Finite Group Databasebeta

← #13160041

Subgroups — #13160041

Order 1728.

1400 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1400 conjugacy classes of subgroups — too many to draw. Only its 162 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×C3⋊D3×D12C211yesmaximal, characteristic
2C2×#88851C211yesmaximal, characteristic, complemented
3#11723431C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592475—31nomaximal
9#10590946—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×#6308C2211yescharacteristic, complemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6291—21no—
16C3⋊D3×D12—22no—
17C3⋊D3×D12—22no—
18C2×C33×C8C2211yescharacteristic, complemented
19#88851—22no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
21C32×C4⋊Dic3C2211yescharacteristic, complemented
22Dic3⋊(C3⋊Dic3)—21no—
23#88803—21no—
24#88794—22no—
25C2×D4×C3⋊D3—31no—
26C2×D3×D12—34no—
27C2×#3754—31no—
28C2×#3744—31no—
29C2×#3744—34no—
30C2×#3744—64no—
31#64106—31no—
32#63857—34no—
33C2×#3716—34no—
34C32×#102—31no—
35#63647D311yescharacteristic, complemented
36#63636—31no—
37C3×#821D314yescomplemented
38C3×(C2×D3)⋊C8—34no—
39#63564—34no—
40C22×C3⋊(C3⋊D3)—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C22×C32×D3—21no—
43C2×D3×C3⋊D3—24no—
44C2×D3×C3⋊D3—42no—
45C2×D3×C3⋊D3—42no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C4⋊(C3⋊(C3⋊D3))D412yes—
50#6308—21no—
51#6308—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12D412yes—
58#6291—24no—
59C33×C8—21no—
60C3×C3⋊(C3⋊C8)—21no—
61#4776—94no—
62#4731—91no—
63C23×C3⋊D3—61no—
64C22×D32—64no—
65C2×C32×D4—31no—
66C2×#638—61no—
67D4×C3⋊D3—62no—
68D4×C3⋊D3—62no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—34no—
71C2×C4⋊(C3⋊D3)—64no—
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×#634—31no—
74C2×#634—34no—
75C2×#634—64no—
76C2×C3×D12C2×D314yes—
77C2×C3×C4×D3—34no—
78C2×#626—64no—
79D3×D12—68no—
80D3×D12—68no—
81C2×C32×C8—24no—
82C2×C32×C8C2×D314yescomplemented
83C2×C32×C8C2×D311yescharacteristic, complemented
84C32×C4⋊C4—31no—
85#3754—62no—
86#3752—61no—
87C4⋊(C3⋊Dic3)—61no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
89#3744—62no—
90#3744—68no—
91#3744—616no—
92C3×C4⋊Dic3C2×D314yescomplemented
93C2×C3×C3⋊C8—34no—
94Dic3⋊Dic3—64no—
95(C2×D3)⋊Dic3—64no—
96#3716—68no—
97C22×C33C2×D411yescharacteristic
98C2×C3⋊(C3⋊D3)—22no—
99C2×C3⋊(C3⋊D3)—41no—
100C2×C3⋊(C3⋊D3)—41no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3SD1612yescomplemented
104C2×C32×D3—22no—
105C2×C32×D3—41no—
106C2×C32×D3—41no—
107D3×C3⋊D3—48no—
108C33×C4C2×D411yescharacteristic
109C33×C4C2×D411yescharacteristic
110C3⋊(C3⋊Dic3)—41no—
111C3×C3⋊Dic3—21no—
112C32×Dic3—41no—
113C2×D3×D4—94no—
114C22×D12—91no—
115C2×#214—94no—
116#924—91no—
117C2×#204—91no—
118C2×#204—94no—
119C2×#204—91no—
120C2×#204—184no—
121#897—94no—
122C3×#102—34no—
123C3×C22⋊C8—91no—
124#821—91no—
125#821C3⋊D311yescharacteristic, complemented
126(C2×D3)⋊C8—34no—
127#809—94no—
128C23×C32—61no—
129C22×C3⋊D3—64no—
130C22×C3⋊D3—61no—
131C22×C3⋊D3—64no—
132C22×C3⋊D3—124no—
133C22×C3⋊D3—122no—
134C22×C3⋊D3—122no—
135C22×C3⋊D3D1211yescharacteristic
136C22×C3×D3—24no—
137C22×C3×D3—34no—
138C2×D32—616no—
139C2×D32—128no—
140C2×D32—128no—
141C32×D4—32no—
142C32×D4—61no—
143C2×C32×C4—24no—
144C2×C32×C4—61no—
145C2×C32×C4C22×D314yes—
146C2×C32×C4C22×D311yescharacteristic
147#638—64no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—64no—
