Finite Group Databasebeta

← #13168310

Subgroups — #13168310

Order 1728.

1490 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 1490 conjugacy classes of subgroups — too many to draw. Only its 236 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×Dic6C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic
3C4×#6294C211yesmaximal, characteristic
4#11723455C212yesmaximal, complemented
5C3×#63873C212yesmaximal, complemented
6#11723034C214yesmaximal, complemented
7#11722910C214yesmaximal
8#10595365—31nomaximal
9D3×#944—32nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2212yes—
14C2×#6294C2212yes—
15C2×#6293C2212yes—
16C2×Dic3×C3⋊D3C2212yes—
17C3⋊D3×Dic6—24no—
18C2×C3×#627C2212yes—
19C2×C3×#624C2212yes—
20C3×#3797—24no—
21C3×C4×C3⋊Dic3C2212yes—
22#88813C2212yes—
23#88807C2218yes—
24C3×#3724C2218yes—
25C2×Q8×C3⋊D3—31no—
26C2×C3×D3×Q8—32no—
27C2×#3797D311yescharacteristic
28C2×D3×Dic6—32no—
29C2×D3×Dic6—61no—
30#64153—32no—
31C3×#944—32no—
32C3×#944—32no—
33#63883—34no—
34C4×#624—31no—
35D3×Dic3⋊C4—38no—
36C4×D3×Dic3—32no—
37#63873D312yescomplemented
38#63866—32no—
39#63866—32no—
40#63866—62no—
41#63840—34no—
42#63839—38no—
43C22×C3×C3⋊D3C2311yescharacteristic
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)C2312yes—
46#6308—22no—
47C2×C3×C3⋊Dic3C2311yescharacteristic
48C2×C3×C3⋊Dic3C2314yes—
49C3×C4×C3⋊D3D414yes—
50C2×C32×Dic3C2314yes—
51C2×C32×Dic3C2312yes—
52C32×Dic6—22no—
53C32×Dic6—24no—
54#6294Q818yescomplemented
55#6293—24no—
56Dic3×C3⋊D3—24no—
57C3×#627—24no—
58C3×#624—24no—
59C2×#944—91no—
60D3×C4⋊Q8—92no—
61C2×C32×Q8—31no—
62C2×C32×Q8—32no—
63Q8×C3⋊D3—64no—
64C2×C4×C3⋊D3—32no—
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×#634—32no—
67C2×#634—31no—
68C2×#634—32no—
69C2×#634—61no—
70C2×#634—62no—
71C2×#634—121no—
72C3×D3×Q8—68no—
73C2×C3×C4×D3—32no—
74C2×C3×C4×D3—34no—
75C2×C3×C4×D3—61no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6—22no—
78C2×C3×Dic6—34no—
79C2×C3×Dic6C2×D312yes—
80C2×C3×Dic6C2×D312yes—
81C2×C3×Dic6C2×D312yes—
82C2×#627—34no—
83C2×#627—62no—
84C2×#627C2×D312yes—
85C2×#624—32no—
86C2×#624C2×D312yes—
87C2×D3×Dic3—34no—
88C2×D3×Dic3—34no—
89C2×D3×Dic3—62no—
90#3797—24no—
91D3×Dic6—68no—
92D3×Dic6—68no—
93#3750—38no—
94C4×C3⋊Dic3C2×D312yes—
95C3×Dic3⋊C4—38no—
96C3×Dic3⋊C4—38no—
97C3×C4×Dic3—32no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—62no—
100#3724C2×D318yes—
101Dic3⋊Dic3—38no—
102Dic3⋊Dic3—38no—
103Dic3⋊Dic3—68no—
104Dic32—34no—
105Dic32—32no—
106C22×C33C2411yescharacteristic
107C2×C3×C3⋊D3C2×D412yes—
108C2×C3×C3⋊D3C2×Q814yes—
109C33×C4C2×D412yes—
110C3⋊(C3⋊Dic3)—22no—
111C3×C3⋊Dic3C2×D412yes—
112C3×C3⋊Dic3C2×Q818yescomplemented
113C32×Dic3—22no—
114C32×Dic3—24no—
115C22×C3×Q8—91no—
116C2×D3×Q8—32no—
117C2×D3×Q8—92no—
118C2×D3×Q8—181no—
119C22×Dic6—91no—
120C3×C4⋊Q8—92no—
121#944—32no—
122#944—32no—
123#944—92no—
124#944—92no—
125#944—94no—
126#944—182no—
127C2×Dic3⋊C4—94no—
128C2×C4×Dic3—91no—
129D3×C4⋊C4—98no—
130#888—98no—
131C42×D3—92no—
132#869—92no—
133C22×C3⋊D3C22×D311yescharacteristic
134C22×C3×D3—32no—
135C22×C3×D3—61no—
136C32×Q8—62no—
