Finite Group Databasebeta

← #13161986

Subgroups — #13161986

Order 1728.

926 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 926 conjugacy classes of subgroups — too many to draw. Only its 159 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×Q8×C3⋊D3C211yesmaximal, characteristic
2#11723689C211yesmaximal, characteristic
3C2×#88808C211yesmaximal, characteristic
4C3×#63662C212yesmaximal, complemented
5#11722076C212yesmaximal, complemented
6#10591552—31nomaximal
7D3×#835—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×Q8×C3⋊D3—22no—
10C3×Q8×C3⋊D3C414yescomplemented
11C2×C3×C4×C3⋊D3—21no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634—21no—
14C2×C3×#634C2211yescharacteristic
15C2×#6294—21no—
16C3×C4⋊(C3⋊Dic3)C2212yes—
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18#88815—21no—
19#88808—22no—
20C2×Q8×C3⋊D3D311yescharacteristic
21C2×C3×D3×Q8—32no—
22C2×C3×D3×Q8—61no—
23#63900—31no—
24D3×C4⋊Dic3—32no—
25C2×#3710—31no—
26C2×D3×C3⋊C8—32no—
27#63662D312yescomplemented
28C3×#835—32no—
29C3×#835—32no—
30C3×#835—62no—
31#63567—32no—
32#63566—32no—
33#63565—32no—
34C22×C3×C3⋊D3D411yescharacteristic
35C33×Q8—21no—
36C33×Q8C2×C412yescomplemented
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4×C3⋊D3—24no—
43C3×C4×C3⋊D3C2×C412yes—
44C3×C4×C3⋊D3D412yes—
45C3×#634—21no—
46C3×#634—24no—
47C3×#634C2×C412yescomplemented
48#6294—42no—
49C3×C3⋊(C3⋊C8)—22no—
50C2×#835—91no—
51D3×#103—92no—
52C2×C32×Q8—21no—
53C2×C32×Q8—22no—
54C2×C32×Q8—41no—
55C2×C32×Q8C2×D312yes—
56C2×C32×Q8C2×D311yescharacteristic
57Q8×C3⋊D3—22no—
58Q8×C3⋊D3Dic314yescomplemented
59C2×C4×C3⋊D3—21no—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×#634—21no—
62C2×#634C2×D311yescharacteristic
63C3×D3×Q8—38no—
64C3×D3×Q8—64no—
65C3×D3×Q8—64no—
66C3×D3×Q8—64no—
67C2×C3×C4×D3—32no—
68C2×C3×C4×D3—62no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—62no—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—62no—
73C2×C3×Dic6—61no—
74C2×C3×Dic6—62no—
75C2×#624—61no—
76C2×D3×Dic3—62no—
77C4⋊(C3⋊Dic3)C2×D312yes—
78C2×C3⋊(C3⋊C8)C2×D312yes—
79C3×C4⋊Dic3—32no—
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3—62no—
82C2×C3×C3⋊C8—32no—
83C2×C3×C3⋊C8—32no—
84C2×C3×C3⋊C8—62no—
85#3724—61no—
86Dic3⋊Dic3—62no—
87#3710—62no—
88D3×C3⋊C8—64no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3C22⋊C412yes—
91C2×C3×C3⋊D3SD1612yes—
92C2×C3×C3⋊D3Q1612yes—
93C33×C4—22no—
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3×C3⋊Dic3—22no—
97C3×C3⋊Dic3—42no—
98C3×C3⋊Dic3C22⋊C412yes—
99C22×C3×Q8—91no—
100C2×D3×Q8—32no—
101C2×D3×Q8—61no—
102C2×C4⋊Dic3—91no—
103C22×C3⋊C8—91no—
104C2×D3×C8—92no—
105D3×C4⋊C4—92no—
106C3×#103—92no—
107#835—32no—
108#835—32no—
109#835—62no—
110#835—92no—
111#816—92no—
112#810—92no—
113C22×C3⋊D3#6711yescharacteristic
114C22×C3×D3—32no—
115C22×C3×D3—61no—
116C32×Q8—21no—
117C32×Q8—22no—
118C32×Q8—22no—
119C32×Q8—24no—
120C32×Q8—22no—
121C32×Q8—42no—
122C32×Q8—42no—
123C32×Q8—42no—
124C32×Q8C4×D314yescomplemented
125C32×Q8C2×Dic312yescomplemented
126C2×C32×C4—21no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4—42no—
133C2×C32×C4—42no—
134C2×C32×C4C22×D312yes—
135C2×C32×C4C22×D311yescharacteristic
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3#6711yescharacteristic
