Finite Group Databasebeta

← #13162031

Subgroups — #13162031

Order 1728.

1000 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 111 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×C4⋊#1113C211yesmaximal, characteristic
2C3×#64155C211yesmaximal, characteristic, complemented
3C2×C3×#3755C212yesmaximal, complemented
4#11722075C212yesmaximal, complemented
5#11722072C211yesmaximal, characteristic, complemented
6#10591652—31nomaximal
7#10591651—32nomaximal
8C22×#1113—21no—
9C2×C33×Q8C2211yescharacteristic, complemented
10C2×C3×C4⋊(C3⋊D3)C2212yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×#6295—21no—
13C4⋊#1113—24no—
14C3×#3755—24no—
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
17C3×#3750—21no—
18C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
19C2×#3802—31no—
20C2×D3×D12—32no—
21#64155D311yescharacteristic, complemented
22C2×#3755D312yescomplemented
23C3×#946—32no—
24C3×#946—61no—
25C2×C3×#215—32no—
26C2×C3×#215—32no—
27C2×C3×#215—62no—
28#63564—32no—
29#63563—32no—
30#63562—32no—
31#63557—31no—
32(C2×D3)⋊(C3⋊C8)—32no—
33C22×C3×C3⋊D3—22no—
34C22×C3×C3⋊D3D411yescharacteristic
35C2×#1113—24no—
36C2×#1113—42no—
37C2×#1113—42no—
38C33×Q8—22no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4⋊(C3⋊D3)—22no—
44C3×C4⋊(C3⋊D3)D414yes—
45C3×C4×C3⋊D3—22no—
46#6295—24no—
47C3×C3⋊(C3⋊C8)—22no—
48#5169—92no—
49#5046—91no—
50C22×D32—62no—
51C22×D32—61no—
52C2×C32×Q8—21no—
53C2×C32×Q8—22no—
54C2×C32×Q8—41no—
55C2×C32×Q8C2×D311yescharacteristic, complemented
56C2×C32×Q8C2×D312yescomplemented
57C2×C4⋊(C3⋊D3)C2×D312yes—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C3×D12—32no—
60C2×C3×D12—32no—
61C2×C3×D12—62no—
62C2×C3×C4×D3—32no—
63C2×C3×C4×D3—61no—
64C2×#626—62no—
65C2×#625—61no—
66#3802—64no—
67D3×D12—64no—
68D3×D12—64no—
69#3755—24no—
70#3752—21no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
72#3750—21no—
73C2×C3⋊(C3⋊C8)C2×D312yescomplemented
74C3×#215—64no—
75C3×#215—64no—
76C3×#215—68no—
77C3×(C2×D3)⋊C4—62no—
78C3×(C2×D3)⋊C4—62no—
79C3×C4⋊Dic3—32no—
80C3×C4⋊Dic3—61no—
81C3×Dic3⋊C4—62no—
82C3×Dic3⋊C4—62no—
83C2×C3×C3⋊C8—32no—
84C2×C3×C3⋊C8—32no—
85C2×C3×C3⋊C8—62no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—24no—
90C2×C3×C3⋊D3—42no—
91C2×C3×C3⋊D3—42no—
92C2×C3×C3⋊D3SD1612yescomplemented
93#1113—48no—
94C33×C4—41no—
95C33×C4C2×D412yes—
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—41no—
98C2×D3×D4—92no—
99C22×D12—91no—
100C2×C3×SD16—92no—
101C3×C22⋊Q8—91no—
102#946—32no—
103#946—61no—
104C2×#215—32no—
105C2×#215—32no—
106C2×#215—62no—
107#832—92no—
108#830—91no—
109#821—92no—
110(C2×D3)⋊C8—92no—
111#809—92no—
112C22×C3⋊D3—22no—
113C22×C3⋊D3#6711yescharacteristic
114C22×C3×D3—32no—
115C22×C3×D3—61no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—122no—
119C2×D32—68no—
120C2×D32—64no—
121C2×D32—124no—
122C2×D32—124no—
123C2×D32—122no—
124C2×D32—122no—
125C32×Q8—22no—
126C32×Q8—24no—
127C32×Q8—22no—
128C32×Q8—22no—
129C32×Q8—42no—
130C32×Q8—42no—
131C32×Q8—44no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—21no—
137C2×C32×C4—41no—
138C2×C32×C4—42no—
139C2×C32×C4—42no—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D312yes—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3#6711yescharacteristic
