Finite Group Databasebeta

← #13168318

Subgroups — #13168318

Order 1728.

1434 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 240 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×#89040C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic
3C4×Dic3×C3⋊D3C211yesmaximal, characteristic
4#11723451C212yesmaximal, complemented
5C3×#63866C212yesmaximal, complemented
6#11723035C214yesmaximal, complemented
7#11722911C214yesmaximal
8D3×#944—32nomaximal
9#10593909—31nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#634C2212yes—
12C2×C32×Dic6C2212yes—
13C2×#6296C2212yes—
14C2×#6294C2212yes—
15#89040—24no—
16C2×Dic3×C3⋊D3C2212yes—
17C2×C3×#627C2212yes—
18C2×C3×#624C2212yes—
19C3×#3797—24no—
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
21C32×C4×Dic3C2211yescharacteristic
22#88806C2218yes—
23Dic3×C3⋊Dic3C2212yes—
24C3×Dic3⋊Dic3C2218yes—
25C2×C3×D3×Q8—32no—
26C2×#3797—31no—
27C2×#3797D311yescharacteristic
28C2×D3×Dic6—32no—
29#64153—32no—
30C42×C3⋊D3—31no—
31C3×#944—32no—
32C3×#869—32no—
33#63919—34no—
34#63900—34no—
35D3×Dic3⋊C4—38no—
36C4×D3×Dic3—32no—
37C4×D3×Dic3—61no—
38#63873—32no—
39#63873—32no—
40#63873—62no—
41#63866D312yescomplemented
42#63839—38no—
43C22×C3×C3⋊D3C2311yescharacteristic
44C2×C33×C4C2311yescharacteristic
45C2×C3⋊(C3⋊Dic3)C2312yes—
46C2×C3×C3⋊Dic3C2314yes—
47C2×C3×C3⋊Dic3C2311yescharacteristic
48C3×C4×C3⋊D3D414yes—
49C3×#634—24no—
50C2×C32×Dic3C2314yes—
51C2×C32×Dic3C2312yes—
52C32×Dic6—24no—
53#6296—24no—
54#6294—24no—
55Dic3×C3⋊D3Q818yescomplemented
56C3×#627—24no—
57C3×#624—24no—
58C2×#944—91no—
59D3×C4⋊Q8—92no—
60C2×C32×Q8—32no—
61C2×C4×C3⋊D3—32no—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C2×#634C2×D312yes—
64C3×D3×Q8—68no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—34no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—32no—
70C2×C3×Dic6—32no—
71C2×C3×Dic6—34no—
72C2×C3×Dic6—62no—
73C2×C3×Dic6C2×D312yes—
74C2×C3×Dic6C2×D312yes—
75C2×#627—34no—
76C2×#627—32no—
77C2×#627C2×D312yes—
78C2×#624—32no—
79C2×#624C2×D312yes—
80C2×D3×Dic3—34no—
81C2×D3×Dic3—34no—
82C2×D3×Dic3—62no—
83#3797—24no—
84#3797—64no—
85D3×Dic6—68no—
86C32×C42—31no—
87#3750—38no—
88C4×C3⋊Dic3—31no—
89C4×C3⋊Dic3—32no—
90C4×C3⋊Dic3—32no—
91C4×C3⋊Dic3—62no—
92C4×C3⋊Dic3—61no—
93C4×C3⋊Dic3—121no—
94C3×C4⋊Dic3—38no—
95C3×Dic3⋊C4—38no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3C2×D312yes—
98#3724—38no—
99#3724—38no—
100#3724—68no—
101Dic3⋊Dic3C2×D318yes—
102Dic32—34no—
103Dic32—62no—
104C22×C33C2411yescharacteristic
105C2×C3×C3⋊D3C2×D412yes—
106C2×C3×C3⋊D3C2×Q814yes—
107C33×C4C2×D412yes—
108C3⋊(C3⋊Dic3)C2×Q814yescomplemented
109C3×C3⋊Dic3—24no—
110C3×C3⋊Dic3C2×D412yes—
111C32×Dic3—24no—
112C32×Dic3C2×Q814yescomplemented
113C22×C3×Q8—91no—
114C2×D3×Q8—32no—
115C2×D3×Q8—92no—
116C22×Dic6—91no—
117C3×C4⋊Q8—92no—
118#944—32no—
119#944—94no—
120#944—92no—
121#944—92no—
122#944—92no—
123#944—182no—
124C2×Dic3⋊C4—94no—
125C2×C4×Dic3—91no—
126D3×C4⋊C4—98no—
127#888—98no—
128C42×D3—92no—
129C42×D3—181no—
130#869—32no—
131C22×C3⋊D3C22×D311yescharacteristic