139C4×C3⋊D3—24no—
140C4×C3⋊D3C2×Dic312yes—
141C4×C3⋊D3#6712yes—
142#634—21no—
143#634—24no—
144#634C2×Dic312yescomplemented
145C2×C3×Dic3—32no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—122no—
152C3×C4×D3—34no—
153C3×C4×D3—34no—
154C3×C4×D3—62no—
155C3×C4×D3—62no—
156C3×C4×D3—68no—
157C3×C4×D3—68no—
158C3×Dic6—34no—
159C3×Dic6—62no—
160C3×Dic6—68no—
161C3×Dic6—68no—
162C3×Dic6—62no—
163C3×Dic6—62no—
164#624—122no—
165D3×Dic3—124no—
166C3⋊(C3⋊C8)—22no—
167C3×C3⋊C8—62no—
168C3×C3⋊C8—62no—
169C3×C3⋊C8—122no—
170C2×#103—271noSylow
171C2×C33C2×C22⋊C411yescharacteristic
172C2×C33C2×SD1611yescharacteristic
173C2×C33C2×Q1611yescharacteristic
174C3×C3⋊D3#10314yescomplemented
175C2×C3×Q8—21no—
176C2×C3×Q8—22no—
177C2×C3×Q8—41no—
178C2×C3×Q8—94no—
179C2×C3×Q8—91no—
180C2×C3×Q8—181no—
181C2×C3×Q8—182no—
182C2×C3×Q8D3212yes—
183C2×C3×Q8D3211yescharacteristic
184C22×C3×C4—91no—
185C22×C3×C4—181no—
186C22×Dic3—181no—
187D3×Q8—38no—
188D3×Q8—64no—
189D3×Q8—64no—
190D3×Q8—64no—
191C2×C4×D3—32no—
192C2×C4×D3—61no—
193C2×C4×D3—62no—
194C2×C4×D3—62no—
195C2×C4×D3—182no—
196C2×Dic6—32no—
197C2×Dic6—61no—
198C2×Dic6—62no—
199C2×Dic6—62no—
200C2×C3×C8—92no—
201C3×C4⋊C4—92no—
202C4⋊Dic3—32no—
203C4⋊Dic3—32no—
204C4⋊Dic3—62no—
205C4⋊Dic3—181no—
206Dic3⋊C4—182no—
207C2×C3⋊C8—32no—
208C2×C3⋊C8—32no—
209C2×C3⋊C8—62no—
210C2×C3⋊C8—182no—
211D3×C8—184no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—41no—
215C22×C32D3×D412yes—
216C22×C32C2×#6711yescharacteristic
217C2×C3⋊D3#21512yes—
218C2×C3⋊D3#21612yes—
219C2×C3⋊D3#21712yes—
220C2×C3×D3—34no—
221C2×C3×D3—34no—
222C2×C3×D3—34no—
223C2×C3×D3—62no—
224C2×C3×D3—62no—
225C2×C3×D3—62no—
226C32×C4—22no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—22no—
230C32×C4—21no—
231C32×C4—22no—
232C32×C4—21no—
233C32×C4—41no—
234C32×C4—41no—
235C32×C4—44no—
236C32×C4—44no—
237C32×C4C2×C4×D312yes—
238C32×C4D3×D412yes—
239C32×C4C22×Dic311yescharacteristic
240C32×C4C2×#6711yescharacteristic
241C3⋊Dic3—22no—
242C3⋊Dic3—42no—
243C3⋊Dic3#21712yes—
244C3×Dic3—34no—
245C3×Dic3—62no—
246C3×Dic3—64no—
247C3×Dic3—64no—
248C3×Dic3—122no—
249C3×Dic3—122no—
250C3×Dic3—242no—
251C22×Q8—91no—
252C22×C8—271no—
253C2×C4⋊C4—271no—
254#103—92no—
255#103—272no—
256C33C2×#10311yescharacteristic, complemented, Sylow
257C23×C3—91no—
258C22×D3—32no—
259C22×D3—61no—
260C3×Q8—22no—
261C3×Q8—22no—
262C3×Q8—21no—
263C3×Q8—22no—
264C3×Q8—24no—
265C3×Q8—42no—
266C3×Q8—42no—
267C3×Q8—42no—
268C3×Q8—92no—
269C3×Q8—184no—
270C3×Q8—181no—
271C3×Q8D3×Dic314yescomplemented
272C3×Q8#62412yescomplemented
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—91no—
282C2×C3×C4—92no—
283C2×C3×C4—92no—
284C2×C3×C4—184no—
285C2×C3×C4—181no—
286C2×C3×C4—182no—
287C2×C3×C4C2×D3212yes—
288C2×C3×C4C2×D3211yescharacteristic
289C2×Dic3—32no—
290C2×Dic3—61no—
291C2×Dic3—62no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294C2×Dic3—62no—
295C2×Dic3—122no—
296C2×Dic3—362no—
297C4×D3—34no—
298C4×D3—34no—
299C4×D3—62no—
300C4×D3—62no—
301C4×D3—68no—
302C4×D3—68no—
303C4×D3—364no—
304Dic6—34no—
305Dic6—62no—
306Dic6—68no—
307Dic6—68no—
308Dic6—62no—
309Dic6—62no—
310C3×C8—182no—
311C3⋊C8—62no—
312C3⋊C8—62no—
313C3⋊C8—122no—
314C2×C32—22no—
