Finite Group Databasebeta

← #13178629

Subgroups — #13178629

Order 1728.

1450 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1450 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×#3830C211yesmaximal, characteristic
2C2×#89040C211yesmaximal, characteristic, complemented
3C3×#64145C212yesmaximal, complemented
4#11723724C214yesmaximal, complemented
5#11723721C214yesmaximal, complemented
6#11723678C212yesmaximal, complemented
7#11723673C211yesmaximal, characteristic, complemented
8#10595171—31nomaximal
9#10595147—32nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638C2212yes—
12C22×C3×C3⋊Dic3C2212yes—
13C3×#3830—24no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634C2212yes—
16C2×C3×#634C2211yescharacteristic
17C2×#6296C2212yes—
18C2×#6294C2212yes—
19#89040—24no—
20C3×#3757C2218yes—
21C3×C4×C3⋊Dic3C2212yes—
22#88815C2212yes—
23#88815C2218yes—
24#88814C2212yes—
25C2×#3830D311yescharacteristic
26C2×C3×#237—32no—
27C2×C3×#237—61no—
28C2×#3797—31no—
29C2×D3×Dic6—32no—
30#64145D312yescomplemented
31C3×#936—32no—
32C3×#936—32no—
33C3×#936—62no—
34#63982—34no—
35#63975—34no—
36#63969—38no—
37#63964—38no—
38#63874—31no—
39#63873—32no—
40#63866—32no—
41#63866—32no—
42#63855—32no—
43C23×C33C2312yes—
44C22×C3×C3⋊D3C2311yescharacteristic
45C33×D4—22no—
46C2×C33×C4C2311yescharacteristic
47C3×#638—24no—
48C2×C3×C3⋊Dic3—24no—
49C2×C3×C3⋊Dic3C2314yes—
50C2×C3×C3⋊Dic3C2314yes—
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C3×C4×C3⋊D3D414yes—
54C3×#634—22no—
55C3×#634—24no—
56#6296—24no—
57#6294—24no—
58#5825—91no—
59#5675—92no—
60C2×C32×D4—21no—
61C2×C32×D4—22no—
62C2×C32×D4—41no—
63C2×C32×D4C2×D311yescharacteristic
64C2×C32×D4C2×D312yes—
65C2×#638C2×D312yes—
66C22×C3⋊Dic3C2×D312yes—
67#3830—24no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×#634C2×D312yes—
70C2×#634C2×D311yescharacteristic
71C2×C3×#67—34no—
72C2×C3×#67—62no—
73C22×C3×Dic3—34no—
74C22×C3×Dic3—62no—
75C3×#237—68no—
76C3×#237—68no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—61no—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—32no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—61no—
83C2×C3×Dic6—62no—
84C2×#627—34no—
85C2×#627—32no—
86C2×#624—32no—
87C2×D3×Dic3—34no—
88#3797—64no—
89D3×Dic6—68no—
90#3757C2×D318yes—
91C4×C3⋊Dic3C2×D312yes—
92C3×#217—38no—
93C3×#217—38no—
94C3×#217—68no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—32no—
97C3×C4×Dic3—62no—
98#3724—38no—
99#3724—32no—
100Dic3⋊Dic3—34no—
101Dic3⋊Dic3—38no—
102Dic3⋊Dic3—38no—
103#3722—32no—
104(C2×D3)⋊Dic3—34no—
105C22×C33—22no—
106C22×C33—24no—
107C22×C33C2411yescharacteristic
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3C2×D412yes—
110C33×C4C2×D412yes—
111C3×C3⋊Dic3—22no—
112C3×C3⋊Dic3—24no—
113C3×C3⋊Dic3—24no—
114C3×C3⋊Dic3C2×D412yes—
115C2×C3×#41—91no—
116C2×D3×Q8—92no—
117C2×#237—32no—
118C2×#237—61no—
119C22×Dic6—91no—
120C3×#124—92no—
121#944—92no—
122#944—92no—
123#936—32no—
124#936—32no—
125#936—62no—
126#935—94no—
127#926—91no—
128#924—94no—
129#883—98no—
130#878—98no—
131#872—92no—
132#869—92no—
133C23×C32—22no—
134C23×C32—24no—
