Finite Group Databasebeta

← D4×#6189 #13150905

Subgroups — D4×#6189 #13150905

Order 1728.

934 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6189C212yesmaximal, complemented
2D4×C3⋊Dic9C211yesmaximal, characteristic
3D4×C9×Dic3C211yesmaximal, characteristic
4C3×D4×Dic9C211yesmaximal, characteristic
5#11721267C212yesmaximal, complemented
6#11721253C212yesmaximal, complemented
7#11721249C212yesmaximal, complemented
8#11720971C211yesmaximal, characteristic, complemented
9#11720970C211yesmaximal, characteristic, complemented
10#11720969C211yesmaximal, characteristic, complemented
11C4×#6189C211yesmaximal, characteristic, complemented
12D4×#627—31nomaximal
13D4×Dic18—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C22×C9×Dic3C2212yes—
17C22×C3×Dic9C2212yes—
18C2×#6189—28no—
19C2×#6189C2212yes—
20C2×#6189C2212yes—
21C2×#6189C2212yes—
22C2×#6189C2211yescharacteristic
23#88551C2212yes—
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25C4×C3⋊Dic9C2211yescharacteristic
26C9×#217C2212yes—
27C3×#3676C2212yes—
28C9×C4⋊Dic3C2211yescharacteristic
29C4×C9×Dic3C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32Dic3⋊Dic9C2211yescharacteristic
33Dic3⋊Dic9C2212yes—
34#88442C2211yescharacteristic
35#88442C2212yes—
36Dic9⋊Dic3C2211yescharacteristic
37Dic9⋊Dic3C2212yes—
38C22×#627—32no—
39D4×C3⋊Dic3—31no—
40C3×D4×Dic3—31no—
41C3×D4×Dic3D311yescharacteristic
42#63982—32no—
43#63964—32no—
44#63964—32no—
45#63919—31no—
46#63918—31no—
47#63918—31no—
48C4×#627—31no—
49C22×Dic18—32no—
50C4×D4×C9—31no—
51D4×Dic9—31no—
52D4×Dic9—61no—
53D4×Dic9D311yescharacteristic
54#63486—32no—
55#63450—31no—
56#63450—31no—
57#63440—32no—
58#63440—32no—
59#63431—31no—
60C4×Dic18—31no—
61C23×C3×C9C2312yes—
62C3×D4×C9Q814yescomplemented
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9—24no—
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C9×Dic3—24no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9—24no—
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76#6189—24no—
77#6189—24no—
78#6189—24no—
79#6189D414yesdirect product factor
80D4×Dic6—91no—
81D4×Dic6—91no—
82C2×C32×D4C2×D311yescharacteristic
83C22×C3⋊Dic3—32no—
84C22×C3×Dic3—32no—
85C22×C3×Dic3C2×D312yes—
86C2×#627—31no—
87C2×#627—32no—
88C2×#627—32no—
89C2×#627—32no—
90C2×#627—68no—
91#3757—32no—
92C4⋊(C3⋊Dic3)—31no—
93C4×C3⋊Dic3—31no—
94C3×#217—32no—
95C3×#217C2×D312yes—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4×Dic3—31no—
99C3×C4×Dic3C2×D311yescharacteristic
100#3724—31no—
101#3724—32no—
102Dic3⋊Dic3—31no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—31no—
105Dic3⋊Dic3—32no—
106C2×D4×C9—21no—
107C2×D4×C9C2×D311yescharacteristic
108C22×C4×C9—32no—
109C22×Dic9—32no—
110C22×Dic9—62no—
111C22×Dic9C2×D312yes—
112C2×Dic18—31no—
113C2×Dic18—32no—
114C2×Dic18—32no—
115C2×Dic18—32no—
