Finite Group Databasebeta

← #13159362

Subgroups — #13159362

Order 1728.

1180 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 166 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#11723934C211yesmaximal, characteristic, complemented
2#11723501C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic
4C3×#63636C211yesmaximal, characteristic, complemented
5C32×#807C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10592571—31nomaximal
9#10590459—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C33×C4⋊C4C2211yescharacteristic
15#88859—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88803—21no—
21#88791C414yescomplemented
22C2×#3745—31no—
23#64102—31no—
24#64102—31no—
25#64102—34no—
26#64102—64no—
27#63898—34no—
28C2×#3714—34no—
29C32×#107—31no—
30C8×C3⋊Dic3—31no—
31#63636—31no—
32#63636D311yescharacteristic, complemented
33C3×#809—34no—
34C3×#807D314yescomplemented
35#63564—34no—
36#63553—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C22×C3×C3⋊D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
45C4×C3⋊(C3⋊D3)—42no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)C2×C412yescomplemented
48C2×C32×Dic3—21no—
49Dic3×C3⋊D3—42no—
50C3×C3⋊(C3⋊C8)—21no—
51C32×C3⋊C8C2×C412yescomplemented
52#4872—94no—
53#4756—91no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C4⋊(C3⋊D3)—34no—
56C2×C4⋊(C3⋊D3)—64no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4×C3⋊D3—61no—
59C2×C4×C3⋊D3—64no—
60C2×C4×C3⋊D3—61no—
61C2×C4×C3⋊D3—68no—
62C2×C3×D12—34no—
63C2×D3×Dic3—64no—
64C2×C32×C8—31no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4—31no—
67C32×C4⋊C4C2×D314yes—
68C32×C4⋊C4C2×D311yescharacteristic
69#3752—61no—
70#3752—64no—
71#3752—61no—
72#3752—68no—
73C4×C3⋊Dic3—31no—
74C4×C3⋊Dic3—34no—
75C4×C3⋊Dic3—64no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77#3745—34no—
78C3×C4⋊Dic3C2×D314yes—
79C2×C3×C3⋊C8—34no—
80C2×C3×C3⋊C8C2×D314yes—
81(C2×D3)⋊Dic3—64no—
82#3714—316no—
83C22×C33C2×D411yescharacteristic
84C2×C3⋊(C3⋊D3)—22no—
85C2×C3⋊(C3⋊D3)—41no—
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3—41no—
90C33×C4—41no—
91C33×C4C22×C411yescharacteristic
92C33×C4#4111yescharacteristic
93C3⋊(C3⋊Dic3)—21no—
94C3⋊(C3⋊Dic3)D812yescomplemented
95C32×Dic3—41no—
96D4×Dic3—91no—
97C2×#213—91no—
98C2×D24—94no—
99#893—94no—
100#893—94no—
101#893—91no—
102#893—184no—
103C3×#107—34no—
104C3×#102—91no—
105#821—91no—
106C8×Dic3—94no—
107C8×Dic3—91no—
108#809—34no—
109#809—94no—
110#807C3⋊D311yescharacteristic, complemented
111C22×C3⋊D3—21no—
112C22×C3⋊D3—68no—
113C22×C3⋊D3—64no—
114C22×C3⋊D3—61no—
115C22×C3×D3—64no—
116C2×C32×C4—28no—
117C2×C32×C4—24no—
118C2×C32×C4—21no—
119C2×C32×C4—24no—
120C2×C32×C4—61no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—34no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—64no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—32no—
131C4⋊(C3⋊D3)—38no—
132C4⋊(C3⋊D3)—68no—
