Finite Group Databasebeta

← #13159941

Subgroups — #13159941

Order 1728.

1004 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 176 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×C32×D3×C8C211yesmaximal, characteristic, complemented
2#11723516C211yesmaximal, characteristic, complemented
3#11723515C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7#11722055C211yesmaximal, characteristic, complemented
8#10592814—31nomaximal
9#10590826—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6291—21no—
14C2×#6290—21no—
15C2×C33×C8C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C32×D3×C8—22no—
20#88804—21no—
21#88803—21no—
22#88791—22no—
23C22×C32×C8—31no—
24#64139—31no—
25#64139—31no—
26C2×#3745—31no—
27C2×C3×D3×C8D314yescomplemented
28#63909—34no—
29#63908—34no—
30C2×#3714—34no—
31#63648—31no—
32#63648—34no—
33#63648—64no—
34#63648D311yescharacteristic, complemented
35#63645—31no—
36C3×#821—34no—
37#63575—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C22×C32×D3D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C2×C3×C3⋊Dic3—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C2×C32×Dic3D411yescharacteristic
49C32×C4×D3—21no—
50C32×C4×D3—21no—
51#6291—42no—
52#6290—42no—
53C33×C8—21no—
54C32×C3⋊C8D412yescomplemented
55#5115—94no—
56#4775—91no—
57C22×C32×C4—31no—
58C2×#638—61no—
59C2×#638—61no—
60C2×C4⋊(C3⋊D3)—31no—
61C2×C4⋊(C3⋊D3)—34no—
62C2×C4⋊(C3⋊D3)—64no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×C4×D3C2×D314yes—
66C2×#626—64no—
67C2×#625—64no—
68C2×C32×C8—24no—
69C2×C32×C8—31no—
70C2×C32×C8—62no—
71C2×C32×C8C2×D314yes—
72C2×C32×C8C2×D311yescharacteristic
73#3752—61no—
74#3752—61no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—34no—
77C4⋊(C3⋊Dic3)—64no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79#3745—62no—
80C3×C4⋊Dic3—34no—
81C2×C3×C3⋊C8C2×D314yes—
82C3×D3×C8—28no—
83#3722—64no—
84(C2×D3)⋊Dic3—64no—
85#3714—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
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—
93C2×C32×D3—21no—
94C2×C32×D3—21no—
95C2×C32×D3D812yescomplemented
96C33×C4C2×D411yescharacteristic
97C33×C4#4111yescharacteristic
98C3⋊(C3⋊Dic3)—41no—
99C3×C3⋊Dic3—41no—
100C32×Dic3—21no—
101C22×C3×C8—34no—
102#938—91no—
103C2×#213—91no—
104#930—94no—
105#930—94no—
106C2×D24—94no—
107C2×D3×C8C3⋊D311yescharacteristic, complemented
108#895—91no—
109C3×#102—91no—
110#821—34no—
111#821—94no—
112#821—91no—
113#821—184no—
114#818—94no—
115#807—91no—
116C23×C32—31no—
117C22×C3⋊D3—21no—
118C22×C3⋊D3—61no—
119C22×C3⋊D3—64no—
120C22×C3⋊D3—124no—
121C22×C3×D3—64no—
122C22×C3×D3D1214yes—
123C2×C32×C4—24no—
124C2×C32×C4—31no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129#638—122no—
130#638—122no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—61no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—124no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—68no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—124no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3D1214yes—
145C3×D12—64no—
146C3×D12—64no—
147C3×C4×D3—24no—
148C3×C4×D3—24no—
149#626—128no—
150#625—128no—
151C32×C8—21no—
152C32×C8—24no—
153C32×C8—32no—
154C32×C8—44no—
155C3×C3⋊C8#6718yescomplemented
156#466—271noSylow
157C2×C33#12111yescharacteristic
158C2×C33C2×D811yescharacteristic
159C2×C33#13511yescharacteristic
160C3⋊(C3⋊D3)—81no—
161C3×C3⋊D3—81no—
162C32×D3—22no—
163C2×C3×D4—91no—
164C22×C3×C4—34no—
165C2×#67—184no—
166C2×#67—184no—
167C2×#67—181no—
168C2×D12—34no—
169C2×D12—94no—
170C2×D12—91no—
171C2×D12—184no—
172C2×D12—181no—
173C2×C4×D3C2×C3⋊D311yescharacteristic
174C2×C3×C8—24no—
175C2×C3×C8—34no—
176C2×C3×C8—68no—
177C2×C3×C8D3214yes—
178C2×C3×C8C2×C3⋊D311yescharacteristic
179C3×C4⋊C4—91no—
180#217—181no—
181#213—182no—
182(C2×D3)⋊C4—184no—
183(C2×D3)⋊C4—184no—
184(C2×D3)⋊C4—181no—
185C4⋊Dic3—34no—
186C4⋊Dic3—94no—
187C4⋊Dic3—91no—
188C4⋊Dic3—184no—
189C2×C3⋊C8C2×C3⋊D311yescharacteristic
190D24—188no—
191D3×C8—22no—
192C22×C32—24no—
193C22×C32—32no—
194C22×C32—61no—
195C22×C32—61no—
196C22×C32C2×D1214yes—
197C22×C32D3×D411yescharacteristic
198C2×C3⋊D3—41no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—128no—
202C2×C3⋊D3—121no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—124no—
207C2×C3⋊D3—124no—
208C2×C3⋊D3—244no—
209C2×C3⋊D3—244no—
210C2×C3×D3—24no—
211C2×C3×D3—24no—