132C22×C3×D3—32no—
133C22×C3×D3—61no—
134C32×Q8—64no—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—32no—
138C2×C32×C4—34no—
139C2×C32×C4—41no—
140C2×C32×C4C22×D312yes—
141C2×C32×C4C22×D311yescharacteristic
142C2×C3⋊Dic3—34no—
143C2×C3⋊Dic3—32no—
144C2×C3⋊Dic3—62no—
145C2×C3⋊Dic3—64no—
146C2×C3⋊Dic3—122no—
147C2×C3⋊Dic3C22×D314yes—
148C2×C3⋊Dic3C22×D311yescharacteristic
149C4×C3⋊D3—38no—
150C4×C3⋊D3#6714yes—
151#634—24no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—64no—
159C2×C3×Dic3C22×D314yes—
160C2×C3×Dic3C22×D314yes—
161C2×C3×Dic3C22×D314yes—
162C3×C4×D3—38no—
163C3×C4×D3—64no—
164C3×C4×D3—68no—
165C3×Dic6—24no—
166C3×Dic6—24no—
167C3×Dic6—44no—
168C3×Dic6—68no—
169C3×Dic6—64no—
170C3×Dic6—64no—
171C3×Dic6—124no—
172#627—24no—
173#627—68no—
174#627—64no—
175#624—24no—
176#624—64no—
177D3×Dic3—316no—
178D3×Dic3—68no—
179D3×Dic3—68no—
180C2×C4⋊Q8—271noSylow
181C2×C33C22×D411yescharacteristic
182C2×C33C22×Q812yes—
183C3×C3⋊D3C4⋊Q814yescomplemented
184C2×C3×Q8—32no—
185C2×C3×Q8—32no—
186C2×C3×Q8—62no—
187C2×C3×Q8—92no—
188C2×C3×Q8—92no—
189C2×C3×Q8—184no—
190C22×C3×C4—92no—
191C22×C3×C4—91no—
192C22×Dic3—92no—
193C22×Dic3—92no—
194D3×Q8—68no—
195D3×Q8—188no—
196C2×C4×D3—32no—
197C2×C4×D3—34no—
198C2×C4×D3—61no—
199C2×C4×D3—94no—
200C2×C4×D3—94no—
201C2×C4×D3—182no—
202C2×Dic6—32no—
203C2×Dic6—32no—
204C2×Dic6—34no—
205C2×Dic6—62no—
206C2×Dic6—94no—
207C2×Dic6—92no—
208C2×Dic6—184no—
209C2×Dic6D3212yes—
210C3×C4⋊C4—98no—
211C3×C42—32no—
212C3×C42—61no—
213C4⋊Dic3—38no—
214Dic3⋊C4—38no—
215Dic3⋊C4—98no—
216Dic3⋊C4—98no—
217Dic3⋊C4—98no—
218Dic3⋊C4—188no—
219C4×Dic3—94no—
220C4×Dic3—92no—
221C4×Dic3—91no—
222C4×Dic3—92no—
223C4×Dic3—182no—
224C4×Dic3—181no—
225C4×Dic3—182no—
226C4×Dic3—361no—
227C4×Dic3D3211yescharacteristic
228C22×C32—22no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32C23×D312yes—
232C22×C32C23×D311yescharacteristic
233C2×C3⋊D3D3×Q814yes—
234C2×C3⋊D3C2×#6712yes—
235C2×C3×D3—34no—
236C2×C3×D3—38no—
237C2×C3×D3—62no—
238C2×C3×D3—64no—
239C32×C4—22no—
240C32×C4—24no—
241C32×C4—34no—
242C32×C4—42no—
243C32×C4—64no—
244C32×C4D3×D414yes—
245C32×C4C2×#6712yes—
246C3⋊Dic3—24no—
247C3⋊Dic3—38no—
248C3⋊Dic3—34no—
249C3⋊Dic3—64no—
250C3⋊Dic3—68no—
251C3⋊Dic3—124no—
252C3⋊Dic3C2×#6712yes—
253C3×Dic3—24no—
254C3×Dic3—24no—
255C3×Dic3—24no—
256C3×Dic3—34no—
257C3×Dic3—44no—
258C3×Dic3—62no—
259C3×Dic3—64no—
260C3×Dic3—64no—
261C3×Dic3—124no—
262C3×Dic3D3×Q818yescomplemented
263C22×Q8—91no—
264C22×Q8—271no—
265C4⋊Q8—92no—
266C4⋊Q8—274no—
267C4⋊Q8—272no—
268C2×C4⋊C4—274no—
269C2×C42—271no—
270C33C2×C4⋊Q811yescharacteristic, complemented, Sylow
271C23×C3—91no—
272C22×D3—32no—
273C22×D3—61no—
274C3×Q8—64no—
275C3×Q8—64no—
276C3×Q8—124no—
277C3×Q8—184no—
278C3×Q8—184no—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—34no—
282C2×C3×C4—34no—
283C2×C3×C4—34no—
284C2×C3×C4—41no—
285C2×C3×C4—62no—
286C2×C3×C4—64no—
287C2×C3×C4—91no—
288C2×C3×C4—94no—
289C2×C3×C4—94no—
290C2×C3×C4—184no—
291C2×C3×C4C2×D3211yescharacteristic
292C2×C3×C4C2×D3212yes—
293C2×Dic3—32no—
