Finite Group Databasebeta

← #13162023

Subgroups — #13162023

Order 1728.

918 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 143 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×#3832C211yesmaximal, characteristic
2#11723667C211yesmaximal, characteristic, complemented
3C2×#88808C211yesmaximal, characteristic, complemented
4C3×#63662C212yesmaximal, complemented
5#11722075C212yesmaximal, complemented
6#10591560—31nomaximal
7#10591556—32nomaximal
8C2×C33×Q8C2211yescharacteristic
9C3×#3832—22no—
10C3×#3832C414yescomplemented
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17#88814—21no—
18#88813—21no—
19#88808—22no—
20C2×#3832D311yescharacteristic
21C2×C3×#239—32no—
22C2×C3×#239—61no—
23#63849—31no—
24#63841—32no—
25C2×#3710—31no—
26C2×D3×C3⋊C8—32no—
27#63662D312yescomplemented
28C3×#835—32no—
29C3×#835—32no—
30C3×#835—62no—
31#63564—32no—
32#63563—32no—
33#63562—32no—
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3D411yescharacteristic
36C33×Q8—21no—
37C33×Q8C2×C412yes—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4⋊(C3⋊D3)—24no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)C2×C412yes—
45C3×C4×C3⋊D3—24no—
46C3×C4×C3⋊D3C2×C412yes—
47C3×C4×C3⋊D3D412yes—
48C3×C3⋊(C3⋊C8)—22no—
49#5233—91no—
50#4804—92no—
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8—41no—
54C2×C32×Q8C2×D312yes—
55C2×C32×Q8C2×D311yescharacteristic
56#3832—22no—
57#3832Dic314yescomplemented
58C2×C4⋊(C3⋊D3)—21no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4×C3⋊D3—21no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C3×#239—38no—
63C3×#239—64no—
64C3×#239—64no—
65C3×#239—64no—
66C2×C3×D12—32no—
67C2×C3×D12—62no—
68C2×C3×D12—61no—
69C2×C3×D12—62no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—62no—
72C2×C3×C4×D3—61no—
73C2×C3×C4×D3—62no—
74C4⋊(C3⋊Dic3)C2×D312yes—
75C2×C3⋊(C3⋊C8)C2×D312yes—
76C3×C4⋊Dic3—32no—
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—62no—
79C2×C3×C3⋊C8—32no—
80C2×C3×C3⋊C8—32no—
81C2×C3×C3⋊C8—62no—
82#3722—61no—
83(C2×D3)⋊Dic3—62no—
84Dic32—62no—
85Dic32—61no—
86#3710—62no—
87D3×C3⋊C8—64no—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—42no—
92C2×C3×C3⋊D3C22⋊C412yes—
93C33×C4—22no—
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3×C3⋊Dic3—42no—
97C3×C3⋊Dic3C22⋊C412yes—
98C2×C3×#41—91no—
99C2×#239—32no—
100C2×#239—61no—
101#926—91no—
102C22×C3⋊C8—91no—
103C2×D3×C8—92no—
104#892—92no—
105C3×#103—92no—
106#835—32no—
107#835—32no—
108#835—62no—
109#832—92no—
110#821—92no—
111#809—92no—
112C22×C3⋊D3—21no—
113C22×C3⋊D3#6711yescharacteristic
114C22×C3×D3—32no—
115C22×C3×D3—61no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C32×Q8—21no—
119C32×Q8—22no—
120C32×Q8—22no—
121C32×Q8—24no—
122C32×Q8—22no—
123C32×Q8—42no—
124C32×Q8—42no—
125C32×Q8—42no—
126C32×Q8C4×D314yes—
127C32×Q8C2×Dic312yes—
128C2×C32×C4—21no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—21no—
133C2×C32×C4—41no—
134C2×C32×C4—42no—
135C2×C32×C4—42no—
136C2×C32×C4C22×D312yes—