137C32×Q8—64no—
138C2×C32×C4—22no—
139C2×C32×C4—21no—
140C2×C32×C4—32no—
141C2×C32×C4—34no—
142C2×C32×C4—41no—
143C2×C32×C4C22×D312yes—
144C2×C32×C4C22×D311yescharacteristic
145C2×C3⋊Dic3—32no—
146C2×C3⋊Dic3—34no—
147C2×C3⋊Dic3—62no—
148C2×C3⋊Dic3—64no—
149C2×C3⋊Dic3—122no—
150C2×C3⋊Dic3C22×D314yes—
151C2×C3⋊Dic3C22×D311yescharacteristic
152C4×C3⋊D3—64no—
153C4×C3⋊D3#6714yes—
154#634—62no—
155#634—64no—
156#634—64no—
157#634—64no—
158#634—124no—
159#634—124no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3C22×D314yes—
168C2×C3×Dic3C22×D314yes—
169C2×C3×Dic3C22×D314yes—
170C3×C4×D3—38no—
171C3×C4×D3—64no—
172C3×C4×D3—68no—
173C3×Dic6—24no—
174C3×Dic6—24no—
175C3×Dic6—24no—
176C3×Dic6—24no—
177C3×Dic6—44no—
178C3×Dic6—68no—
179#627—24no—
180#627—68no—
181#627—68no—
182#624—24no—
183#624—38no—
184D3×Dic3—316no—
185D3×Dic3—68no—
186D3×Dic3—68no—
187C2×C4⋊Q8—271noSylow
188C2×C33C22×D411yescharacteristic
189C2×C33C22×Q812yes—
190C3×C3⋊D3C4⋊Q814yescomplemented
191C2×C3×Q8—32no—
192C2×C3×Q8—32no—
193C2×C3×Q8—32no—
194C2×C3×Q8—61no—
195C2×C3×Q8—62no—
196C2×C3×Q8—92no—
197C2×C3×Q8—184no—
198C22×C3×C4—92no—
199C22×C3×C4—91no—
200C22×Dic3—92no—
201C22×Dic3—92no—
202D3×Q8—68no—
203D3×Q8—188no—
204D3×Q8—188no—
205C2×C4×D3—32no—
206C2×C4×D3—34no—
207C2×C4×D3—61no—
208C2×C4×D3—94no—
209C2×C4×D3—94no—
210C2×C4×D3—182no—
211C2×Dic6—34no—
212C2×Dic6—94no—
213C2×Dic6—92no—
214C2×Dic6—91no—
215C2×Dic6—92no—
216C2×Dic6—182no—
217C2×Dic6—181no—
218C2×Dic6—182no—
219C2×Dic6—184no—
220C2×Dic6—361no—
221C2×Dic6D3212yes—
222C2×Dic6D3211yescharacteristic
223C3×C4⋊C4—98no—
224C3×C42—92no—
225C4⋊Dic3—98no—
226Dic3⋊C4—38no—
227Dic3⋊C4—38no—
228Dic3⋊C4—98no—
229Dic3⋊C4—98no—
230Dic3⋊C4—188no—
231C4×Dic3—32no—
232C4×Dic3—32no—
233C4×Dic3—62no—
234C4×Dic3—94no—
235C4×Dic3—92no—
236C22×C32—22no—
237C22×C32—21no—
238C22×C32—41no—
239C22×C32C23×D312yes—
240C22×C32C23×D311yescharacteristic
241C2×C3⋊D3D3×Q814yes—
242C2×C3⋊D3C2×#6712yes—
243C2×C3×D3—34no—
244C2×C3×D3—38no—
245C2×C3×D3—62no—
246C2×C3×D3—64no—
247C32×C4—24no—
248C32×C4—22no—
249C32×C4—42no—
250C32×C4—62no—
251C32×C4—64no—
252C32×C4D3×D414yes—
253C32×C4C2×#6712yes—
254C3⋊Dic3—62no—
255C3⋊Dic3—64no—
256C3⋊Dic3—64no—
257C3⋊Dic3—124no—
258C3⋊Dic3—124no—
259C3⋊Dic3D3×Q818yescomplemented
260C3⋊Dic3C2×#6712yes—
261C3×Dic3—24no—
262C3×Dic3—24no—
263C3×Dic3—24no—
264C3×Dic3—24no—
265C3×Dic3—34no—
266C3×Dic3—38no—
267C3×Dic3—38no—
268C3×Dic3—44no—
269C3×Dic3—62no—
270C3×Dic3—68no—
271C22×Q8—91no—
272C22×Q8—271no—
273C4⋊Q8—92no—
274C4⋊Q8—272no—
275C4⋊Q8—274no—
276C2×C4⋊C4—274no—
277C2×C42—271no—
278C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
279C23×C3—91no—
280C22×D3—32no—
281C22×D3—61no—
282C3×Q8—64no—
283C3×Q8—64no—
284C3×Q8—64no—
285C3×Q8—64no—
286C3×Q8—124no—
287C3×Q8—184no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—34no—
291C2×C3×C4—34no—
292C2×C3×C4—34no—
293C2×C3×C4—41no—
294C2×C3×C4—62no—
295C2×C3×C4—64no—
296C2×C3×C4—91no—
297C2×C3×C4—94no—
298C2×C3×C4—94no—
299C2×C3×C4—184no—
300C2×C3×C4C2×D3212yes—
301C2×C3×C4C2×D3211yescharacteristic
302C2×Dic3—34no—