144C4⋊(C3⋊D3)—22no—
145C4⋊(C3⋊D3)#6714yes—
146C4×C3⋊D3—22no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C3×D12—34no—
152C3×D12—34no—
153C3×D12—64no—
154C3×D12—62no—
155C3×D12—62no—
156C3×D12—122no—
157C3×C4×D3—62no—
158C3×C4×D3—62no—
159C3×C4×D3—122no—
160#626—68no—
161#625—64no—
162C3⋊(C3⋊C8)—22no—
163C3×C3⋊C8—64no—
164C3×C3⋊C8—62no—
165C3×C3⋊C8—62no—
166#450—271noSylow
167C2×C33#12011yescharacteristic
168C2×C33C2×SD1611yescharacteristic
169C2×C33AGL(1,ℤ/8)11yescharacteristic
170C3×C3⋊D3—22no—
171C3×C3⋊D3—44no—
172C23×D3—182no—
173C23×D3—181no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—41no—
177C2×C3×Q8D3211yescharacteristic, complemented
178C2×C3×Q8D3212yescomplemented
179C2×C3×D4—92no—
180C22×C3×C4—91no—
181C2×#67—182no—
182D3×D4—184no—
183D3×D4—184no—
184C2×D12—32no—
185C2×D12—32no—
186C2×D12—62no—
187C2×D12—184no—
188C2×D12—181no—
189C2×C4×D3—32no—
190C2×C4×D3—61no—
191C3×SD16—184no—
192C2×C3×C8—92no—
193C3×C4⋊C4—91no—
194C3×C4⋊C4—181no—
195C3×C22⋊C4—181no—
196#215—64no—
197#215—64no—
198#215—68no—
199(C2×D3)⋊C4—62no—
200(C2×D3)⋊C4—62no—
201C4⋊Dic3—32no—
202C4⋊Dic3—61no—
203Dic3⋊C4—62no—
204Dic3⋊C4—62no—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—32no—
207C2×C3⋊C8—62no—
208C22×C32—22no—
209C22×C32—21no—
210C22×C32—41no—
211C22×C32D3×D412yes—
212C22×C32C2×#6711yescharacteristic
213C2×C3⋊D3—21no—
214C2×C3⋊D3—21no—
215C2×C3⋊D3—24no—
216C2×C3⋊D3—42no—
217C2×C3⋊D3—42no—
218C2×C3⋊D3#21512yescomplemented
219C2×C3×D3—34no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—62no—
223C2×C3×D3—64no—
224C2×C3×D3—64no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—124no—
228C2×C3×D3—122no—
229C2×C3×D3—122no—
230C2×C3×D3—122no—
231C2×C3×D3—122no—
232C2×C3×D3—242no—
233C2×C3×D3—242no—
234D32—1216no—
235D32—128no—
236C32×C4—22no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—42no—
240C32×C4—42no—
241C32×C4—44no—
242C32×C4—44no—
243C32×C4—41no—
244C32×C4—42no—
245C32×C4D3×D412yes—
246C32×C4D3×D412yes—
247C32×C4C2×#6712yes—
248C3⋊Dic3—21no—
249C3⋊Dic3—41no—
250C3×Dic3—62no—
251C3×Dic3—121no—
252C3×Dic3—122no—
253C3×Dic3—122no—
254C22×D4—271no—
255C2×SD16—92no—
256C22⋊Q8—91no—
257#102—272no—
258C22⋊C8—271no—
259C33#45011yescharacteristic, complemented, Sylow
260C23×C3—91no—
261C23×C3—182no—
262C22×D3—32no—
263C22×D3—61no—
264C22×D3—62no—
265C22×D3—62no—
266C22×D3—122no—
267C22×D3—184no—
268C22×D3—188no—
269C22×D3—364no—
270C22×D3—364no—
271C22×D3—362no—
272C22×D3—362no—
273C3×Q8—24no—
274C3×Q8—22no—
275C3×Q8—22no—
276C3×Q8—22no—
277C3×Q8—42no—
278C3×Q8—42no—
279C3×Q8—44no—
280C3×D4—94no—
281C3×D4—182no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—91no—
291C2×C3×C4—182no—
292C2×C3×C4—181no—
293C2×C3×C4C2×D3211yescharacteristic
294C2×C3×C4C2×D3212yes—
295#67—188no—
296C2×Dic3—32no—
297C2×Dic3—61no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300D12—34no—
301D12—34no—
302D12—64no—
303D12—62no—
304D12—62no—
305D12—122no—
306D12—184no—
307C4×D3—62no—
308C4×D3—62no—
309C4×D3—122no—
310C3×C8—182no—
311C3⋊C8—64no—
312C3⋊C8—62no—
313C3⋊C8—62no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—22no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32D3×SD1612yes—
324C2×C32#91412yes—
325C2×C32#93712yes—
326C2×C32#93711yescharacteristic