315C2×C32—21no—
316C2×C32—22no—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—21no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32D3×C22⋊C412yes—
324C2×C32D3×SD1612yes—
325C2×C32D3×Q1612yes—
326C2×C32C2×#21511yescharacteristic
327C2×C32C2×#21611yescharacteristic
328C2×C32C2×#21711yescharacteristic
329C3⋊D3#83514yescomplemented
330C3×D3—38no—
331C3×D3—64no—
332C2×Q8—94no—
333C2×Q8—91no—
334C2×Q8—181no—
335C2×Q8—182no—
336C2×Q8#111311yescharacteristic
337C22×C4—91no—
338C22×C4—181no—
339C22×C4—541no—
340C2×C8—92no—
341C2×C8—542no—
342C4⋊C4—92no—
343C4⋊C4—541no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—41no—
347C22×C3—92no—
348C22×C3—92no—
349C22×C3—92no—
350C22×C3D3×#6712yes—
351C22×C3#381111yescharacteristic
352C2×D3—34no—
353C2×D3—34no—
354C2×D3—34no—
355C2×D3—62no—
356C2×D3—62no—
357C2×D3—62no—
358C3×C4—24no—
359C3×C4—24no—
360C3×C4—22no—
361C3×C4—21no—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—21no—
365C3×C4—44no—
366C3×C4—44no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—92no—
370C3×C4—182no—
371C3×C4—362no—
372C3×C4C2×D3×Dic312yes—
373C3×C4C2×#62411yescharacteristic
374C3×C4D3×#6712yes—
375C3×C4#381111yescharacteristic
376Dic3—34no—
377Dic3—62no—
378Dic3—64no—
379Dic3—64no—
380Dic3—122no—
381Dic3—122no—
382Dic3—242no—
383C32—22no—
384C32—21no—
385C32—41no—
386C32D3×#10312yescomplemented
387C32C2×#83511yescharacteristic, complemented
388C23—91no—
389Q8—21no—
390Q8—92no—
391Q8—184no—
392Q8—181no—
393Q8#629412yescomplemented
394C2×C4—21no—
395C2×C4—91no—
396C2×C4—92no—
397C2×C4—92no—
398C2×C4—184no—
399C2×C4—181no—
400C2×C4—182no—
401C2×C4—1082no—
402C2×C4C2×#111311yescharacteristic
403C8—182no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—22no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—94no—
414C2×C3D3×#21512yes—
415C2×C3#6394011yescharacteristic
416C2×C3D3×#21612yes—
417C2×C3#6394411yescharacteristic
418C2×C3D3×#21712yes—
419C2×C3#6397411yescharacteristic
420D3—38no—
421D3—64no—
422C22—92no—
423C22—92no—
424C22—92no—
425C22#8904911yescharacteristic
426C4—22no—
427C4—92no—
428C4—182no—
429C4—362no—
430C4C2×#629411yescharacteristic
431C4#8904911yescharacteristic
432C3—21no—
433C3—22no—
434C3—41no—
435C3D3×#83512yescomplemented
436C3#1059155211yescharacteristic, complemented
437C2—94no—
438C2#1172370811yescharacteristic
439C2#1172371011yescharacteristic
440C2#1172372311yescharacteristic
subgroupG/H
centerC22#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C33×C4C23
hypercenterC2×Q8#1113
hypocenterC33C2×#103
LCSG ▷ 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
4. C3×#63662C2semidirect
5. #11722076C2semidirect
10. C3×Q8×C3⋊D3C4semidirect
27. #63662D3semidirect
36. C33×Q8C2×C4semidirect
47. C3×#634C2×C4semidirect
58. Q8×C3⋊D3Dic3semidirect
124. C32×Q8C4×D3semidirect
125. C32×Q8C2×Dic3semidirect
144. #634C2×Dic3semidirect
174. C3×C3⋊D3#103semidirect
256. C33C2×#103semidirect
271. C3×Q8D3×Dic3semidirect
272. C3×Q8#624semidirect
329. C3⋊D3#835semidirect
386. C32D3×#103semidirect
387. C32C2×#835semidirect
393. Q8#6294semidirect
435. C3D3×#835semidirect
436. C3#10591552semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
#117236892
C2×#888082
C3×#636622
#117220762
#105915523
D3×#8353