Finite Group Databasebeta

← #13178628

Subgroups — #13178628

Order 1728.

1762 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 1762 conjugacy classes of subgroups — too many to draw. Only its 215 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×D4×C3⋊D3C211yesmaximal, characteristic
2C2×#89040C211yesmaximal, characteristic, complemented
3C3×#64145C212yesmaximal, complemented
4#11723723C214yesmaximal, complemented
5#11723718C214yesmaximal, complemented
6C4×#6294C211yesmaximal, characteristic, complemented
7#11723677C212yesmaximal, complemented
8#10595170—31nomaximal
9D3×#936—32nomaximal
10C23×C3×C3⋊D3C2212yes—
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638C2212yes—
13C3×D4×C3⋊D3—24no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2212yescomplemented
17C2×#6296C2212yescomplemented
18C2×#6294C2212yes—
19C2×#6294C2212yes—
20#89040—24no—
21C3×#3757C2218yescomplemented
22C3×C4×C3⋊Dic3C2212yescomplemented
23#88814C2218yescomplemented
24#88813C2212yescomplemented
25C2×D4×C3⋊D3D311yescharacteristic
26C2×C3×D3×D4—32no—
27C2×C3×D3×D4—61no—
28C2×#3797—31no—
29C2×D3×Dic6—32no—
30#64145D312yescomplemented
31C3×#936—32no—
32C3×#936—32no—
33C3×#936—62no—
34#63974—34no—
35D3×#217—38no—
36#63959—38no—
37#63953—34no—
38C4×#624—31no—
39C4×D3×Dic3—32no—
40#63864—32no—
41#63863—32no—
42#63860—32no—
43C23×C33C2312yes—
44C22×C3×C3⋊D3—24no—
45C22×C3×C3⋊D3—28no—
46C22×C3×C3⋊D3C2312yes—
47C22×C3×C3⋊D3C2311yescharacteristic
48C33×D4—22no—
49C2×C33×C4C2311yescharacteristic
50C3×#638—24no—
51C2×C3×C3⋊Dic3C2314yes—
52C2×C3×C3⋊Dic3C2314yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C3×C4⋊(C3⋊D3)—22no—
55C3×C4×C3⋊D3D414yes—
56C3×#634—24no—
57#6296—24no—
58#6294—24no—
59#6294—24no—
60C2×#936—91no—
61D3×#124—92no—
62C23×C3⋊D3C2×D312yes—
63C23×C3×D3—34no—
64C23×C3×D3—62no—
65C2×C32×D4—21no—
66C2×C32×D4—22no—
67C2×C32×D4—41no—
68C2×C32×D4C2×D311yescharacteristic
69C2×C32×D4C2×D312yes—
70C2×#638C2×D312yes—
71D4×C3⋊D3—24no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×#634C2×D312yescomplemented
75C2×C3×#67—34no—
76C2×C3×#67—62no—
77C3×D3×D4—68no—
78C3×D3×D4—68no—
79C2×C3×D12—32no—
80C2×C3×D12—61no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—61no—
83C2×C3×Dic6—32no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6—62no—
86C2×#627—34no—
87C2×#627—32no—
88C2×#624—32no—
89C2×#624—32no—
90C2×D3×Dic3—34no—
91C2×D3×Dic3—34no—
92#3797—64no—
93D3×Dic6—68no—
94#3757C2×D318yescomplemented
95C4×C3⋊Dic3C2×D312yescomplemented
96C3×#217—38no—
97C3×#217—38no—
98C3×#217—68no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—32no—
101C3×C4×Dic3—62no—
102#3722—38no—
103(C2×D3)⋊Dic3—38no—
104(C2×D3)⋊Dic3—38no—
105Dic32—34no—
