Finite Group Databasebeta

← #13160044

Subgroups — #13160044

Order 1728.

1328 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 160 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×#89026C211yesmaximal, characteristic
2C2×C32×#204C211yesmaximal, characteristic, complemented
3#11723456C211yesmaximal, characteristic, complemented
4C2×#88793C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592868—31nomaximal
9#10590943—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic, complemented
15C2×#6290—21no—
16#89026—22no—
17#89026—22no—
18C2×C33×C8C2211yescharacteristic, complemented
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
21C32×#204—22no—
22#88807—21no—
23#88804—21no—
24#88793—22no—
25C2×D4×C3⋊D3—31no—
26C2×#3802—34no—
27C2×C32×SD16—31no—
28#64155—31no—
29C2×#3755—31no—
30C2×C3×#204D314yescomplemented
31#63871—34no—
32C2×#3715—34no—
33#63659—31no—
34#63648D311yescharacteristic, complemented
35#63647—31no—
36#63647—34no—
37#63647—64no—
38C3×#821—34no—
39#63562—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C22×C3×C3⋊D3—21no—
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)D411yescharacteristic
48C4×C3⋊(C3⋊D3)—21no—
49C4×C3⋊(C3⋊D3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)D412yes—
53C2×C32×Dic3—21no—
54C32×D12—21no—
55C32×D12D412yes—
56C32×Dic6—21no—
57C32×Dic6—21no—
58#6290—24no—
59C33×C8—21no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61#5169—94no—
62#4776—91no—
63C23×C3⋊D3—61no—
64C22×D32—64no—
65C2×C32×Q8—31no—
66C2×C32×D4—31no—
67C2×#638—61no—
68D4×C3⋊D3—62no—
69D4×C3⋊D3—62no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4×C3⋊D3—31no—
72C2×C4×C3⋊D3—34no—
73C2×C4×C3⋊D3—64no—
74C2×C3×D12—34no—
75C2×C3×D12C2×D314yes—
76C2×C3×Dic6C2×D314yescomplemented
77C2×#625—64no—
78#3802—68no—
79#3802—68no—
80C32×SD16—62no—
81C2×C32×C8—24no—
82C2×C32×C8C2×D314yescomplemented
83C2×C32×C8C2×D311yescharacteristic, complemented
84#3755—62no—
85#3752—61no—
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
87#3750—61no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—34no—
90C2×C3⋊(C3⋊C8)—64no—
91C3×C4⋊Dic3—34no—
92C3×#204—28no—
93Dic3⋊Dic3—64no—
94#3722—64no—
95#3715—68no—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—21no—
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)SD1612yescomplemented
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3—41no—
103C2×C32×D3—22no—
104C2×C32×D3—41no—
105C2×C32×D3—41no—
106D3×C3⋊D3—48no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3⋊(C3⋊Dic3)—21no—
110C3×C3⋊Dic3—41no—
111C32×Dic3—41no—
112C2×D3×D4—94no—
113C2×D3×D4—91no—
114C2×C3×SD16—34no—
115#946—94no—
116C2×#215—94no—
117C2×#214—91no—
118C2×#204C3⋊D311yescharacteristic, complemented
119#897—91no—
120C3×#102—91no—
121#832—94no—
122#821—34no—
123(C2×D3)⋊C8—94no—
124(C2×D3)⋊C8—91no—
125(C2×D3)⋊C8—184no—
126#809—91no—
127C23×C32—61no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—31no—
130C22×C3⋊D3—34no—
131C22×C3⋊D3—64no—
132C22×C3⋊D3—64no—
133C22×C3⋊D3—122no—
134C22×C3⋊D3—122no—
135C22×C3×D3—24no—
136C22×C3×D3—64no—
137C2×D32—616no—
138C2×D32—128no—
139C2×D32—128no—
140C32×Q8—61no—
141C32×Q8—61no—
142C32×D4—32no—
143C32×D4—61no—
144C2×C32×C4—24no—
145C2×C32×C4—61no—
146C2×C32×C4C22×D314yes—
147C2×C32×C4C22×D311yescharacteristic
148#638—64no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—31no—