135C23×C32—42no—
136C23×C32C22×D312yes—
137C23×C32C22×D314yes—
138C22×C3⋊D3C22×D311yescharacteristic
139C22×C3×D3—32no—
140C22×C3×D3—61no—
141C32×D4—22no—
142C32×D4—24no—
143C32×D4—24no—
144C32×D4—44no—
145C32×D4—44no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D311yescharacteristic
151#638—24no—
152C2×C3⋊Dic3—24no—
153C2×C3⋊Dic3C22×D314yes—
154C2×C3⋊Dic3C22×D312yes—
155C2×C3⋊Dic3C22×D314yes—
156C2×C3⋊Dic3C22×D311yescharacteristic
157C4×C3⋊D3#6714yes—
158#634—22no—
159#634—24no—
160C3×#67—68no—
161C3×#67—68no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3—68no—
173C2×C3×Dic3—68no—
174C3×C4×D3—38no—
175C3×C4×D3—64no—
176C3×Dic6—64no—
177C3×Dic6—64no—
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×C33—42no—
187C2×C33C22×D411yescharacteristic
188C2×C332−1+412yes—
189C3×C3⋊D3—22no—
190C3×#41—184no—
191C2×C3×Q8—91no—
192C2×C3×Q8—92no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—41no—
196C2×C3×D4—92no—
197C2×C3×D4D3211yescharacteristic
198C2×C3×D4D3212yes—
199C22×C3×C4—91no—
200C22×C3×C4—92no—
201C2×#67—34no—
202C2×#67—62no—
203C22×Dic3—34no—
204C22×Dic3—62no—
205C22×Dic3—92no—
206D3×Q8—188no—
207#237—68no—
208#237—68no—
209C2×C4×D3—32no—
210C2×C4×D3—61no—
211C2×C4×D3—94no—
212C2×Dic6—32no—
213C2×Dic6—32no—
214C2×Dic6—32no—
215C2×Dic6—61no—
216C2×Dic6—62no—
217C2×Dic6—94no—
218C2×Dic6—92no—
219C2×Dic6—184no—
220C3×C22⋊C4—98no—
221C3×C42—92no—
222#217—38no—
223#217—38no—
224#217—68no—
225#217—92no—
226(C2×D3)⋊C4—94no—
227C4⋊Dic3—98no—
228C4⋊Dic3—92no—
229Dic3⋊C4—94no—
230Dic3⋊C4—98no—
231Dic3⋊C4—98no—
232C4×Dic3—32no—
233C4×Dic3—32no—
234C4×Dic3—62no—
235C22×C32—22no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—24no—
239C22×C32—28no—
240C22×C32—22no—
241C22×C32—21no—
242C22×C32—41no—
243C22×C32—44no—
244C22×C32—44no—
245C22×C32—44no—
246C22×C32—48no—
247C22×C32—84no—
248C22×C32C23×D312yes—
249C22×C32C23×D311yescharacteristic
250C2×C3⋊D3—22no—
251C2×C3⋊D3C2×#6712yes—
252C2×C3×D3—34no—
253C2×C3×D3—62no—
254C2×C3×D3—64no—
255C2×C3×D3—122no—
256C32×C4—24no—
257C32×C4—22no—
258C32×C4—42no—
259C32×C4D3×D414yes—
260C32×C4C2×#6712yes—
261C3⋊Dic3—22no—
262C3⋊Dic3—24no—
263C3⋊Dic3—24no—
264C3⋊Dic3C2×#6712yes—
265C3×Dic3—34no—
266C3×Dic3—62no—
267C3×Dic3—64no—
268C3×Dic3—64no—
269C3×Dic3—64no—
270C3×Dic3—64no—
271C3×Dic3—64no—
272C3×Dic3—64no—
273C3×Dic3—124no—
274C3×Dic3—124no—
275C2×#41—91no—
276C22×Q8—271no—
277C4⋊Q8—272no—
278#124—92no—
279#123—274no—
280C22⋊Q8—274no—
281#117—271no—
282C33#53611yescharacteristic, complemented, Sylow
283C23×C3—22no—
284C23×C3—24no—
285C23×C3—42no—
286C23×C3—91no—
287C23×C3C2×D3212yes—
288C23×C3C2×D3214yes—
289C22×D3—32no—
290C22×D3—61no—
291C3×Q8—184no—
292C3×Q8—182no—
293C3×D4—24no—
294C3×D4—24no—
295C3×D4—22no—
296C3×D4—44no—
297C3×D4—44no—
298C3×D4—184no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—41no—
302C2×C3×C4—91no—
303C2×C3×C4—94no—
304C2×C3×C4—92no—
305C2×C3×C4—94no—
306C2×C3×C4—94no—
307C2×C3×C4—184no—
308C2×C3×C4C2×D3212yes—
309C2×C3×C4C2×D3211yescharacteristic
310#67—68no—
311#67—68no—
312C2×Dic3—32no—
313C2×Dic3—34no—
314C2×Dic3—34no—
