Finite Group Databasebeta

← #13178714

Subgroups — #13178714

Order 1728.

1418 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1418 conjugacy classes of subgroups — too many to draw. Only its 199 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
2C2×#89040C211yesmaximal, characteristic, complemented
3C3×#64153C212yesmaximal, complemented
4#11723686C214yesmaximal, complemented
5#11723677C212yesmaximal, complemented
6#11723673C211yesmaximal, characteristic, complemented
7#11723671C214yesmaximal, complemented
8#10595356—32nomaximal
9#10595350—31nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832—24no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C3×C4×C3⋊D3C2212yes—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2212yes—
17C2×#6296C2212yes—
18C2×#6294C2212yes—
19#89040—24no—
20C3×#3750C2218yes—
21C3×C4×C3⋊Dic3C2212yes—
22#88815C2212yes—
23#88814C2218yes—
24#88814C2212yes—
25C2×#3832D311yescharacteristic
26C2×C3×#239—32no—
27C2×C3×#239—61no—
28C2×#3797—31no—
29C2×D3×Dic6—32no—
30#64153D312yescomplemented
31C3×#944—32no—
32C3×#944—32no—
33C3×#944—62no—
34#63897—34no—
35#63884—38no—
36#63874—31no—
37#63865—34no—
38#63864—32no—
39#63863—32no—
40#63860—32no—
41#63855—32no—
42#63851—38no—
43C22×C3×C3⋊D3C2312yes—
44C22×C3×C3⋊D3C2311yescharacteristic
45C33×Q8—22no—
46C2×C33×C4C2312yes—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2314yes—
49C2×C3×C3⋊Dic3C2314yes—
50C2×C3×C3⋊Dic3C2311yescharacteristic
51C3×C4⋊(C3⋊D3)—24no—
52C3×C4⋊(C3⋊D3)—22no—
53C3×C4×C3⋊D3—24no—
54C3×C4×C3⋊D3D414yes—
55C3×#634—24no—
56#6296—24no—
57#6294—24no—
58#5825—91no—
59#5724—92no—
60C2×C32×Q8—21no—
61C2×C32×Q8—22no—
62C2×C32×Q8—41no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C2×D312yes—
65#3832—24no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4⋊(C3⋊D3)C2×D312yes—
68C2×C4×C3⋊D3C2×D312yes—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×#634C2×D312yes—
71C3×#239—68no—
72C3×#239—68no—
73C2×C3×D12—32no—
74C2×C3×D12—34no—
75C2×C3×D12—62no—
76C2×C3×D12—61no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—34no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3—62no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—32no—
83C2×C3×Dic6—62no—
84C2×#627—34no—
85C2×#627—32no—
86C2×#624—32no—
87C2×D3×Dic3—34no—
88#3797—64no—
89D3×Dic6—68no—
90#3750C2×D318yes—
91C4×C3⋊Dic3C2×D312yes—
92C3×Dic3⋊C4—38no—
93C3×Dic3⋊C4—38no—
94C3×Dic3⋊C4—68no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—32no—
97C3×C4×Dic3—62no—
98#3724—32no—
99Dic3⋊Dic3—34no—
100#3722—38no—
101#3722—32no—
102(C2×D3)⋊Dic3—38no—
103(C2×D3)⋊Dic3—38no—
104(C2×D3)⋊Dic3—34no—
105C22×C33C2411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—24no—
109C2×C3×C3⋊D3C2×D412yes—
110C33×C4—22no—
111C33×C4C2×D412yes—
112C3×C3⋊Dic3—24no—
113C3×C3⋊Dic3—24no—
114C3×C3⋊Dic3C2×D412yes—
115C2×C3×#41—91no—
116C2×#239—32no—
117C2×#239—61no—
118C2×D3×Q8—92no—
119C22×Dic6—91no—
120C3×C4⋊Q8—92no—
121#947—92no—
122#944—32no—
123#944—32no—
124#944—62no—
125#936—92no—
126#935—94no—
127#926—91no—
128#924—94no—
129#896—98no—
130#894—98no—
131#875—92no—
132#872—92no—
133C22×C3⋊D3C22×D312yes—
134C22×C3⋊D3C22×D311yescharacteristic