116C2×Dic18—68no—
117C9×C4⋊C4—31no—
118C9×C22⋊C4—32no—
119C42×C9—31no—
120#3676—32no—
121#3676—62no—
122#3676C2×D312yes—
123C4⋊Dic9—31no—
124C4⋊Dic9—31no—
125C4⋊Dic9—32no—
126C4⋊Dic9—61no—
127C4⋊Dic9C2×D311yescharacteristic
128Dic9⋊C4—31no—
129Dic9⋊C4—32no—
130Dic9⋊C4—31no—
131Dic9⋊C4—32no—
132C4×Dic9—31no—
133C4×Dic9—61no—
134C4×Dic9C2×D311yescharacteristic
135C22×C3×C9—22no—
136C22×C3×C9—22no—
137C22×C3×C9C2×Q814yescomplemented
138C22×C3×C9C2411yescharacteristic
139C3×C4×C9C2×Q812yescomplemented
140C3⋊Dic9—22no—
141C3⋊Dic9—21no—
142C3⋊Dic9C2×D412yes—
143C9×Dic3—22no—
144C9×Dic3—21no—
145C9×Dic3C2×D412yes—
146C3×Dic9—22no—
147C3×Dic9—21no—
148C3×Dic9C2×D412yes—
149C22×Dic6—92no—
150C22×Dic6—92no—
151C3×C4×D4—31no—
152C3×C4×D4—91no—
153D4×Dic3—31no—
154D4×Dic3—91no—
155D4×Dic3—91no—
156D4×Dic3—181no—
157D4×Dic3D911yescharacteristic
158#924—92no—
159#924—92no—
160#888—91no—
161#888—91no—
162#888—91no—
163#888—91no—
164#878—92no—
165#878—92no—
166#878—92no—
167#878—92no—
168#869—91no—
169#869—91no—
170C4×Dic6—91no—
171C4×Dic6—91no—
172C23×C32C22×D312yes—
173C32×D4Dic614yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—32no—
178C2×C3⋊Dic3—64no—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—64no—
184C2×C3×Dic3C22×D311yescharacteristic
185C2×C3×Dic3C22×D312yes—
186C2×C3×Dic3C22×D311yescharacteristic
187#627—34no—
188#627—64no—
189#627—64no—
190#627—64no—
191C23×C9—22no—
192C23×C9C22×D312yes—
193D4×C9—24no—
194D4×C9Dic614yescomplemented
195C2×C4×C9—21no—
196C2×C4×C9—31no—
197C2×C4×C9—31no—
198C2×C4×C9—32no—
199C2×C4×C9—64no—
200C2×C4×C9C22×D311yescharacteristic
201C2×Dic9—24no—
202C2×Dic9—31no—
203C2×Dic9—31no—
204C2×Dic9—32no—
205C2×Dic9—61no—
206C2×Dic9—61no—
207C2×Dic9—62no—
208C2×Dic9—64no—
209C2×Dic9—124no—
210C2×Dic9C22×D311yescharacteristic
211C2×Dic9C22×D312yes—
212C2×Dic9C22×D311yescharacteristic
213Dic18—34no—
214Dic18—64no—
215Dic18—64no—
216Dic18—64no—
217D4×Q8—271noSylow
218C2×C3×C9—24no—
219C2×C3×C9C22×D411yescharacteristic
220C2×C3×C9C22×Q811yescharacteristic
221C2×C3×C92−1+411yescharacteristic
222C2×C3×D4—21no—
223C2×C3×D4C2×D911yescharacteristic
224C2×C3×D4D3211yescharacteristic
225C22×C3×C4—32no—
226C22×C3×C4—92no—
227C22×Dic3—32no—
228C22×Dic3—92no—
229C22×Dic3—92no—
230C22×Dic3—182no—
231C22×Dic3C2×D912yes—
232C2×Dic6—91no—
233C2×Dic6—91no—
234C2×Dic6—92no—
235C2×Dic6—92no—
236C2×Dic6—92no—
237C2×Dic6—92no—
238C2×Dic6—92no—
239C2×Dic6—92no—
240C2×Dic6—188no—
241C2×Dic6—188no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—91no—
244C3×C22⋊C4—32no—
245C3×C22⋊C4—92no—
246C3×C42—31no—
247C3×C42—91no—
248#217—32no—
249#217—92no—
250#217—92no—
251#217—182no—
252#217C2×D912yes—
253C4⋊Dic3—31no—
254C4⋊Dic3—91no—
255C4⋊Dic3—92no—
256C4⋊Dic3—91no—
257C4⋊Dic3—91no—
258C4⋊Dic3—91no—
259C4⋊Dic3—92no—
260C4⋊Dic3—181no—