315C2×Dic3—34no—
316C2×Dic3—34no—
317C2×Dic3—34no—
318C2×Dic3—62no—
319C2×Dic3—61no—
320C2×Dic3—64no—
321C2×Dic3—64no—
322C2×Dic3—68no—
323C2×Dic3—68no—
324C2×Dic3—184no—
325C4×D3—38no—
326C4×D3—64no—
327C4×D3—188no—
328Dic6—64no—
329Dic6—64no—
330Dic6—64no—
331Dic6—64no—
332Dic6—124no—
333Dic6—188no—
334Dic6—184no—
335C2×C32—24no—
336C2×C32—22no—
337C2×C32—22no—
338C2×C32—21no—
339C2×C32—41no—
340C2×C32—42no—
341C2×C32—42no—
342C2×C32—44no—
343C2×C32—44no—
344C2×C32—84no—
345C2×C32—84no—
346C2×C32C2×D3×D412yes—
347C2×C32#101014yes—
348C2×C32#101012yes—
349C2×C32C22×#6711yescharacteristic
350C3⋊D3—22no—
351C3×D3—64no—
352C3×D3—122no—
353#41—184no—
354C2×Q8—92no—
355C2×Q8—91no—
356C2×Q8—272no—
357C2×Q8—544no—
358C2×D4—92no—
359C2×D4#111311yescharacteristic
360C22×C4—91no—
361C22×C4—92no—
362C22×C4—272no—
363C4⋊C4—272no—
364C4⋊C4—278no—
365C22⋊C4—98no—
366C22⋊C4—272no—
367C42—92no—
368C22×C3—24no—
369C22×C3—28no—
370C22×C3—24no—
371C22×C3—24no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3—48no—
378C22×C3—44no—
379C22×C3—41no—
380C22×C3—84no—
381C22×C3—92no—
382C22×C3—182no—
383C22×C3C22×D3212yes—
384C22×C3C22×D3211yescharacteristic
385C2×D3—34no—
386C2×D3—62no—
387C2×D3—64no—
388C2×D3—122no—
389C3×C4—22no—
390C3×C4—24no—
391C3×C4—42no—
392C3×C4—92no—
393C3×C4—184no—
394C3×C4—184no—
395C3×C4—182no—
396C3×C4D3×#6714yes—
397C3×C4#381112yes—
398Dic3—34no—
399Dic3—62no—
400Dic3—64no—
401Dic3—64no—
402Dic3—64no—
403Dic3—64no—
404Dic3—64no—
405Dic3—64no—
406Dic3—124no—
407Dic3—124no—
408C32—22no—
409C32—21no—
410C32—41no—
411C32#567512yescomplemented
412C32#582511yescharacteristic, complemented
413C23—91no—
414C23C2×#111312yes—
415Q8—184no—
416Q8—182no—
417Q8—544no—
418D4—22no—
419D4—184no—
420C2×C4—91no—
421C2×C4—94no—
422C2×C4—94no—
423C2×C4—94no—
424C2×C4—92no—
425C2×C4—184no—
426C2×C4—544no—
427C2×C4C2×#111311yescharacteristic
428C2×C3—24no—
429C2×C3—22no—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—44no—
433C2×C3—44no—
434C2×C3—42no—
435C2×C3—42no—
436C2×C3—41no—
437C2×C3—84no—
438C2×C3—84no—
439C2×C3—182no—
440C2×C3#6430914yes—
441C2×C3#6430912yes—
442C2×C3C2×D3×#6712yes—
443C2×C3C2×#381111yescharacteristic
444D3—64no—
445D3—122no—
446C22—24no—
447C22—22no—
448C22—92no—
449C22—182no—
450C22C22×#111311yescharacteristic
451C4—92no—
452C4—184no—
453C4—182no—
454C4—184no—
455C4#8904912yes—
456C3—21no—
457C3—22no—
458C3—41no—
459C3#1059514712yescomplemented
460C3#1059517111yescharacteristic, complemented
461C2—42no—
462C2—182no—
463C2#1172462012yes—
464C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC2×C3×#3830C2
hypercenterC2×D4#1113
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#89040C2semidirect
3. C3×#64145C2semidirect
4. #11723724C2semidirect
5. #11723721C2semidirect
6. #11723678C2semidirect
7. #11723673C2semidirect
30. #64145D3semidirect
282. C33#536semidirect
411. C32#5675semidirect
412. C32#5825semidirect
459. C3#10595147semidirect
460. C3#10595171semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#890402
C3×#641452
#117237242
#117237212
#117236782
#117236732
#105951713
#105951473