106Dic32—32no—
107C22×C33—22no—
108C22×C33—24no—
109C22×C33C2411yescharacteristic
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3—24no—
112C2×C3×C3⋊D3—48no—
113C2×C3×C3⋊D3C2×D412yes—
114C2×C3×C3⋊D3#4114yes—
115C33×C4C2×D412yes—
116C3×C3⋊Dic3—24no—
117C3×C3⋊Dic3—24no—
118C3×C3⋊Dic3C2×D412yes—
119C22×C3×D4—91no—
120C2×D3×Q8—92no—
121C2×D3×D4—32no—
122C2×D3×D4—61no—
123C22×Dic6—91no—
124C3×#124—92no—
125C2×#217—94no—
126#947—92no—
127#936—32no—
128#936—32no—
129#936—62no—
130#936—92no—
131#936—94no—
132C2×C4×Dic3—91no—
133#885—98no—
134D3×C22⋊C4—98no—
135#875—92no—
136C42×D3—92no—
137C23×C32—22no—
138C23×C32—24no—
139C23×C32—42no—
140C23×C32C22×D314yes—
141C23×C32C22×D312yes—
142C22×C3⋊D3—24no—
143C22×C3⋊D3—28no—
144C22×C3⋊D3C22×D312yes—
145C22×C3⋊D3C22×D311yescharacteristic
146C22×C3×D3—32no—
147C22×C3×D3—34no—
148C22×C3×D3—61no—
149C22×C3×D3—62no—
150C22×C3×D3—616no—
151C22×C3×D3—68no—
152C22×C3×D3—68no—
153C22×C3×D3—128no—
154C32×D4—22no—
155C32×D4—24no—
156C32×D4—24no—
157C32×D4—44no—
158C32×D4—44no—
159C2×C32×C4—22no—
160C2×C32×C4—21no—
161C2×C32×C4—41no—
162C2×C32×C4C22×D312yes—
163C2×C32×C4C22×D311yescharacteristic
164#638—24no—
165C2×C3⋊Dic3C22×D314yes—
166C2×C3⋊Dic3C22×D314yes—
167C2×C3⋊Dic3C22×D311yescharacteristic
168C4⋊(C3⋊D3)—22no—
169C4×C3⋊D3#6714yes—
170#634—24no—
171C3×#67—68no—
172C3×#67—68no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—34no—
175C2×C3×Dic3—34no—
176C2×C3×Dic3—34no—
177C2×C3×Dic3—34no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—64no—
180C2×C3×Dic3—64no—
181C3×D12—64no—
182C3×D12—64no—
183C3×C4×D3—38no—
184C3×C4×D3—64no—
185C3×Dic6—64no—
186C3×Dic6—64no—
187C3×Dic6—124no—
188#627—68no—
189#627—64no—
190#624—64no—
191#624—64no—
192D3×Dic3—68no—
193D3×Dic3—68no—
194C2×#124—271noSylow
195C2×C33—42no—
196C2×C33C22×D411yescharacteristic
197C2×C33C2×#4112yes—
198C3×C3⋊D3—42no—
199C3×C3⋊D3#12414yescomplemented
200C24×C3—92no—
201C23×D3—34no—
202C23×D3—62no—
203C2×C3×Q8—92no—
204C2×C3×D4—21no—
205C2×C3×D4—22no—
206C2×C3×D4—41no—
207C2×C3×D4—92no—
208C2×C3×D4—91no—
209C2×C3×D4—184no—
210C2×C3×D4D3211yescharacteristic
211C2×C3×D4D3212yes—
212C22×C3×C4—91no—
213C2×#67—34no—
214C2×#67—62no—
215C22×Dic3—92no—
216C22×Dic3—92no—
217D3×Q8—188no—
218D3×D4—68no—
219D3×D4—68no—
220C2×D12—32no—
221C2×D12—61no—
222C2×C4×D3—32no—
223C2×C4×D3—61no—
224C2×C4×D3—94no—
225C2×C4×D3—94no—
226C2×Dic6—32no—
227C2×Dic6—32no—
228C2×Dic6—62no—
229C2×Dic6—94no—
230C2×Dic6—92no—
231C2×Dic6—184no—
232C3×C22⋊C4—98no—
233C3×C42—92no—
234#217—38no—
235#217—38no—
236#217—68no—
237#217—98no—
238(C2×D3)⋊C4—98no—
239(C2×D3)⋊C4—98no—
240C4×Dic3—32no—
241C4×Dic3—32no—
242C4×Dic3—62no—
243C4×Dic3—94no—
244C4×Dic3—92no—