261C4⋊Dic3C2×D911yescharacteristic
262Dic3⋊C4—91no—
263Dic3⋊C4—91no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—91no—
267Dic3⋊C4—91no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270C4×Dic3—31no—
271C4×Dic3—91no—
272C4×Dic3—91no—
273C4×Dic3—181no—
274C4×Dic3C2×D911yescharacteristic
275C22×C32—22no—
276C22×C32—22no—
277C22×C32C2×Dic614yes—
278C22×C32C23×D311yescharacteristic
279C32×C4C2×Dic612yes—
280C3⋊Dic3—32no—
281C3⋊Dic3—62no—
282C3⋊Dic3—61no—
283C3×Dic3—22no—
284C3×Dic3—21no—
285C3×Dic3—32no—
286C3×Dic3—62no—
287C3×Dic3—61no—
288C3×Dic3D3×D412yes—
289C22×C9—22no—
290C22×C9—22no—
291C22×C9—21no—
292C22×C9—24no—
293C22×C9—42no—
294C22×C9—42no—
295C22×C9C2×Dic614yescomplemented
296C22×C9C23×D311yescharacteristic
297C4×C9—22no—
298C4×C9—32no—
299C4×C9—62no—
300C4×C9—61no—
301C4×C9C2×Dic612yescomplemented
302Dic9—22no—
303Dic9—21no—
304Dic9—32no—
305Dic9—62no—
306Dic9—62no—
307Dic9—61no—
308Dic9—121no—
309Dic9—122no—
310Dic9D3×D412yes—
311C22×Q8—272no—
312C4⋊Q8—271no—
313C4⋊Q8—271no—
314C4⋊Q8—271no—
315C22⋊Q8—272no—
316C22⋊Q8—272no—
317C22⋊Q8—272no—
318C4×Q8—271no—
319C4×D4—31no—
320C4×D4—91no—
321C4×D4—271no—
322C3×C9D4×Q811yescharacteristic, complemented, Sylow
323C23×C3—22no—
324C23×C3C22×D912yes—
325C23×C3C2×D3212yes—
326C3×D4—24no—
327C3×D4Dic1814yescomplemented
328C3×D4#62714yes—
329C2×C3×C4—21no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—32no—
333C2×C3×C4—64no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4—92no—
337C2×C3×C4—184no—
338C2×C3×C4C22×D911yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×Dic3—24no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—32no—
344C2×Dic3—64no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—91no—
348C2×Dic3—92no—
349C2×Dic3—91no—
350C2×Dic3—92no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—182no—
354C2×Dic3—184no—
355C2×Dic3—184no—
356C2×Dic3—364no—
357C2×Dic3C22×D911yescharacteristic
358C2×Dic3C22×D912yes—
359C2×Dic3C22×D911yescharacteristic
360Dic6—94no—
361Dic6—94no—
362Dic6—184no—
363Dic6—184no—
364Dic6—184no—
365Dic6—184no—
366Dic6—184no—
367Dic6—184no—
368C2×C32—24no—
369C2×C32C22×Dic611yescharacteristic
370C2×C32C2×D3×D411yescharacteristic
371C2×C32#101011yescharacteristic
372C2×C9—24no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—44no—
377C2×C9C22×Dic611yescharacteristic
378C2×C9C2×D3×D411yescharacteristic
379C2×C9#101011yescharacteristic
380C2×Q8—271no—
381C2×Q8—272no—
382C2×Q8—272no—
383C2×Q8—272no—
384C2×Q8—548no—
385C2×D4D3×D911yescharacteristic
386C22×C4—32no—
387C22×C4—92no—
388C22×C4—272no—
389C4⋊C4—31no—
390C4⋊C4—91no—
391C4⋊C4—271no—
392C4⋊C4—271no—
393C4⋊C4—271no—
394C4⋊C4—272no—
395C4⋊C4—272no—
396C4⋊C4—272no—
397C4⋊C4—271no—
398C22⋊C4—32no—
399C22⋊C4—92no—
400C22⋊C4—272no—
401C42—31no—
402C42—91no—