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3#6711yescharacteristic
140C4⋊(C3⋊D3)—24no—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)C2×Dic312yes—
143C4×C3⋊D3—24no—
144C4×C3⋊D3C2×Dic312yes—
145C4×C3⋊D3#6712yes—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—122no—
151C3×D12—34no—
152C3×D12—62no—
153C3×D12—68no—
154C3×D12—68no—
155C3×D12—62no—
156C3×D12—62no—
157C3×C4×D3—34no—
158C3×C4×D3—34no—
159C3×C4×D3—62no—
160C3×C4×D3—62no—
161C3×C4×D3—68no—
162C3×C4×D3—68no—
163C3⋊(C3⋊C8)—22no—
164C3×C3⋊C8—62no—
165C3×C3⋊C8—62no—
166C3×C3⋊C8—122no—
167#416—271noSylow
168C2×C33C2×C22⋊C411yescharacteristic
169C2×C33#13512yes—
170C3×C3⋊D3—22no—
171C3×C3⋊D3—42no—
172C3×#41—94no—
173C3×#41—182no—
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—41no—
177C2×C3×Q8D3212yes—
178C2×C3×Q8D3211yescharacteristic
179C2×C3×D4—91no—
180C2×C3×D4—181no—
181C22×C3×C4—91no—
182C22×C3×C4—181no—
183#239—38no—
184#239—64no—
185#239—64no—
186#239—64no—
187C2×D12—32no—
188C2×D12—61no—
189C2×D12—62no—
190C2×D12—62no—
191C2×C4×D3—32no—
192C2×C4×D3—61no—
193C2×C4×D3—62no—
194C2×C4×D3—62no—
195C2×C3×C8—92no—
196C3×C4⋊C4—92no—
197#217—181no—
198(C2×D3)⋊C4—182no—
199C4⋊Dic3—32no—
200C4⋊Dic3—32no—
201C4⋊Dic3—62no—
202C4×Dic3—182no—
203C4×Dic3—181no—
204C2×C3⋊C8—32no—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—62no—
207C2×C3⋊C8—182no—
208D3×C8—184no—
209C22×C32—22no—
210C22×C32—21no—
211C22×C32—41no—
212C22×C32D3×D412yes—
213C22×C32C2×#6711yescharacteristic
214C2×C3⋊D3—22no—
215C2×C3⋊D3—22no—
216C2×C3⋊D3—42no—
217C2×C3⋊D3#21712yes—
218C2×C3×D3—34no—
219C2×C3×D3—62no—
220C2×C3×D3—62no—
221C2×C3×D3—62no—
222C2×C3×D3—64no—
223C2×C3×D3—64no—
224C2×C3×D3—122no—
225C2×C3×D3—124no—
226C2×C3×D3—122no—
227C2×C3×D3—122no—
228C32×C4—22no—
229C32×C4—24no—
230C32×C4—24no—
231C32×C4—22no—
232C32×C4—21no—
233C32×C4—22no—
234C32×C4—21no—
235C32×C4—41no—
236C32×C4—41no—
237C32×C4—44no—
238C32×C4—44no—
239C32×C4C2×C4×D312yes—
240C32×C4D3×D412yes—
241C32×C4C22×Dic311yescharacteristic
242C32×C4C2×#6711yescharacteristic
243C3⋊Dic3—42no—
244C3⋊Dic3#21712yes—
245C3×Dic3—34no—
246C3×Dic3—62no—
247C3×Dic3—122no—
248C3×Dic3—122no—
249C3×Dic3—242no—
250C2×#41—91no—
251C22×C8—271no—
252#117—271no—
253#103—92no—
254#102—272no—
255C33#41611yescharacteristic, complemented, Sylow
256C23×C3—91no—
257C23×C3—181no—
258C22×D3—32no—
259C22×D3—61no—
260C22×D3—62no—
261C22×D3—62no—
262C3×Q8—22no—
263C3×Q8—22no—
264C3×Q8—21no—
265C3×Q8—22no—
266C3×Q8—24no—
267C3×Q8—42no—
268C3×Q8—42no—
269C3×Q8—42no—
270C3×Q8D3×Dic314yes—
271C3×Q8#62412yes—
272C3×D4—92no—
273C3×D4—184no—
274C3×D4—181no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—91no—
284C2×C3×C4—92no—
285C2×C3×C4—92no—
286C2×C3×C4—184no—
287C2×C3×C4—182no—
288C2×C3×C4C2×D3212yes—
289C2×C3×C4C2×D3211yescharacteristic
290C2×Dic3—32no—
291C2×Dic3—61no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294C2×Dic3—122no—
295D12—34no—
296D12—62no—
297D12—68no—
298D12—68no—
299D12—62no—
300D12—62no—
301C4×D3—34no—
302C4×D3—34no—
303C4×D3—62no—