150C2×C3⋊Dic3—124no—
151C2×C3⋊Dic3D1211yescharacteristic
152C4⋊(C3⋊D3)—32no—
153C4⋊(C3⋊D3)—38no—
154C4⋊(C3⋊D3)—68no—
155C4⋊(C3⋊D3)—61no—
156C4⋊(C3⋊D3)—64no—
157C4⋊(C3⋊D3)—124no—
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160#634—68no—
161#634—64no—
162#634—61no—
163#634—61no—
164#634—64no—
165#634—124no—
166C2×C3×Dic3—24no—
167C2×C3×Dic3—34no—
168C3×D12—24no—
169C3×D12D1218yes—
170C3×C4×D3—64no—
171C3×C4×D3—64no—
172#626—616no—
173C32×C8—28no—
174C32×C8—24no—
175C32×C8—21no—
176C3⋊(C3⋊C8)—21no—
177C3×C3⋊C8—64no—
178#450—271noSylow
179C2×C33#12011yescharacteristic
180C2×C33C2×SD1611yescharacteristic
181C2×C33AGL(1,ℤ/8)11yescharacteristic
182C3⋊(C3⋊D3)—42no—
183C3×C3⋊D3—22no—
184C32×D3—42no—
185C23×D3—181no—
186C23×D3—184no—
187C2×C3×D4—34no—
188C22×C3×C4—91no—
189C2×#67—184no—
190D3×D4—188no—
191D3×D4—188no—
192C2×D12—94no—
193C2×D12—91no—
194C2×D12—184no—
195C2×D12—182no—
196C2×D12—181no—
197C2×D12—182no—
198C2×D12C2×C3⋊D311yescharacteristic
199C2×C4×D3—34no—
200C2×Dic6—94no—
201C2×Dic6—91no—
202C2×Dic6—184no—
203C2×C3×C8—24no—
204C2×C3×C8—91no—
205C2×C3×C8D3214yescomplemented
206C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
207C3×C4⋊C4—34no—
208#217—181no—
209#214—188no—
210(C2×D3)⋊C4—184no—
211C4⋊Dic3—184no—
212C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
213Dic3⋊C4—181no—
214C2×C3⋊C8—34no—
215#204—188no—
216#204—182no—
217#204—1816no—
218#204—182no—
219C22×C32—24no—
220C22×C32—62no—
221C22×C32—121no—
222C22×C32—121no—
223C22×C32C2×D1211yescharacteristic
224C22×C32D3×D414yes—
225C2×C3⋊D3—21no—
226C2×C3⋊D3—21no—
227C2×C3⋊D3—62no—
228C2×C3⋊D3—68no—
229C2×C3⋊D3—128no—
230C2×C3⋊D3—128no—
231C2×C3⋊D3—121no—
232C2×C3⋊D3—121no—
233C2×C3⋊D3—124no—
234C2×C3⋊D3—124no—
235C2×C3⋊D3—244no—
236C2×C3⋊D3—244no—
237C2×C3⋊D3#20412yescomplemented
238C2×C3×D3—28no—
239C2×C3×D3—38no—
240C2×C3×D3—44no—
241C2×C3×D3—44no—
242C2×C3×D3—64no—
243C2×C3×D3—64no—
244D32—1232no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—121no—
248C32×C4C2×D1214yes—
249C32×C4D3×D414yes—
250C32×C4D3×D411yescharacteristic
251C32×C4D3×D411yescharacteristic
252C3⋊Dic3—21no—
253C3⋊Dic3—128no—
254C3⋊Dic3—124no—
255C3⋊Dic3—121no—
256C3×Dic3—44no—
257C3×Dic3—64no—
258C22×D4—271no—
259C2×SD16—271no—
260C2×SD16—271no—
261C22⋊Q8—271no—
262#102—31no—
263#102—271no—
264C22⋊C8—91no—
265C33#45011yescharacteristic, complemented, Sylow
266C23×C3—64no—
267C23×C3—91no—
268C22×D3—21no—
269C22×D3—34no—
270C22×D3—184no—
271C22×D3—1816no—
272C22×D3—184no—
273C22×D3—181no—
274C22×D3—364no—
275C22×D3—368no—
276C22×D3—362no—
277C22×D3—362no—
278C22×D3—368no—
279C3×D4—38no—
280C3×D4—64no—
281C2×C3×C4—24no—
282C2×C3×C4—64no—
283C2×C3×C4—91no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4C2×D3214yes—
287C2×C3×C4C22×C3⋊D311yescharacteristic
288#67—1816no—
289C2×Dic3—21no—
290C2×Dic3—34no—
291C2×Dic3—184no—
292C2×Dic3—181no—
293C2×Dic3—364no—
294D12—21no—
295D12—98no—
296D12—92no—
297D12—188no—
298D12—184no—
299D12—184no—
300D12—181no—
301D12—364no—
302D12C4⋊(C3⋊D3)12yes—
303C4×D3—64no—
304C4×D3—64no—
305Dic6—188no—
306Dic6—184no—
307Dic6—184no—
308Dic6—181no—
309Dic6—181no—
310Dic6—364no—
311C3×C8—28no—
312C3×C8—24no—
313C3×C8—21no—
314C3×C8—181no—
315C3⋊C8—64no—
316C2×C32—24no—
317C2×C32—24no—
318C2×C32—24no—
319C2×C32—122no—
320C2×C32#88214yes—
321C2×C32#88211yescharacteristic
322C2×C32C2×#20411yescharacteristic
323C2×C32#90811yescharacteristic
324C2×C32#91114yes—
325C2×C32D3×SD1614yes—