303C2×Dic3—34no—
304C2×Dic3—32no—
305C2×Dic3—64no—
306C2×Dic3—61no—
307C2×Dic3—98no—
308C2×Dic3—92no—
309C2×Dic3—94no—
310C2×Dic3—182no—
311C2×Dic3—184no—
312C2×Dic3—184no—
313C2×Dic3—362no—
314C2×Dic3C2×D3212yes—
315C2×Dic3C2×D3214yes—
316C4×D3—38no—
317C4×D3—64no—
318C4×D3—68no—
319C4×D3—916no—
320C4×D3—188no—
321C4×D3—188no—
322Dic6—24no—
323Dic6—22no—
324Dic6—68no—
325Dic6—184no—
326Dic6—188no—
327Dic6—188no—
328Dic6—184no—
329Dic6—184no—
330Dic6—182no—
331Dic6—364no—
332Dic6—364no—
333C2×C32—24no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—41no—
338C2×C32—42no—
339C2×C32C2×D3×D412yes—
340C2×C32C2×D3×Q814yes—
341C2×C32C2×D3×Q812yes—
342C2×C32C22×#6711yescharacteristic
343C3⋊D3#94414yescomplemented
344C3×D3—38no—
345C3×D3—64no—
346C2×Q8—31no—
347C2×Q8—32no—
348C2×Q8—92no—
349C2×Q8—184no—
350C2×Q8—272no—
351C2×Q8—271no—
352C2×Q8—544no—
353C22×C4—92no—
354C22×C4—91no—
355C22×C4—272no—
356C22×C4—272no—
357C4⋊C4—98no—
358C4⋊C4—278no—
359C42—92no—
360C42—272no—
361C22×C3—22no—
362C22×C3—21no—
363C22×C3—41no—
364C22×C3—94no—
365C22×C3—92no—
366C22×C3C22×D3212yes—
367C22×C3C22×D3211yescharacteristic
368C2×D3—38no—
369C2×D3—34no—
370C2×D3—62no—
371C2×D3—64no—
372C3×C4—22no—
373C3×C4—24no—
374C3×C4—42no—
375C3×C4—64no—
376C3×C4—64no—
377C3×C4—64no—
378C3×C4—64no—
379C3×C4—98no—
380C3×C4—92no—
381C3×C4—124no—
382C3×C4D3×#6714yes—
383C3×C4#381112yes—
384Dic3—24no—
385Dic3—22no—
386Dic3—38no—
387Dic3—38no—
388Dic3—34no—
389Dic3—62no—
390Dic3—68no—
391Dic3—184no—
392Dic3—184no—
393Dic3—182no—
394Dic3—364no—
395Dic3—364no—
396C32—22no—
397C32—21no—
398C32—41no—
399C32D3×C4⋊Q812yescomplemented
400C32C2×#94411yescharacteristic, complemented
401C23—91no—
402Q8—64no—
403Q8—62no—
404Q8—184no—
405Q8—544no—
406Q8—542no—
407C2×C4—32no—
408C2×C4—34no—
409C2×C4—94no—
410C2×C4—91no—
411C2×C4—94no—
412C2×C4—184no—
413C2×C4—278no—
414C2×C4—272no—
415C2×C4—544no—
416C2×C4D3311yescharacteristic
417C2×C3—24no—
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—42no—
422C2×C3—41no—
423C2×C3—94no—
424C2×C3D32×Q814yes—
425C2×C3D32×Q812yes—
426C2×C3C2×D3×#6712yes—
427C2×C3C2×#381111yescharacteristic
428D3—38no—
429D3—64no—
430C22—94no—
431C22—92no—
432C22C2×D3311yescharacteristic
433C4—64no—
434C4—62no—
435C4—98no—
436C4—92no—
437C4—542no—
438C4#8895212yes—
439C3—21no—
440C3—22no—
441C3—41no—
442C3D3×#94412yescomplemented
443C3#1059536511yescharacteristic, complemented
444C2—94no—
445C2#1172437812yes—
446C2C2×#8895211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4D33
hypocenterC33C2×C4⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11723455C2semidirect
5. C3×#63873C2semidirect
6. #11723034C2semidirect
37. #63873D3semidirect
54. #6294Q8semidirect
112. C3×C3⋊Dic3C2×Q8semidirect
190. C3×C3⋊D3C4⋊Q8semidirect
259. C3⋊Dic3D3×Q8semidirect
278. C33C2×C4⋊Q8semidirect
343. C3⋊D3#944semidirect
399. C32D3×C4⋊Q8semidirect
400. C32C2×#944semidirect
442. C3D3×#944semidirect
443. C3#10595365semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×C3×#37972
C4×#62942
#117234552
C3×#638732
#117230342
#117229102
#105953653
D3×#9443