294C2×Dic3—34no—
295C2×Dic3—34no—
296C2×Dic3—61no—
297C2×Dic3—64no—
298C2×Dic3—94no—
299C2×Dic3—92no—
300C2×Dic3—98no—
301C2×Dic3—182no—
302C2×Dic3—184no—
303C2×Dic3—184no—
304C2×Dic3—362no—
305C2×Dic3C2×D3214yes—
306C2×Dic3C2×D3212yes—
307C4×D3—38no—
308C4×D3—64no—
309C4×D3—68no—
310C4×D3—916no—
311C4×D3—188no—
312C4×D3—188no—
313Dic6—24no—
314Dic6—68no—
315Dic6—64no—
316Dic6—64no—
317Dic6—124no—
318Dic6—188no—
319Dic6—184no—
320C2×C32—22no—
321C2×C32—21no—
322C2×C32—24no—
323C2×C32—22no—
324C2×C32—41no—
325C2×C32—42no—
326C2×C32C2×D3×D412yes—
327C2×C32C2×D3×Q814yes—
328C2×C32C2×D3×Q812yes—
329C2×C32C22×#6711yescharacteristic
330C3⋊D3#94414yescomplemented
331C3×D3—38no—
332C3×D3—64no—
333C2×Q8—32no—
334C2×Q8—92no—
335C2×Q8—92no—
336C2×Q8—184no—
337C2×Q8—272no—
338C2×Q8—544no—
339C22×C4—92no—
340C22×C4—91no—
341C22×C4—272no—
342C22×C4—272no—
343C4⋊C4—98no—
344C4⋊C4—278no—
345C42—31no—
346C42—272no—
347C42—271no—
348C22×C3—22no—
349C22×C3—21no—
350C22×C3—41no—
351C22×C3—94no—
352C22×C3—92no—
353C22×C3C22×D3211yescharacteristic
354C22×C3C22×D3212yes—
355C2×D3—38no—
356C2×D3—34no—
357C2×D3—62no—
358C2×D3—64no—
359C3×C4—24no—
360C3×C4—22no—
361C3×C4—38no—
362C3×C4—42no—
363C3×C4—64no—
364C3×C4—64no—
365C3×C4—64no—
366C3×C4—92no—
367C3×C4—124no—
368C3×C4—184no—
369C3×C4#380212yes—
370C3×C4D3×#6714yes—
371Dic3—24no—
372Dic3—34no—
373Dic3—62no—
374Dic3—64no—
375Dic3—64no—
376Dic3—94no—
377Dic3—98no—
378Dic3—124no—
379Dic3—184no—
380Dic3—188no—
381Dic3—364no—
382Dic3#379714yescomplemented
383C32—22no—
384C32—21no—
385C32—41no—
386C32D3×C4⋊Q812yescomplemented
387C32C2×#94411yescharacteristic, complemented
388C23—91no—
389Q8—64no—
390Q8—184no—
391Q8—184no—
392Q8—544no—
393C2×C4—32no—
394C2×C4—34no—
395C2×C4—91no—
396C2×C4—94no—
397C2×C4—94no—
398C2×C4—184no—
399C2×C4—278no—
400C2×C4—272no—
401C2×C4—544no—
402C2×C4D3311yescharacteristic
403C2×C3—22no—
404C2×C3—24no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—42no—
408C2×C3—41no—
409C2×C3—94no—
410C2×C3C2×#379712yes—
411C2×C3C2×#380211yescharacteristic
412C2×C3D32×Q814yes—
413C2×C3C2×D3×#6712yes—
414D3—38no—
415D3—64no—
416C22—94no—
417C22—92no—
418C22C2×D3311yescharacteristic
419C4—34no—
420C4—64no—
421C4—92no—
422C4—184no—
423C4—274no—
424C4#8895312yes—
425C3—21no—
426C3—22no—
427C3—41no—
428C3#1059390911yescharacteristic, complemented
429C3D3×#94412yescomplemented
430C2—94no—
431C2D3×#379712yes—
432C2C2×#8895311yescharacteristic
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. #11723451C2semidirect
5. C3×#63866C2semidirect
6. #11723035C2semidirect
41. #63866D3semidirect
55. Dic3×C3⋊D3Q8semidirect
108. C3⋊(C3⋊Dic3)C2×Q8semidirect
112. C32×Dic3C2×Q8semidirect
183. C3×C3⋊D3C4⋊Q8semidirect
262. C3×Dic3D3×Q8semidirect
270. C33C2×C4⋊Q8semidirect
330. C3⋊D3#944semidirect
382. Dic3#3797semidirect
386. C32D3×C4⋊Q8semidirect
387. C32C2×#944semidirect
428. C3#10593909semidirect
429. C3D3×#944semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×#37972
C4×Dic3×C3⋊D32
#117234512
C3×#638662
#117230352
#117229112
D3×#9443
#105939093