326C3⋊D3—22no—
327C3⋊D3—122no—
328C3⋊D3—128no—
329C3⋊D3—248no—
330C3×D3—48no—
331C3×D3—68no—
332C24—541no—
333C2×Q8—271no—
334C2×D4—31no—
335C2×D4—271no—
336C2×D4—541no—
337C2×D4—542no—
338C2×D4—542no—
339C22×C4—91no—
340SD16—542no—
341SD16—542no—
342C2×C8—91no—
343C2×C8D3×C3⋊D311yescharacteristic, complemented
344C4⋊C4—31no—
345C4⋊C4—541no—
346C22⋊C4—541no—
347C22×C3—24no—
348C22×C3—68no—
349C22×C3—92no—
350C22×C3—124no—
351C22×C3—124no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3D3×D1214yes—
355C22×C3D4×C3⋊D311yescharacteristic
356C2×D3—22no—
357C2×D3—38no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—64no—
361C2×D3—64no—
362C2×D3—182no—
363C2×D3—188no—
364C2×D3—368no—
365C2×D3—368no—
366C2×D3—3632no—
367C2×D3—364no—
368C2×D3—364no—
369C2×D3—361no—
370C2×D3—361no—
371C2×D3—724no—
372C2×D3—724no—
373C3×C4—24no—
374C3×C4—24no—
375C3×C4—124no—
376C3×C4—181no—
377C3×C4D3×D1214yes—
378C3×C4#381114yes—
379C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
380C3×C4D4×C3⋊D311yescharacteristic
381Dic3—41no—
382Dic3—64no—
383Dic3—368no—
384Dic3—361no—
385Dic3—364no—
386C32—24no—
387C32#473111yescharacteristic, complemented
388C32#477614yescomplemented
389C23—61no—
390C23—91no—
391C23—544no—
392C23—541no—
393C23—1082no—
394C23—1082no—
395Q8—541no—
396Q8—541no—
397D4—32no—
398D4—61no—
399D4—272no—
400D4—544no—
401D4—541no—
402C2×C4—61no—
403C2×C4—91no—
404C2×C4—181no—
405C2×C4—181no—
406C2×C4—541no—
407C2×C4C2×D3×C3⋊D311yescharacteristic
408C8—21no—
409C8—181no—
410C2×C3—24no—
411C2×C3—24no—
412C2×C3—24no—
413C2×C3—128no—
414C2×C3—182no—
415C2×C3D3×#20414yes—
416C2×C3#6379514yes—
417C2×C3#6392314yes—
418C2×C3#6409111yescharacteristic
419C2×C3#6412011yescharacteristic
420C2×C3SD16×C3⋊D311yescharacteristic
421D3—42no—
422D3—68no—
423D3—362no—
424D3—368no—
425D3—728no—
426C22—62no—
427C22—92no—
428C22—121no—
429C22—121no—
430C22—181no—
431C22—181no—
432C22—542no—
433C22—1088no—
434C22—1081no—
435C22—1081no—
436C22C3⋊D3×D1211yescharacteristic
437C4—121no—
438C4—181no—
439C4—1081no—
440C4D3×C4⋊(C3⋊D3)11yescharacteristic
441C4#8903911yescharacteristic
442C3—24no—
443C3#1059094614yescomplemented
444C3#1059247511yescharacteristic, complemented
445C2—122no—
446C2—182no—
447C2—1082no—
448C2C3⋊D3×#20411yescharacteristic
449C2#1172332011yescharacteristic
450C2#1172354211yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3⋊D3×D12C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#450
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
2. C2×#88851C2semidirect
3. #11723431C2semidirect
4. C2×#88794C2semidirect
5. C3×#63647C2semidirect
6. C32×#821C2semidirect
7. #11722035C2semidirect
12. C2×#6308C22semidirect
18. C2×C33×C8C22semidirect
20. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. C32×C4⋊Dic3C22semidirect
35. #63647D3semidirect
37. C3×#821D3semidirect
82. C2×C32×C8C2×D3semidirect
83. C2×C32×C8C2×D3semidirect
88. C2×C3⋊(C3⋊C8)C2×D3semidirect
92. C3×C4⋊Dic3C2×D3semidirect
103. C2×C3×C3⋊D3SD16semidirect
125. #821C3⋊D3semidirect
205. C2×C3×C8D32semidirect
206. C2×C3×C8C2×C3⋊D3semidirect
212. C4⋊Dic3C2×C3⋊D3semidirect
237. C2×C3⋊D3#204semidirect
265. C33#450semidirect
343. C2×C8D3×C3⋊D3semidirect
387. C32#4731semidirect
388. C32#4776semidirect
443. C3#10590946semidirect
444. C3#10592475semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C2×#888512
#117234312
C2×#887942
C3×#636472
C32×#8212
#117220352
#105924753
#105909463