Finite Group Databasebeta

← #13159692

Subgroups — #13159692

Order 1728.

1016 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 154 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×#88851C211yesmaximal, characteristic, complemented
2#11723484C211yesmaximal, characteristic, complemented
3#11723439C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic
5C3×#63648C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592543—31nomaximal
9#10590860—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6289—21no—
14C2×C33×C8C2211yescharacteristic
15#88851—22no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88803—21no—
21#88803—21no—
22#88791—22no—
23C2×#3745—31no—
24C2×#3744—31no—
25C2×#3744—34no—
26C2×#3744—64no—
27#64084—31no—
28C4×C4⋊(C3⋊D3)—31no—
29#63894—34no—
30#63860—34no—
31C2×#3714—34no—
32C32×C4⋊C8—31no—
33#63648D311yescharacteristic, complemented
34#63643—31no—
35C3×#821—34no—
36C3×Dic3⋊C8D314yescomplemented
37#63567—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))D412yes—
44#6308—21no—
45#6308—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51#6289—42no—
52C33×C8—21no—
53C32×C3⋊C8—21no—
54#4844—94no—
55#4787—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4×C3⋊D3—61no—
61C2×#634—31no—
62C2×#634—34no—
63C2×#634—64no—
64C2×C3×D12—34no—
65C2×#624—64no—
66C2×C32×C8—24no—
67C2×C32×C8—31no—
68C2×C32×C8C2×D314yes—
69C2×C32×C8C2×D311yescharacteristic
70C32×C42—31no—
71#3752—61no—
72#3752—61no—
73#3752—61no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75#3745—62no—
76#3744—62no—
77#3744—68no—
78#3744—616no—
79C3×C4⋊Dic3—34no—
80C3×C4×Dic3C2×D314yes—
81C2×C3×C3⋊C8C2×D314yes—
82#3722—64no—
83(C2×D3)⋊Dic3—64no—
84(C2×D3)⋊Dic3—64no—
85#3714—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—22no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C33×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—41no—
97C32×Dic3—21no—
98C32×Dic3—21no—
99#936—91no—
100C2×#213—91no—
101C2×D24—94no—
102C2×#204—94no—
103C2×#204—91no—
104C2×#204—184no—
105#893—91no—
106#875—94no—
107C4×D12—94no—
108C3×C4⋊C8—34no—
109C3×#102—91no—
110#821—34no—
111#816—94no—
112Dic3⋊C8C3⋊D311yescharacteristic, complemented
113#810—91no—
114C22×C3⋊D3—21no—
115C22×C3⋊D3—61no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—124no—
118C22×C3×D3—64no—
119C2×C32×C4—24no—
120C2×C32×C4—31no—
121C2×C32×C4—31no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—124no—
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—32no—
131C4⋊(C3⋊D3)—38no—
132C4⋊(C3⋊D3)—68no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—124no—
136C4×C3⋊D3—122no—
137#634—68no—
138#634—61no—
139#634—64no—
140#634—61no—
141#634—64no—
142#634—124no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3D1214yes—
145C2×C3×Dic3D1214yes—
146C3×D12—64no—
147C3×D12—64no—
148#624—128no—
149C32×C8—28no—
150C32×C8—24no—
151C32×C8—21no—
152C32×C8—61no—
153C3×C3⋊C8—24no—
154#463—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33#13511yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3⋊(C3⋊D3)—42no—
159C3×C3⋊D3—81no—
160C2×C3×D4—91no—
161C2×D12—34no—
162C2×D12—94no—
163C2×D12—91no—
164C2×D12—184no—
165C2×C4×D3—184no—
166C2×C4×D3—181no—
167C2×Dic6—94no—
168C2×Dic6—91no—
169C2×Dic6—184no—
170C2×C3×C8—24no—
171C2×C3×C8—34no—
172C2×C3×C8D3214yes—
173C2×C3×C8C2×C3⋊D311yescharacteristic
174C3×C4⋊C4—91no—
175C3×C42—34no—
176#217—181no—
177#217—181no—
178#213—182no—
179(C2×D3)⋊C4—184no—
180(C2×D3)⋊C4—184no—
181(C2×D3)⋊C4—184no—
182(C2×D3)⋊C4—181no—
183C4⋊Dic3—34no—
184C4×Dic3C2×C3⋊D311yescharacteristic
185C2×C3⋊C8C2×C3⋊D311yescharacteristic
186D24—188no—
187#204—188no—
188#204—182no—
189#204—1816no—
190C22×C32—24no—
191C22×C32C2×D1214yes—
192C22×C32D3×D411yescharacteristic
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—62no—
197C2×C3⋊D3—68no—
198C2×C3⋊D3—128no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—121no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—244no—
204C2×C3⋊D3—244no—
205C2×C3×D3—124no—
206C2×C3×D3—124no—
207C2×C3×D3—124no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—61no—
211C32×C4—61no—
