no—
360D3—188no—
361D3—182no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—272no—
366C22—541no—
367C22—541no—
368C22#8902611yescharacteristic
369C4—121no—
370C4—361no—
371C4—361no—
372C4—541no—
373C4#8902011yescharacteristic
374C4#8903911yescharacteristic
375C3—24no—
376C3#1059084114yescomplemented
377C3#1059247711yescharacteristic, complemented
378C2—241no—
379C2—542no—
380C2#1172332111yescharacteristic
381C2#1172334311yescharacteristic
382C2#1172352711yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723531C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#465
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×C8×C3⋊(C3⋊D3)C2semidirect
3. #11723449C2semidirect
4. C2×#88792C2semidirect
5. C3×#63643C2semidirect
6. C32×#821C2semidirect
7. #11722053C2semidirect
33. #63643D3semidirect
35. C3×#821D3semidirect
53. C3⋊(C3⋊(C3⋊C8))D4semidirect
89. C2×C3⋊(C3⋊D3)SD16semidirect
110. #821C3⋊D3semidirect
229. C33#465semidirect
330. C32#4778semidirect
331. C32#4807semidirect
376. C3#10590841semidirect
377. C3#10592477semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊(C3⋊D3)2
#117235312
#117234492
C2×#887922
C3×#636432
C32×#8212
#117220532
#105924773
#105908413