403C42—271no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—24no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3C2×Dic1814yescomplemented
413C22×C3C23×D911yescharacteristic
414C22×C3C2×#62714yes—
415C22×C3C22×D3211yescharacteristic
416C3×C4—22no—
417C3×C4—32no—
418C3×C4—61no—
419C3×C4—62no—
420C3×C4—92no—
421C3×C4—182no—
422C3×C4—181no—
423C3×C4C2×Dic1812yescomplemented
424C3×C4C2×#62712yes—
425Dic3—22no—
426Dic3—21no—
427Dic3—32no—
428Dic3—61no—
429Dic3—62no—
430Dic3—92no—
431Dic3—92no—
432Dic3—182no—
433Dic3—181no—
434Dic3—182no—
435Dic3—182no—
436Dic3—181no—
437Dic3—362no—
438Dic3—361no—
439Dic3D4×D912yes—
440C32D4×Dic611yescharacteristic
441C9—21no—
442C9D4×Dic611yescharacteristic, complemented
443C23C2×D3×D912yes—
444Q8—274no—
445Q8—544no—
446Q8—544no—
447Q8—544no—
448D4#618914yesdirect product factor
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—32no—
452C2×C4—64no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—92no—
456C2×C4—184no—
457C2×C4—271no—
458C2×C4—271no—
459C2×C4—272no—
460C2×C4—544no—
461C2×C4C2×D3×D911yescharacteristic
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—44no—
468C2×C3C22×Dic1811yescharacteristic
469C2×C3C2×D4×D911yescharacteristic
470C2×C3#6371611yescharacteristic
471C2×C3#6430911yescharacteristic
472C2×C3D32×D411yescharacteristic
473C2×C3C22×#62711yescharacteristic
474C22—22no—
475C22—22no—
476C22C2×#618914yescomplemented
477C22C22×D3×D911yescharacteristic
478C4—32no—
479C4—61no—
480C4—62no—
481C4—92no—
482C4—181no—
483C4—182no—
484C4—272no—
485C4—541no—
486C4—542no—
487C4C2×#618912yescomplemented
488C3—21no—
489C3D4×Dic1811yescharacteristic, complemented
490C3D4×#62711yescharacteristic
491C2—24no—
492C2#1172249411yescharacteristic
493C2D3×D4×D911yescharacteristic
494C2C22×#618911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×C32×D4C2×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9D4×Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C22×#6189C2semidirect
5. #11721267C2semidirect
6. #11721253C2semidirect
7. #11721249C2semidirect
8. #11720971C2semidirect
9. #11720970C2semidirect
10. #11720969C2semidirect
11. C4×#6189C2semidirect
62. C3×D4×C9Q8semidirect
79. #6189D4direct (N × H)
137. C22×C3×C9C2×Q8semidirect
139. C3×C4×C9C2×Q8semidirect
194. D4×C9Dic6semidirect
295. C22×C9C2×Dic6semidirect
301. C4×C9C2×Dic6semidirect
322. C3×C9D4×Q8semidirect
327. C3×D4Dic18semidirect
412. C22×C3C2×Dic18semidirect
423. C3×C4C2×Dic18semidirect
442. C9D4×Dic6semidirect
448. D4#6189direct (N × H)
476. C22C2×#6189semidirect
487. C4C2×#6189semidirect
489. C3D4×Dic18semidirect

Maximal subgroups

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

subgroupindex
C22×#61892
D4×C3⋊Dic92
D4×C9×Dic32
C3×D4×Dic92
#117212672
#117212532
#117212492
#117209712
#117209702
#117209692
C4×#61892
D4×#6273
D4×Dic183