327C2×C32C2×#21511yescharacteristic
328C2×C32#94911yescharacteristic
329C3⋊D3—22no—
330C3⋊D3—44no—
331C3×D3—64no—
332C3×D3—122no—
333C3×D3—124no—
334C3×D3—124no—
335C3×D3—244no—
336C24—541no—
337C2×Q8#111311yescharacteristic, complemented
338C2×D4—92no—
339C2×D4—541no—
340C2×D4—544no—
341C22×C4—91no—
342SD16—184no—
343C2×C8—92no—
344C4⋊C4—91no—
345C4⋊C4—181no—
346C22⋊C4—181no—
347C22×C3—21no—
348C22×C3—22no—
349C22×C3—41no—
350C22×C3—92no—
351C22×C3—181no—
352C22×C3—181no—
353C22×C3—184no—
354C22×C3—362no—
355C22×C3—362no—
356C22×C3D3×#6712yes—
357C22×C3#381111yescharacteristic
358C2×D3—34no—
359C2×D3—62no—
360C2×D3—62no—
361C2×D3—62no—
362C2×D3—64no—
363C2×D3—64no—
364C2×D3—121no—
365C2×D3—121no—
366C2×D3—124no—
367C2×D3—122no—
368C2×D3—122no—
369C2×D3—122no—
370C2×D3—122no—
371C2×D3—242no—
372C2×D3—242no—
373C2×D3—3616no—
374C2×D3—368no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—42no—
379C3×C4—44no—
380C3×C4—44no—
381C3×C4—42no—
382C3×C4—41no—
383C3×C4—42no—
384C3×C4—181no—
385C3×C4—361no—
386C3×C4D3×#6712yes—
387C3×C4D3×#6712yes—
388C3×C4#381112yes—
389Dic3—62no—
390Dic3—121no—
391Dic3—122no—
392Dic3—122no—
393C32—22no—
394C32—21no—
395C32—41no—
396C32#504611yescharacteristic, complemented
397C32#516912yescomplemented
398C23—91no—
399C23—182no—
400C23—544no—
401C23—1082no—
402C23—1082no—
403Q8—22no—
404D4—94no—
405D4—182no—
406D4—544no—
407C2×C4—21no—
408C2×C4—91no—
409C2×C4—182no—
410C2×C4—181no—
411C2×C4C2×#111311yescharacteristic
412C8—182no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—21no—
416C2×C3—22no—
417C2×C3—21no—
418C2×C3—22no—
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—182no—
423C2×C3—364no—
424C2×C3D3×#21512yes—
425C2×C3#6393912yes—
426C2×C3#6394011yescharacteristic
427C2×C3#6394111yescharacteristic
428C2×C3#6398312yes—
429C2×C3#6398311yescharacteristic
430D3—64no—
431D3—122no—
432D3—124no—
433D3—124no—
434D3—244no—
435C22—92no—
436C22—181no—
437C22—181no—
438C22—184no—
439C22—362no—
440C22—362no—
441C22—1088no—
442C22#8904911yescharacteristic
443C4—41no—
444C4—181no—
445C4—361no—
446C4#8904912yes—
447C3—21no—
448C3—22no—
449C3—41no—
450C3#1059165112yescomplemented
451C3#1059165211yescharacteristic, complemented
452C2—182no—
453C2—364no—
454C2#1172370811yescharacteristic
455C2#1172370911yescharacteristic
456C2#1172372911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊#1113C2
hypercenterC2×Q8#1113
hypocenterC33#450
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#64155C2semidirect
3. C2×C3×#3755C2semidirect
4. #11722075C2semidirect
5. #11722072C2semidirect
9. C2×C33×Q8C22semidirect
16. C3×C4⋊(C3⋊Dic3)C22semidirect
18. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. #64155D3semidirect
22. C2×#3755D3semidirect
55. C2×C32×Q8C2×D3semidirect
56. C2×C32×Q8C2×D3semidirect
71. C4⋊(C3⋊Dic3)C2×D3semidirect
73. C2×C3⋊(C3⋊C8)C2×D3semidirect
92. C2×C3×C3⋊D3SD16semidirect
177. C2×C3×Q8D32semidirect
178. C2×C3×Q8D32semidirect
218. C2×C3⋊D3#215semidirect
259. C33#450semidirect
337. C2×Q8#1113semidirect
396. C32#5046semidirect
397. C32#5169semidirect
450. C3#10591651semidirect
451. C3#10591652semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C3×#641552
C2×C3×#37552
#117220752
#117220722
#105916523
#105916513