212C2×C3×D3—124no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215C2×C3×D3D2418yescomplemented
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—61no—
219C32×C4#23514yes—
220C32×C4#23511yescharacteristic
221C32×C4D3×D411yescharacteristic
222C32×C4C2×#6714yes—
223C3⋊Dic3—41no—
224C3⋊Dic3—121no—
225C3⋊Dic3—124no—
226C3⋊Dic3—244no—
227C3×Dic3—24no—
228C3×Dic3—124no—
229C2×D8—271no—
230C22×C8—31no—
231#121—271no—
232#121—271no—
233#107—271no—
234#102—91no—
235#102—271no—
236C33#46611yescharacteristic, complemented, Sylow
237C23×C3—34no—
238C23×C3—181no—
239C22×D3—64no—
240C22×D3—184no—
241C22×D3—181no—
242C22×D3—364no—
243C22×D3C4⋊(C3⋊D3)11yescharacteristic
244C3×D4—181no—
245C3×D4—181no—
246C2×C3×C4—24no—
247C2×C3×C4—34no—
248C2×C3×C4—64no—
249C2×C3×C4—64no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—368no—
254#67—368no—
255#67—362no—
256C2×Dic3—64no—
257C2×Dic3—184no—
258C2×Dic3—181no—
259C2×Dic3—364no—
260C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
261D12—64no—
262D12—64no—
263D12—188no—
264D12—184no—
265D12—184no—
266D12—181no—
267D12—181no—
268D12—364no—
269D12—362no—
270C4×D3—21no—
271C4×D3—21no—
272C3×C8—24no—
273C3×C8—21no—
274C3×C8—38no—
275C3×C8—44no—
276C3⋊C8#63812yescomplemented
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—62no—
281C2×C32#88411yescharacteristic
282C2×C32C2×D2414yes—
283C2×C32#90414yes—
284C2×C32D3×D811yescharacteristic
285C2×C32#91611yescharacteristic
286C2×C32#93014yes—
287C3⋊D3—81no—
288C3⋊D3—248no—
289C3⋊D3—241no—
290C3⋊D3—244no—
291C3×D3—28no—
292C3×D3—244no—
293C2×D4—91no—
294C2×D4—271no—
295C2×D4—541no—
296C2×D4—541no—
297C22×C4—31no—
298D8—542no—
299C2×C8—31no—
300C2×C8—62no—
301C2×C8D3×C3⋊D311yescharacteristic
302C4⋊C4—91no—
303C4⋊C4—271no—
304C22⋊C4—541no—
305C22⋊C4—541no—
306C22×C3—24no—
307C22×C3—38no—
308C22×C3—64no—
309C22×C3—64no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3—361no—
313C22×C3D3×D1214yes—
314C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—124no—
318C2×D3—124no—
319C2×D3—124no—
320C2×D3—368no—
321C2×D3—364no—
322C2×D3—364no—
323C2×D3—361no—
324C2×D3—361no—
325C2×D3—361no—
326C2×D3—364no—
327C2×D3—724no—
328C2×D3—724no—
329C2×D3#374512yescomplemented
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—64no—
333C3×C4—361no—
334C3×C4#379814yes—
335C3×C4D3×#6714yes—
336C3×C4#382811yescharacteristic
337C3×C4C2×#63811yescharacteristic
338Dic3—21no—
339Dic3—124no—
340Dic3—364no—
341Dic3—361no—
342Dic3—724no—
343C32—24no—
344C32#477511yescharacteristic, complemented
345C32#511514yescomplemented
346C23—31no—
347C23—181no—
348C23—541no—
349D4—181no—
350D4—181no—
351D4—541no—
352D4—541no—
353D4—1082no—
354D4—1082no—
355C2×C4—31no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—181no—
359C2×C4—541no—
360C2×C4C2×D3×C3⋊D311yescharacteristic
361C8—21no—
362C8—32no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—68no—
367C2×C3—721no—
368C2×C3D3×D2414yes—
369C2×C3#6380614yes—
370C2×C3#6390614yes—
371C2×C3C2×#374511yescharacteristic
372C2×C3#6411311yescharacteristic
373C2×C3#6413911yescharacteristic
374D3—22no—
375D3—244no—
376D3—728no—
377D3—721no—
378D3—724no—
379C22—32no—
380C22—61no—
381C22—61no—
382C22—361no—
383C22—361no—
384C22—361no—
385C22—1081no—
386C22—1081no—
387C22—1081no—
388C22D3×C4⋊(C3⋊D3)11yescharacteristic
389C4—61no—
390C4—361no—
391C4—1081no—
392C4#8902011yescharacteristic
393C4D3×#63811yescharacteristic
394C3—24no—
395C3#1059082614yescomplemented
396C3#1059281411yescharacteristic, complemented
397C2—62no—
398C2—721no—
399C2—2161no—
400C2D3×#374511yescharacteristic
401C2#1172333511yescharacteristic
402C2#1172351411yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#466
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×C32×D3×C8C2semidirect
2. #11723516C2semidirect
3. #11723515C2semidirect
4. C2×#88791C2semidirect
5. #11722191C2semidirect
6. C3×#63648C2semidirect
7. #11722055C2semidirect
27. C2×C3×D3×C8D3semidirect
34. #63648D3semidirect
54. C32×C3⋊C8D4semidirect
95. C2×C32×D3D8semidirect
107. C2×D3×C8C3⋊D3semidirect
155. C3×C3⋊C8#67semidirect
215. C2×C3×D3D24semidirect
236. C33#466semidirect
276. C3⋊C8#638semidirect
329. C2×D3#3745semidirect
344. C32#4775semidirect
345. C32#5115semidirect
395. C3#10590826semidirect
396. C3#10592814semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×C82
#117235162
#117235152
C2×#887912
#117221912
C3×#636482
#117220552
#105928143
#105908263