151C2×C3⋊Dic3—34no—
152C2×C3⋊Dic3—64no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)D1212yes—
155C4×C3⋊D3—61no—
156C4×C3⋊D3—61no—
157C4×C3⋊D3—64no—
158C4×C3⋊D3—64no—
159C4×C3⋊D3—124no—
160C4×C3⋊D3—124no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—64no—
163C3×D12—24no—
164C3×D12—38no—
165C3×D12—64no—
166C3×D12#6718yes—
167C3×Dic6—24no—
168C3×Dic6—24no—
169#625—616no—
170C32×C8—21no—
171C32×C8—24no—
172C32×C8—44no—
173C3⋊(C3⋊C8)—61no—
174C3⋊(C3⋊C8)—64no—
175C3⋊(C3⋊C8)—124no—
176#450—271noSylow
177C2×C33#12011yescharacteristic
178C2×C33C2×SD1611yescharacteristic
179C2×C33AGL(1,ℤ/8)11yescharacteristic
180C3⋊(C3⋊D3)—22no—
181C3×C3⋊D3—42no—
182C32×D3—42no—
183C23×D3—181no—
184C23×D3—184no—
185C2×C3×Q8—34no—
186C2×C3×D4—34no—
187C2×C3×D4—91no—
188C2×#67—181no—
189C2×#67—184no—
190D3×D4—182no—
191D3×D4—188no—
192D3×D4—188no—
193D3×D4—182no—
194C2×D12—34no—
195C2×D12C2×C3⋊D311yescharacteristic
196C2×C4×D3—94no—
197C2×C4×D3—91no—
198C2×C4×D3—184no—
199C2×Dic6C2×C3⋊D311yescharacteristic, complemented
200C3×SD16—68no—
201C2×C3×C8—24no—
202C2×C3×C8D3214yescomplemented
203C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
204C3×C4⋊C4—91no—
205#215—188no—
206#214—182no—
207(C2×D3)⋊C4—181no—
208(C2×D3)⋊C4—184no—
209C4⋊Dic3—34no—
210C4⋊Dic3—181no—
211Dic3⋊C4—184no—
212C2×C3⋊C8—94no—
213C2×C3⋊C8—91no—
214C2×C3⋊C8—184no—
215#204—22no—
216C22×C32—24no—
217C22×C32—62no—
218C22×C32—121no—
219C22×C32—121no—
220C22×C32D3×D414yes—
221C22×C32D3×D411yescharacteristic
222C2×C3⋊D3—22no—
223C2×C3⋊D3—32no—
224C2×C3⋊D3—38no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3—68no—
228C2×C3⋊D3—61no—
229C2×C3⋊D3—61no—
230C2×C3⋊D3—64no—
231C2×C3⋊D3—64no—
232C2×C3⋊D3—124no—
233C2×C3⋊D3—124no—
234C2×C3⋊D3—128no—
235C2×C3×D3—28no—
236C2×C3×D3—44no—
237C2×C3×D3—44no—
238C2×C3×D3—68no—
239C2×C3×D3—124no—
240C2×C3×D3—124no—
241D32—1232no—
242C32×C4—24no—
243C32×C4—24no—
244C32×C4—121no—
245C32×C4C2×D1211yescharacteristic
246C32×C4D3×D414yes—
247C32×C4D3×D411yescharacteristic
248C32×C4C2×#6714yes—
249C3⋊Dic3—41no—
250C3⋊Dic3—61no—
251C3⋊Dic3—64no—
252C3⋊Dic3—124no—
253C3×Dic3—44no—
254C3×Dic3—124no—
255C22×D4—271no—
256C2×SD16—31no—
257C2×SD16—271no—
258C22⋊Q8—271no—
259#102—91no—
260#102—271no—
261C22⋊C8—271no—
262C33#45011yescharacteristic, complemented, Sylow
263C23×C3—64no—
264C23×C3—181no—
265C22×D3—21no—
266C22×D3—64no—
267C22×D3—94no—
268C22×D3—91no—
269C22×D3—184no—
270C22×D3—184no—
271C22×D3—1816no—
272C22×D3—362no—
273C22×D3—362no—
274C22×D3—368no—
275C22×D3—368no—
276C3×Q8—64no—
277C3×Q8—64no—
278C3×D4—38no—
279C3×D4—64no—
280C3×D4—92no—
281C3×D4—181no—
282C2×C3×C4—24no—
283C2×C3×C4—64no—
284C2×C3×C4—181no—
285C2×C3×C4C2×D3214yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287#67—184no—
288#67—1816no—
289C2×Dic3—21no—
290C2×Dic3—64no—
291C2×Dic3—94no—
292C2×Dic3—91no—
293C2×Dic3—184no—
294D12—21no—
295D12—38no—
296D12—64no—
297D12#63812yes—
298C4×D3—184no—
299C4×D3—184no—
300C4×D3—181no—
301C4×D3—181no—
302C4×D3—364no—
303C4×D3—364no—
304Dic6—21no—
305Dic6—21no—
306C3×C8—24no—
307C3×C8—21no—
308C3×C8—44no—
309C3⋊C8—184no—
310C3⋊C8—181no—
311C3⋊C8—364no—
312C2×C32—24no—
313C2×C32—24no—
314C2×C32—24no—
315C2×C32—122no—
316C2×C32#88211yescharacteristic
317C2×C32#91111yescharacteristic
318C2×C32D3×SD1614yes—
319C2×C32D3×SD1611yescharacteristic
320C2×C32#91414yes—