212C32×C4#23511yescharacteristic
213C32×C4D3×D414yes—
214C32×C4D3×D411yescharacteristic
215C32×C4#23914yes—
216C3⋊Dic3—41no—
217C3⋊Dic3—128no—
218C3⋊Dic3—121no—
219C3⋊Dic3—124no—
220C3×Dic3—24no—
221C3×Dic3—24no—
222C3×Dic3—124no—
223C2×SD16—271no—
224C2×D8—271no—
225#124—271no—
226C4×D4—271no—
227C4⋊C8—31no—
228#103—271no—
229#102—91no—
230C33#46311yescharacteristic, complemented, Sylow
231C23×C3—181no—
232C22×D3—64no—
233C22×D3—184no—
234C22×D3—181no—
235C22×D3—364no—
236C3×D4—181no—
237C3×D4—181no—
238C2×C3×C4—24no—
239C2×C3×C4—34no—
240C2×C3×C4—34no—
241C2×C3×C4—181no—
242C2×C3×C4C2×D3214yes—
243C2×C3×C4C22×C3⋊D311yescharacteristic
244C2×Dic3—64no—
245C2×Dic3—184no—
246C2×Dic3—181no—
247C2×Dic3—364no—
248C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
249C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
250D12—64no—
251D12—64no—
252D12—98no—
253D12—92no—
254D12—188no—
255D12—184no—
256D12—181no—
257D12—364no—
258C4×D3—362no—
259C4×D3—368no—
260Dic6—188no—
261Dic6—184no—
262Dic6—184no—
263Dic6—181no—
264Dic6—181no—
265Dic6—364no—
266C3×C8—28no—
267C3×C8—24no—
268C3×C8—21no—
269C3×C8—64no—
270C3⋊C8—21no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32#88411yescharacteristic
275C2×C32#89514yes—
276C2×C32#90414yes—
277C2×C32#90814yes—
278C2×C32#91111yescharacteristic
279C2×C32#91611yescharacteristic
280C3⋊D3—81no—
281C3⋊D3—122no—
282C3⋊D3—128no—
283C3⋊D3—248no—
284C3×D3—244no—
285C2×Q8—271no—
286C2×D4—91no—
287C2×D4—271no—
288C22×C4—541no—
289SD16—542no—
290D8—542no—
291C2×C8—31no—
292C2×C8D3×C3⋊D311yescharacteristic
293C4⋊C4—91no—
294C22⋊C4—541no—
295C22⋊C4—541no—
296C22⋊C4—541no—
297C42—31no—
298C22×C3—24no—
299C22×C3—361no—
300C22×C3—361no—
301C22×C3—361no—
302C22×C3D3×D1214yes—
303C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
304C2×D3—124no—
305C2×D3—124no—
306C2×D3—124no—
307C2×D3—182no—
308C2×D3—188no—
309C2×D3—368no—
310C2×D3—364no—
311C2×D3—364no—
312C2×D3—361no—
313C2×D3—361no—
314C2×D3—724no—
315C2×D3—724no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—64no—
319C3×C4—64no—
320C3×C4—361no—
321C3×C4#379914yes—
322C3×C4#381114yes—
323C3×C4D4×C3⋊D311yescharacteristic
324C3×C4#383211yescharacteristic
325Dic3—21no—
326Dic3—21no—
327Dic3—124no—
328Dic3—368no—
329Dic3—361no—
330Dic3—364no—
331C32—24no—
332C32#478711yescharacteristic, complemented
333C32#484414yescomplemented
334C23—181no—
335C23—541no—
336Q8—541no—
337Q8—541no—
338D4—181no—
339D4—181no—
340D4—272no—
341D4—541no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—181no—
345C2×C4—541no—
346C2×C4—1082no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—21no—
349C8—61no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—721no—
354C2×C3#6379514yes—
355C2×C3#6380614yes—
356C2×C3#6391014yes—
357C2×C3#6410411yescharacteristic
358C2×C3#6411311yescharacteristic
359C2×C3#6411711yescharacteristic
360D3—244no—
361D3—362no—
362D3—368no—
363D3—728no—
364C22—361no—
365C22—361no—
366C22—361no—
367C22—542no—
368C22—1081no—
369C22—1081no—
370C22D3×C4⋊(C3⋊D3)11yescharacteristic
371C4—61no—
372C4—61no—
373C4—361no—
374C4—1081no—
375C4#8901811yescharacteristic
376C4#8903911yescharacteristic
377C3—24no—
378C3#1059086014yescomplemented
379C3#1059254311yescharacteristic, complemented
380C2—721no—
381C2—1082no—
382C2#1172331611yescharacteristic
383C2#1172333511yescharacteristic
384C2#1172348611yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#88791C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#463
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×#88851C2semidirect
2. #11723484C2semidirect
3. #11723439C2semidirect
5. C3×#63648C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. #11722042C2semidirect
33. #63648D3semidirect
36. C3×Dic3⋊C8D3semidirect
112. Dic3⋊C8C3⋊D3semidirect
230. C33#463semidirect
332. C32#4787semidirect
333. C32#4844semidirect
378. C3#10590860semidirect
379. C3#10592543semidirect

Maximal subgroups

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

subgroupindex
C2×#888512
#117234842
#117234392
C2×#887912
C3×#636482
C32×Dic3⋊C82
#117220422
#105925433
#105908603