135C22×C3×D3—32no—
136C22×C3×D3—34no—
137C22×C3×D3—61no—
138C22×C3×D3—62no—
139C32×Q8—22no—
140C32×Q8—24no—
141C32×Q8—24no—
142C32×Q8—44no—
143C32×Q8—44no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—21no—
147C2×C32×C4—24no—
148C2×C32×C4—41no—
149C2×C32×C4—42no—
150C2×C32×C4C22×D312yes—
151C2×C32×C4C22×D314yes—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3C22×D314yes—
155C2×C3⋊Dic3C22×D314yes—
156C2×C3⋊Dic3C22×D311yescharacteristic
157C4⋊(C3⋊D3)—24no—
158C4⋊(C3⋊D3)—22no—
159C4×C3⋊D3—24no—
160C4×C3⋊D3#6714yes—
161#634—24no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—64no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—64no—
170C3×D12—68no—
171C3×D12—68no—
172C3×D12—64no—
173C3×D12—64no—
174C3×C4×D3—38no—
175C3×C4×D3—64no—
176C3×C4×D3—68no—
177C3×C4×D3—68no—
178C3×Dic6—64no—
179C3×Dic6—64no—
180C3×Dic6—124no—
181#627—68no—
182#627—64no—
183#624—64no—
184D3×Dic3—68no—
185#536—271noSylow
186C2×C33C22×D411yescharacteristic
187C2×C332−1+412yes—
188C3×C3⋊D3—22no—
189C3×C3⋊D3—42no—
190C3×#41—184no—
191C2×C3×Q8—21no—
192C2×C3×Q8—22no—
193C2×C3×Q8—41no—
194C2×C3×Q8—92no—
195C2×C3×Q8D3211yescharacteristic
196C2×C3×Q8D3212yes—
197C2×C3×D4—92no—
198C2×C3×D4—91no—
199C22×C3×C4—91no—
200C22×C3×C4—92no—
201C22×Dic3—92no—
202#239—68no—
203#239—68no—
204D3×Q8—188no—
205C2×D12—32no—
206C2×D12—34no—
207C2×D12—62no—
208C2×D12—61no—
209C2×C4×D3—32no—
210C2×C4×D3—34no—
211C2×C4×D3—61no—
212C2×C4×D3—62no—
213C2×C4×D3—94no—
214C2×Dic6—32no—
215C2×Dic6—32no—
216C2×Dic6—62no—
217C2×Dic6—94no—
218C2×Dic6—92no—
219C2×Dic6—184no—
220C3×C4⋊C4—98no—
221C3×C42—92no—
222#217—98no—
223#217—92no—
224(C2×D3)⋊C4—94no—
225(C2×D3)⋊C4—98no—
226(C2×D3)⋊C4—98no—
227C4⋊Dic3—92no—
228Dic3⋊C4—38no—
229Dic3⋊C4—38no—
230Dic3⋊C4—68no—
231Dic3⋊C4—94no—
232C4×Dic3—32no—
233C4×Dic3—32no—
234C4×Dic3—62no—
235C22×C32—22no—
236C22×C32—21no—
237C22×C32—41no—
238C22×C32C23×D312yes—
239C22×C32C23×D311yescharacteristic
240C2×C3⋊D3—22no—
241C2×C3⋊D3—22no—
242C2×C3⋊D3—24no—
243C2×C3⋊D3C2×#6712yes—
244C2×C3×D3—34no—
245C2×C3×D3—62no—
246C2×C3×D3—64no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—68no—
250C2×C3×D3—122no—
251C2×C3×D3—124no—
252C32×C4—22no—
253C32×C4—24no—
254C32×C4—24no—
255C32×C4—24no—
256C32×C4—22no—
257C32×C4—42no—
258C32×C4—44no—
259C32×C4—44no—
260C32×C4D3×D414yes—
261C32×C4C2×#6712yes—
262C3⋊Dic3—24no—
263C3⋊Dic3—24no—
264C3⋊Dic3C2×#6712yes—
265C3×Dic3—34no—
266C3×Dic3—62no—
267C3×Dic3—68no—
268C3×Dic3—64no—
269C3×Dic3—64no—
270C3×Dic3—64no—
271C3×Dic3—64no—
272C3×Dic3—124no—
273C2×#41—91no—
274C22×Q8—271no—
275C4⋊Q8—92no—
276#124—272no—
277#123—274no—
278C22⋊Q8—274no—
279#117—271no—
280C33#53611yescharacteristic, complemented, Sylow
281C23×C3—91no—
282C23×C3—92no—
283C22×D3—32no—
284C22×D3—34no—
285C22×D3—61no—
286C22×D3—62no—
287C3×Q8—24no—
288C3×Q8—24no—
289C3×Q8—22no—
290C3×Q8—44no—
291C3×Q8—44no—
292C3×Q8—184no—
293C3×D4—184no—
294C3×D4—182no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—24no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—91no—
302C2×C3×C4—94no—
303C2×C3×C4—94no—
304C2×C3×C4—94no—
305C2×C3×C4—184no—
306C2×C3×C4C2×D3212yes—