133C4⋊(C3⋊D3)—68no—
134C4⋊(C3⋊D3)—61no—
135C4⋊(C3⋊D3)—64no—
136C4⋊(C3⋊D3)C4×D312yescomplemented
137C4×C3⋊D3—122no—
138C4×C3⋊D3—128no—
139C4×C3⋊D3—122no—
140C4×C3⋊D3—1216no—
141C2×C3×Dic3—24no—
142C3×D12—38no—
143C3×D12—64no—
144D3×Dic3—128no—
145C32×C8—32no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—64no—
148C3×C3⋊C8C4×D318yescomplemented
149C4×D8—271noSylow
150C2×C33C4×D411yescharacteristic
151C2×C33C2×D811yescharacteristic
152C2×C33#13511yescharacteristic
153C3⋊(C3⋊D3)—42no—
154C3×C3⋊D3—42no—
155C2×C3×D4—91no—
156C22×Dic3—181no—
157C2×D12—34no—
158C2×D12—94no—
159C2×D12—91no—
160C2×D12—184no—
161C2×C4×D3—184no—
162C2×C4×D3—181no—
163C2×C4×D3—184no—
164C2×C4×D3—188no—
165C2×C3×C8—34no—
166C2×C3×C8—91no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—34no—
169C3×C4⋊C4D3214yes—
170C3×C4⋊C4C2×C3⋊D311yescharacteristic
171#217—181no—
172#213—94no—
173(C2×D3)⋊C4—184no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—184no—
176(C2×D3)⋊C4—188no—
177C4⋊Dic3C2×C3⋊D311yescharacteristic
178C4×Dic3—94no—
179C4×Dic3—91no—
180C4×Dic3—184no—
181C2×C3⋊C8—34no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183D24—916no—
184C22×C32—24no—
185C22×C32D3×D414yes—
186C22×C32D3×D411yescharacteristic
187C2×C3⋊D3—22no—
188C2×C3⋊D3—41no—
189C2×C3⋊D3—41no—
190C2×C3⋊D3—616no—
191C2×C3⋊D3—62no—
192C2×C3⋊D3—68no—
193C2×C3⋊D3—128no—
194C2×C3⋊D3—128no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—121no—
198C2×C3⋊D3—124no—
199C2×C3×D3—68no—
200C2×C3×D3—124no—
201C2×C3×D3—124no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—48no—
205C32×C4—44no—
206C32×C4—41no—
207C32×C4—121no—
208C32×C4C2×C4×D314yes—
209C32×C4C2×C4×D311yescharacteristic
210C32×C4#23711yescharacteristic
211C32×C4#23914yes—
212C3⋊Dic3—32no—
213C3⋊Dic3—38no—
214C3⋊Dic3—68no—
215C3⋊Dic3—61no—
216C3⋊Dic3—64no—
217C3⋊Dic3—124no—
218C3×Dic3—44no—
219C2×D8—271no—
220C4×D4—271no—
221C4×D4—271no—
222#107—31no—
223#102—91no—
224#102—271no—
225C4×C8—271no—
226C33C4×D811yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—64no—
229C22×D3—188no—
230C22×D3—184no—
231C22×D3—181no—
232C3×D4—92no—
233C3×D4—181no—
234C2×C3×C4—28no—
235C2×C3×C4—24no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—64no—
239C2×C3×C4C2×D3214yes—
240C2×C3×C4C22×C3⋊D311yescharacteristic
241C2×Dic3—21no—
242C2×Dic3—94no—
243C2×Dic3—91no—
244C2×Dic3—94no—
245C2×Dic3—91no—
246C2×Dic3—184no—
247C2×Dic3—184no—
248C2×Dic3—362no—
249D12—38no—
250D12—64no—
251D12—98no—
252D12—92no—
253D12—188no—
254D12—188no—
255D12—184no—
256D12—181no—
257C4×D3—362no—
258C4×D3—368no—
259C4×D3—368no—
260C4×D3—3616no—
261C3×C8—38no—
262C3×C8—181no—
263C3⋊C8—64no—
264C3⋊C8C4×C3⋊D312yescomplemented
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32#67⋊C411yescharacteristic
269C2×C32#89314yes—
270C2×C32D3×D814yes—
271C2×C32D3×D811yescharacteristic
272C2×C32#91611yescharacteristic
273C2×C32#91914yes—
274C3⋊D3—42no—
275C3⋊D3—1216no—
276C3⋊D3—128no—