245C22×C32—22no—
246C22×C32—24no—
247C22×C32—24no—
248C22×C32—24no—
249C22×C32—28no—
250C22×C32—22no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32—44no—
254C22×C32—44no—
255C22×C32—44no—
256C22×C32—48no—
257C22×C32—84no—
258C22×C32C23×D312yes—
259C22×C32C23×D311yescharacteristic
260C2×C3⋊D3—22no—
261C2×C3⋊D3—24no—
262C2×C3⋊D3—48no—
263C2×C3⋊D3#23714yes—
264C2×C3⋊D3C2×#6712yes—
265C2×C3×D3—34no—
266C2×C3×D3—38no—
267C2×C3×D3—62no—
268C2×C3×D3—64no—
269C2×C3×D3—64no—
270C2×C3×D3—64no—
271C2×C3×D3—68no—
272C2×C3×D3—124no—
273C2×C3×D3—1216no—
274C2×C3×D3—1216no—
275C32×C4—24no—
276C32×C4—22no—
277C32×C4—42no—
278C32×C4D3×D414yes—
279C32×C4C2×#6712yes—
280C3⋊Dic3—24no—
281C3⋊Dic3—24no—
282C3⋊Dic3C2×#6712yes—
283C3×Dic3—34no—
284C3×Dic3—62no—
285C3×Dic3—64no—
286C3×Dic3—64no—
287C3×Dic3—64no—
288C3×Dic3—64no—
289C3×Dic3—124no—
290C3×Dic3—124no—
291C22×Q8—271no—
292C22×D4—91no—
293#124—92no—
294#124—272no—
295#124—274no—
296C2×C22⋊C4—274no—
297C2×C42—271no—
298C33C2×#12411yescharacteristic, complemented, Sylow
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—42no—
302C23×C3—91no—
303C23×C3—92no—
304C23×C3—184no—
305C23×C3—188no—
306C23×C3C2×D3212yes—
307C23×C3C2×D3214yes—
308C22×D3—32no—
309C22×D3—34no—
310C22×D3—61no—
311C22×D3—62no—
312C22×D3—68no—
313C22×D3—616no—
314C22×D3—68no—
315C22×D3—128no—
316C3×Q8—184no—
317C3×D4—24no—
318C3×D4—24no—
319C3×D4—22no—
320C3×D4—44no—
321C3×D4—44no—
322C3×D4—184no—
323C3×D4—182no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
327C2×C3×C4—91no—
328C2×C3×C4—94no—
329C2×C3×C4—94no—
330C2×C3×C4—94no—
331C2×C3×C4C2×D3212yes—
332C2×C3×C4C2×D3211yescharacteristic
333#67—68no—
334#67—68no—
335C2×Dic3—32no—
336C2×Dic3—34no—
337C2×Dic3—34no—
338C2×Dic3—34no—
339C2×Dic3—34no—
340C2×Dic3—61no—
341C2×Dic3—64no—
342C2×Dic3—64no—
343C2×Dic3—184no—
344C2×Dic3—184no—
345D12—64no—
346D12—64no—
347C4×D3—38no—
348C4×D3—64no—
349C4×D3—188no—
350C4×D3—188no—
351Dic6—64no—
352Dic6—64no—
353Dic6—124no—
354Dic6—188no—
355Dic6—184no—
356C2×C32—24no—
357C2×C32—22no—
358C2×C32—22no—
359C2×C32—21no—
360C2×C32—41no—
361C2×C32—42no—
362C2×C32—42no—
363C2×C32—44no—
364C2×C32—44no—
365C2×C32—84no—
366C2×C32—84no—
367C2×C32C2×D3×D412yes—
368C2×C32C2×#23712yes—
369C2×C32D3×#4114yes—
370C2×C32C22×#6711yescharacteristic
371C3⋊D3—42no—
372C3⋊D3#93614yescomplemented
373C3×D3—38no—
374C3×D3—64no—
375C3×D3—124no—
376C3×D3—124no—
377C24—92no—
378C2×Q8—92no—
379C2×Q8—272no—
380C2×Q8—544no—
381C2×D4—92no—
382C2×D4—91no—
383C2×D4—184no—
384C2×D4#111311yescharacteristic
385C22×C4—91no—
386C22×C4—272no—
387C22×C4—272no—
388C22⋊C4—98no—
389C22⋊C4—278no—
390C42—92no—
391C42—272no—
392C22×C3—24no—
393C22×C3—28no—