304C4×D3—62no—
305C4×D3—68no—
306C4×D3—68no—
307C3×C8—182no—
308C3⋊C8—62no—
309C3⋊C8—62no—
310C3⋊C8—122no—
311C2×C32—22no—
312C2×C32—22no—
313C2×C32—22no—
314C2×C32—22no—
315C2×C32—21no—
316C2×C32—41no—
317C2×C32—42no—
318C2×C32D3×C22⋊C412yes—
319C2×C32#91612yes—
320C2×C32#91912yes—
321C2×C32#95012yes—
322C2×C32C2×#21711yescharacteristic
323C3⋊D3—22no—
324C3⋊D3—42no—
325C3×D3—64no—
326C3×D3—122no—
327C3×D3—124no—
328C3×D3—124no—
329#41—94no—
330#41—182no—
331C2×Q8#111311yescharacteristic
332C2×D4—91no—
333C2×D4—181no—
334C22×C4—91no—
335C22×C4—181no—
336C2×C8—92no—
337C2×C8—542no—
338C4⋊C4—92no—
339C22⋊C4—541no—
340C42—541no—
341C22×C3—21no—
342C22×C3—22no—
343C22×C3—41no—
344C22×C3—92no—
345C22×C3—182no—
346C22×C3—182no—
347C22×C3—362no—
348C22×C3D3×#6712yes—
349C22×C3#381111yescharacteristic
350C2×D3—34no—
351C2×D3—62no—
352C2×D3—62no—
353C2×D3—62no—
354C2×D3—64no—
355C2×D3—64no—
356C2×D3—122no—
357C2×D3—124no—
358C2×D3—122no—
359C2×D3—122no—
360C3×C4—24no—
361C3×C4—24no—
362C3×C4—22no—
363C3×C4—21no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—44no—
368C3×C4—44no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—92no—
372C3×C4—362no—
373C3×C4C2×D3×Dic312yes—
374C3×C4C2×#62411yescharacteristic
375C3×C4D3×#6712yes—
376C3×C4#381111yescharacteristic
377Dic3—34no—
378Dic3—62no—
379Dic3—122no—
380Dic3—122no—
381Dic3—242no—
382C32—22no—
383C32—21no—
384C32—41no—
385C32#480412yescomplemented
386C32#523311yescharacteristic, complemented
387C23—91no—
388C23—181no—
389Q8—21no—
390Q8#629412yes—
391D4—92no—
392D4—184no—
393D4—181no—
394C2×C4—21no—
395C2×C4—91no—
396C2×C4—92no—
397C2×C4—92no—
398C2×C4—184no—
399C2×C4—182no—
400C2×C4C2×#111311yescharacteristic
401C8—182no—
402C2×C3—22no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—42no—
408C2×C3—41no—
409C2×C3—182no—
410C2×C3—362no—
411C2×C3#6394712yes—
412C2×C3#6394912yes—
413C2×C3#6395012yes—
414C2×C3D3×#21712yes—
415C2×C3#6397411yescharacteristic
416D3—64no—
417D3—122no—
418D3—124no—
419D3—124no—
420C22—92no—
421C22—182no—
422C22—182no—
423C22—362no—
424C22#8904911yescharacteristic
425C4—22no—
426C4—92no—
427C4—362no—
428C4C2×#629411yescharacteristic
429C4#8904911yescharacteristic
430C3—21no—
431C3—22no—
432C3—41no—
433C3#1059155612yescomplemented
434C3#1059156011yescharacteristic, complemented
435C2—182no—
436C2—362no—
437C2#1172371312yes—
438C2#1172372311yescharacteristic
subgroupG/H
centerC22#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
cosocleC2×C33×C4C23
hypercenterC2×Q8#1113
hypocenterC33#416
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
2. #11723667C2semidirect
3. C2×#88808C2semidirect
4. C3×#63662C2semidirect
5. #11722075C2semidirect
10. C3×#3832C4semidirect
27. #63662D3semidirect
57. #3832Dic3semidirect
255. C33#416semidirect
385. C32#4804semidirect
386. C32#5233semidirect
433. C3#10591556semidirect
434. C3#10591560semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
#117236672
C2×#888082
C3×#636622
#117220752
#105915603
#105915563