321C2×C32#93714yes—
322C3⋊D3—42no—
323C3⋊D3—62no—
324C3⋊D3—68no—
325C3⋊D3—128no—
326C3×D3—48no—
327C3×D3—128no—
328C24—541no—
329C2×Q8—31no—
330C2×D4—31no—
331C2×D4—91no—
332C2×D4—541no—
333C2×D4—542no—
334C2×D4—542no—
335C22×C4—271no—
336SD16—62no—
337SD16—542no—
338C2×C8—271no—
339C2×C8D3×C3⋊D311yescharacteristic, complemented
340C4⋊C4—91no—
341C4⋊C4—541no—
342C22⋊C4—541no—
343C22×C3—24no—
344C22×C3—68no—
345C22×C3—124no—
346C22×C3—124no—
347C22×C3—182no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3#380214yes—
351C22×C3D4×C3⋊D311yescharacteristic
352C2×D3—22no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—68no—
356C2×D3—98no—
357C2×D3—92no—
358C2×D3—124no—
359C2×D3—124no—
360C2×D3—188no—
361C2×D3—184no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—184no—
365C2×D3—364no—
366C2×D3—364no—
367C2×D3—368no—
368C2×D3—3632no—
369C3×C4—24no—
370C3×C4—24no—
371C3×C4—124no—
372C3×C4—361no—
373C3×C4D3×D1214yes—
374C3×C4D3×#6714yes—
375C3×C4D4×C3⋊D311yescharacteristic
376C3×C4C2×#63811yescharacteristic
377Dic3—41no—
378Dic3—124no—
379Dic3—181no—
380Dic3—184no—
381Dic3—364no—
382C32—24no—
383C32#477611yescharacteristic, complemented
384C32#516914yescomplemented
385C23—61no—
386C23—181no—
387C23—271no—
388C23—544no—
389C23—1082no—
390C23—1082no—
391Q8—61no—
392Q8—61no—
393D4—32no—
394D4—61no—
395D4—92no—
396D4—181no—
397D4—544no—
398C2×C4—61no—
399C2×C4—181no—
400C2×C4—271no—
401C2×C4—541no—
402C2×C4—541no—
403C2×C4C2×D3×C3⋊D311yescharacteristic
404C8—21no—
405C8—541no—
406C2×C3—24no—
407C2×C3—24no—
408C2×C3—24no—
409C2×C3—128no—
410C2×C3—362no—
411C2×C3#6379614yes—
412C2×C3#6379814yes—
413C2×C3#6392214yes—
414C2×C3SD16×C3⋊D311yescharacteristic
415C2×C3#6412311yescharacteristic
416C2×C3#6414611yescharacteristic
417D3—42no—
418D3—128no—
419D3—182no—
420D3—188no—
421D3—368no—
422C22—62no—
423C22—121no—
424C22—121no—
425C22—182no—
426C22—272no—
427C22—361no—
428C22—361no—
429C22—541no—
430C22—541no—
431C22—1088no—
432C22#8902611yescharacteristic
433C4—121no—
434C4—361no—
435C4—541no—
436C4C3⋊D3×D1211yescharacteristic
437C4D3×#63811yescharacteristic
438C3—24no—
439C3#1059094314yescomplemented
440C3#1059286811yescharacteristic, complemented
441C2—122no—
442C2—362no—
443C2—542no—
444C2#1172332111yescharacteristic
445C2#1172332311yescharacteristic
446C2#1172354011yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89026C2
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×C32×#204C2semidirect
3. #11723456C2semidirect
4. C2×#88793C2semidirect
5. #11722190C2semidirect
6. C3×#63648C2semidirect
7. #11722031C2semidirect
14. C2×C32×Dic6C22semidirect
18. C2×C33×C8C22semidirect
19. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
20. C3×C4⋊(C3⋊Dic3)C22semidirect
30. C2×C3×#204D3semidirect
34. #63648D3semidirect
76. C2×C3×Dic6C2×D3semidirect
82. C2×C32×C8C2×D3semidirect
83. C2×C32×C8C2×D3semidirect
86. C4⋊(C3⋊Dic3)C2×D3semidirect
99. C2×C3⋊(C3⋊D3)SD16semidirect
118. C2×#204C3⋊D3semidirect
199. C2×Dic6C2×C3⋊D3semidirect
202. C2×C3×C8D32semidirect
203. C2×C3×C8C2×C3⋊D3semidirect
262. C33#450semidirect
339. C2×C8D3×C3⋊D3semidirect
383. C32#4776semidirect
384. C32#5169semidirect
439. C3#10590943semidirect
440. C3#10592868semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
C2×C32×#2042
#117234562
C2×#887932
#117221902
C3×#636482
#117220312
#105928683
#105909433