394C22×C3—24no—
395C22×C3—24no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—21no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—48no—
402C22×C3—44no—
403C22×C3—41no—
404C22×C3—84no—
405C22×C3—94no—
406C22×C3—92no—
407C22×C3—184no—
408C22×C3—182no—
409C22×C3—368no—
410C22×C3C22×D3212yes—
411C22×C3C22×D3211yescharacteristic
412C2×D3—38no—
413C2×D3—34no—
414C2×D3—62no—
415C2×D3—64no—
416C2×D3—64no—
417C2×D3—68no—
418C2×D3—64no—
419C2×D3—124no—
420C2×D3—1216no—
421C2×D3—1216no—
422C3×C4—24no—
423C3×C4—22no—
424C3×C4—42no—
425C3×C4—92no—
426C3×C4—184no—
427C3×C4—184no—
428C3×C4D3×#6714yes—
429C3×C4#381112yes—
430Dic3—34no—
431Dic3—62no—
432Dic3—64no—
433Dic3—64no—
434Dic3—64no—
435Dic3—64no—
436Dic3—124no—
437Dic3—124no—
438C32—22no—
439C32—21no—
440C32—41no—
441C32D3×#12412yescomplemented
442C32C2×#93611yescharacteristic, complemented
443C23—91no—
444C23—92no—
445C23—184no—
446C23—188no—
447C23C2×#111312yes—
448Q8—184no—
449Q8—544no—
450D4—22no—
451D4—184no—
452D4—182no—
453C2×C4—91no—
454C2×C4—94no—
455C2×C4—94no—
456C2×C4—94no—
457C2×C4—544no—
458C2×C4—544no—
459C2×C4C2×#111311yescharacteristic
460C2×C3—24no—
461C2×C3—22no—
462C2×C3—21no—
463C2×C3—22no—
464C2×C3—44no—
465C2×C3—44no—
466C2×C3—42no—
467C2×C3—42no—
468C2×C3—41no—
469C2×C3—84no—
470C2×C3—84no—
471C2×C3—94no—
472C2×C3—362no—
473C2×C3D3×#23714yes—
474C2×C3#6431212yes—
475C2×C3C2×D3×#6712yes—
476C2×C3C2×#381111yescharacteristic
477D3—38no—
478D3—64no—
479D3—124no—
480D3—124no—
481C22—24no—
482C22—22no—
483C22—94no—
484C22—92no—
485C22—184no—
486C22—182no—
487C22—368no—
488C22C22×#111311yescharacteristic
489C4—92no—
490C4—184no—
491C4—184no—
492C4#8904912yes—
493C3—21no—
494C3—22no—
495C3—41no—
496C3D3×#93612yescomplemented
497C3#1059517011yescharacteristic, complemented
498C2—42no—
499C2—94no—
500C2—362no—
501C2#1172462212yes—
502C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC2×C3×D4×C3⋊D3C2
hypercenterC2×D4#1113
hypocenterC33C2×#124
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. #11723723C2semidirect
5. #11723718C2semidirect
6. C4×#6294C2semidirect
7. #11723677C2semidirect
16. C2×C3×#634C22semidirect
17. C2×#6296C22semidirect
21. C3×#3757C22semidirect
22. C3×C4×C3⋊Dic3C22semidirect
23. #88814C22semidirect
24. #88813C22semidirect
30. #64145D3semidirect
74. C2×#634C2×D3semidirect
94. #3757C2×D3semidirect
95. C4×C3⋊Dic3C2×D3semidirect
199. C3×C3⋊D3#124semidirect
298. C33C2×#124semidirect
372. C3⋊D3#936semidirect
441. C32D3×#124semidirect
442. C32C2×#936semidirect
496. C3D3×#936semidirect
497. C3#10595170semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#890402
C3×#641452
#117237232
#117237182
C4×#62942
#117236772
#105951703
D3×#9363