307C2×C3×C4C2×D3214yes—
308C2×C3×C4C2×D3212yes—
309C2×C3×C4C2×D3211yescharacteristic
310C2×Dic3—32no—
311C2×Dic3—34no—
312C2×Dic3—34no—
313C2×Dic3—34no—
314C2×Dic3—34no—
315C2×Dic3—64no—
316C2×Dic3—61no—
317C2×Dic3—64no—
318C2×Dic3—184no—
319D12—68no—
320D12—68no—
321D12—64no—
322D12—64no—
323C4×D3—38no—
324C4×D3—64no—
325C4×D3—68no—
326C4×D3—68no—
327C4×D3—188no—
328Dic6—64no—
329Dic6—64no—
330Dic6—124no—
331Dic6—188no—
332Dic6—184no—
333C2×C32—24no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—41no—
338C2×C32—42no—
339C2×C32C2×D3×D412yes—
340C2×C32#100714yes—
341C2×C32#101012yes—
342C2×C32C22×#6711yescharacteristic
343C3⋊D3—22no—
344C3⋊D3—42no—
345C3×D3—64no—
346C3×D3—122no—
347C3×D3—124no—
348C3×D3—124no—
349#41—184no—
350C2×Q8—92no—
351C2×Q8—272no—
352C2×Q8—544no—
353C2×Q8#111311yescharacteristic
354C2×D4—92no—
355C2×D4—91no—
356C22×C4—91no—
357C22×C4—92no—
358C22×C4—272no—
359C4⋊C4—98no—
360C4⋊C4—272no—
361C22⋊C4—272no—
362C22⋊C4—278no—
363C42—92no—
364C22×C3—22no—
365C22×C3—21no—
366C22×C3—41no—
367C22×C3—92no—
368C22×C3—182no—
369C22×C3—184no—
370C22×C3—182no—
371C22×C3C22×D3212yes—
372C22×C3C22×D3211yescharacteristic
373C2×D3—34no—
374C2×D3—62no—
375C2×D3—64no—
376C2×D3—64no—
377C2×D3—68no—
378C2×D3—64no—
379C2×D3—122no—
380C2×D3—124no—
381C3×C4—24no—
382C3×C4—24no—
383C3×C4—22no—
384C3×C4—24no—
385C3×C4—22no—
386C3×C4—44no—
387C3×C4—44no—
388C3×C4—42no—
389C3×C4—92no—
390C3×C4—184no—
391C3×C4—184no—
392C3×C4D3×#6714yes—
393C3×C4#381112yes—
394Dic3—34no—
395Dic3—62no—
396Dic3—68no—
397Dic3—64no—
398Dic3—64no—
399Dic3—64no—
400Dic3—64no—
401Dic3—124no—
402C32—21no—
403C32—22no—
404C32—41no—
405C32#572412yescomplemented
406C32#582511yescharacteristic, complemented
407C23—91no—
408C23—92no—
409Q8—22no—
410Q8—184no—
411Q8—544no—
412D4—184no—
413D4—182no—
414C2×C4—91no—
415C2×C4—94no—
416C2×C4—94no—
417C2×C4—94no—
418C2×C4—184no—
419C2×C4—544no—
420C2×C4C2×#111312yes—
421C2×C4C2×#111311yescharacteristic
422C2×C3—24no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—22no—
426C2×C3—42no—
427C2×C3—41no—
428C2×C3—182no—
429C2×C3—362no—
430C2×C3#6431514yes—
431C2×C3#6431612yes—
432C2×C3C2×D3×#6712yes—
433C2×C3C2×#381111yescharacteristic
434D3—64no—
435D3—122no—
436D3—124no—
437D3—124no—
438C22—92no—
439C22—182no—
440C22—184no—
441C22—182no—
442C22C22×#111311yescharacteristic
443C4—22no—
444C4—92no—
445C4—184no—
446C4—184no—
447C4#8904912yes—
448C3—21no—
449C3—22no—
450C3—41no—
451C3#1059535011yescharacteristic, complemented
452C3#1059535612yescomplemented
453C2—182no—
454C2—362no—
455C2#1172462412yes—
456C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC2×C3×#3832C2
hypercenterC2×Q8#1113
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#89040C2semidirect
3. C3×#64153C2semidirect
4. #11723686C2semidirect
5. #11723677C2semidirect
6. #11723673C2semidirect
7. #11723671C2semidirect
30. #64153D3semidirect
280. C33#536semidirect
405. C32#5724semidirect
406. C32#5825semidirect
451. C3#10595350semidirect
452. C3#10595356semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×#890402
C3×#641532
#117236862
#117236772
#117236732
#117236712
#105953563
#105953503