277C3⋊D3—122no—
278C3×D3—128no—
279C2×D4—91no—
280C2×D4—271no—
281C22×C4—541no—
282C22×C4—541no—
283D8—274no—
284C2×C8—31no—
285C2×C8—91no—
286C4⋊C4—31no—
287C4⋊C4D3×C3⋊D311yescharacteristic
288C22⋊C4—541no—
289C22⋊C4—541no—
290C42—271no—
291C22×C3—24no—
292C22×C3—182no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3#381114yes—
296C22×C3D4×C3⋊D311yescharacteristic
297C2×D3—68no—
298C2×D3—124no—
299C2×D3—124no—
300C2×D3—1816no—
301C2×D3—182no—
302C2×D3—188no—
303C2×D3—368no—
304C2×D3—368no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—364no—
308C2×D3—364no—
309C3×C4—24no—
310C3×C4—24no—
311C3×C4—48no—
312C3×C4—41no—
313C3×C4—44no—
314C3×C4—124no—
315C3×C4#379614yes—
316C3×C4C4×D3214yes—
317C3×C4C2×C4×C3⋊D311yescharacteristic
318C3×C4#383211yescharacteristic
319Dic3—41no—
320Dic3—92no—
321Dic3—98no—
322Dic3—188no—
323Dic3—181no—
324Dic3—184no—
325Dic3—364no—
326C32—24no—
327C32#475611yescharacteristic, complemented
328C32#487214yescomplemented
329C23—181no—
330C23—541no—
331D4—92no—
332D4—181no—
333D4—272no—
334D4—541no—
335C2×C4—21no—
336C2×C4—61no—
337C2×C4—271no—
338C2×C4—271no—
339C2×C4—1082no—
340C2×C4—1082no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—32no—
343C8—181no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—362no—
348C2×C3#6388114yes—
349C2×C3#6392614yes—
350C2×C3#6395014yes—
351C2×C3#6410211yescharacteristic
352C2×C3D8×C3⋊D311yescharacteristic
353C2×C3#6412811yescharacteristic
354D3—128no—
355D3—3616no—
356D3—362no—
357D3—368no—
358C22—182no—
359C22—361no—
360C22—361no—
361C22—542no—
362C22—1081no—
363C22—1081no—
364C22#8903911yescharacteristic
365C4—41no—
366C4—121no—
367C4—272no—
368C4—541no—
369C4#8901711yescharacteristic
370C4C4×D3×C3⋊D311yescharacteristic
371C3—24no—
372C3#1059045914yescomplemented
373C3#1059257111yescharacteristic, complemented
374C2—362no—
375C2—1082no—
376C2#1172347211yescharacteristic
377C2#1172354711yescharacteristic
378C2#1172358911yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#88791C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C4×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723934C2semidirect
2. #11723501C2semidirect
4. C3×#63636C2semidirect
5. C32×#807C2semidirect
6. #11722035C2semidirect
7. #11722018C2semidirect
21. #88791C4semidirect
32. #63636D3semidirect
34. C3×#807D3semidirect
44. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
47. C3×C4⋊(C3⋊D3)C2×C4semidirect
51. C32×C3⋊C8C2×C4semidirect
94. C3⋊(C3⋊Dic3)D8semidirect
110. #807C3⋊D3semidirect
136. C4⋊(C3⋊D3)C4×D3semidirect
148. C3×C3⋊C8C4×D3semidirect
226. C33C4×D8semidirect
264. C3⋊C8C4×C3⋊D3semidirect
327. C32#4756semidirect
328. C32#4872semidirect
372. C3#10590459semidirect
373. C3#10592571semidirect

Maximal subgroups

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

subgroupindex
#117239342
#117235012
C2×#887912
C3×#636362
C32×#8072
#117220352
#117220182
#105925713
#105904593