Finite Group Databasebeta

← D3×#63785 #13166547

Subgroups — D3×#63785 #13166547

Order 1728.

748 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D3×#3832C211yesmaximal, characteristic
2C4×D3×#157C211yesmaximal, characteristic, complemented
3D3×#3788C211yesmaximal, characteristic, complemented
4#11723110C211yesmaximal, characteristic, complemented
5C3×#63785C211yesmaximal, characteristic, complemented
6#11722713C211yesmaximal, characteristic, complemented
7#11722712C211yesmaximal, characteristic, complemented
8C2×#63785—31nomaximal
9C2×D3×#157—21no—
10#89080C411yescharacteristic, complemented
11C3×#3832C2211yescharacteristic
12C32×D3×Q8C411yescharacteristic, complemented
13D3×C4⋊(C3⋊D3)—21no—
14D3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
15C4×D3×C3⋊D3—21no—
16C4×D3×C3⋊D3C2211yescharacteristic
17#89018—21no—
18#89018C2211yescharacteristic
19#89017—21no—
20#89017C411yescharacteristic, complemented
21C4×C3⋊#157C2211yescharacteristic, complemented
22#88989C2211yescharacteristic, complemented
23C3×C4×#157C2211yescharacteristic, complemented
24C3×#3788C2211yescharacteristic, complemented
25#88926—21no—
26D3×#622—21no—
27Dic3×#157—21no—
28C2×#3832—31no—
29D3×#239—62no—
30C2×C4×#157—31no—
31C2×#3788—31no—
32#63785—31no—
33#63785—31no—
34#63785—31no—
35#63785D311yescharacteristic, direct product factor
36C2×D3×C3⋊D3—21no—
37C2×D3×C3⋊D3D411yescharacteristic
38C2×C3⋊#157—21no—
39C2×C3×#157—21no—
40D3×#157—24no—
41C33×Q8C2×C411yescharacteristic, complemented
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic, complemented
44C4×C3⋊(C3⋊D3)—21no—
45C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)C2×C411yescharacteristic, complemented
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3C2311yescharacteristic
50C32×C4×D3—21no—
51C32×C4×D3C2×C411yescharacteristic
52C32×Dic6—21no—
53C32×Dic6C2×C411yescharacteristic, complemented
54#6292—21no—
55#6292—21no—
56#6292—21no—
57#6289—21no—
58#6289D411yescharacteristic
59Dic3×C3⋊D3—21no—
60Dic3×C3⋊D3D411yescharacteristic
61D3×C3⋊Dic3D411yescharacteristic
62C3⋊#622—21no—
63C3×#622—21no—
64D3×#104—91nomaximal
65C22×#157—61no—
66C2×C32×Q8—31no—
67#3832—31no—
68#3832—31no—
69#3832—31no—
70#3832—61no—
71#3832—61no—
72#3832—62no—
73#3832—122no—
74#3832C2×D311yescharacteristic
75C2×C4⋊(C3⋊D3)—31no—
76C2×C4⋊(C3⋊D3)—61no—
77C2×C4×C3⋊D3—31no—
78C2×C4×C3⋊D3—61no—
79C3×#239—62no—
80C3×D3×Q8—22no—
81D3×D12—62no—
82D3×D12—64no—
83C4×D32—62no—
84C4×D32—64no—
85#3799—62no—
86#3799—64no—
87#3796—62no—
88#3796—64no—
89C2×#622—61no—
90C4×#157—31no—
91C4×#157—61no—
92C4×#157C2×D311yescharacteristic, complemented
93#3788—31no—
94#3788—61no—
95#3788C2×D311yescharacteristic, complemented
96C2×C3⋊(C3⋊D3)—21no—
97C2×C3⋊(C3⋊D3)C22⋊C411yescharacteristic
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3C2×D411yescharacteristic
100C2×C32×D3C22⋊C411yescharacteristic
101D3×C3⋊D3—22no—
102D3×C3⋊D3—42no—
103C3⋊#157—22no—
104C3×#157—22no—
105C33×C4—21no—
106C33×C4C22×C411yescharacteristic
107C3⋊(C3⋊Dic3)C22⋊C411yescharacteristic
108C3×C3⋊Dic3C2×D411yescharacteristic
109C32×Dic3—21no—
110C32×Dic3C22⋊C411yescharacteristic
111D3×#41—91no—
112C2×#239—182no—
113D3×M16—91no—
114C42×D3—91no—
115C3×#104—91no—
116#837—91no—
117#825—91no—
118#805—91no—
119C22×C3⋊D3—31no—
120C22×C3⋊D3—61no—
121C2×D32—62no—
122C2×D32—64no—
123C2×#157—21no—
124C2×#157—64no—
125C2×#157—61no—
126C32×Q8—22no—
127C32×Q8—31no—
128C32×Q8—42no—
129C32×Q8—61no—
130C32×Q8C4×D311yescharacteristic, complemented
131C2×C32×C4—31no—
132C2×C32×C4—61no—
133C2×C3⋊Dic3—31no—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—31no—
136C4⋊(C3⋊D3)—62no—
137C4⋊(C3⋊D3)—61no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—124no—
143C4⋊(C3⋊D3)—122no—
144C4⋊(C3⋊D3)C4×D311yescharacteristic, complemented
145C4×C3⋊D3—21no—
146C4×C3⋊D3—31no—
147C4×C3⋊D3—31no—
148C4×C3⋊D3—31no—
149C4×C3⋊D3—62no—
150C4×C3⋊D3—61no—
151C4×C3⋊D3—61no—
152C4×C3⋊D3—61no—
153C4×C3⋊D3—64no—
154C4×C3⋊D3—124no—
155C4×C3⋊D3—122no—
156C4×C3⋊D3C22×D311yescharacteristic
157C3×D12—62no—
158C3×D12—64no—
159C3×C4×D3—24no—
160C3×C4×D3—22no—
161C3×C4×D3—62no—
162C3×C4×D3—64no—
163C3×Dic6—24no—
164C3×Dic6—22no—
165#626—64no—
166#626—64no—
167#626—64no—
168#624—62no—
169#624—64no—
170D3×Dic3—62no—
171D3×Dic3—62no—
172D3×Dic3—64no—
173#622—21no—
174#622—61no—
175C2×#104—271noSylow
176C2×C33C2×C22⋊C411yescharacteristic
177C3⋊(C3⋊D3)—21no—
178C3⋊(C3⋊D3)—41no—
179C3×C3⋊D3—21no—
180C3×C3⋊D3—41no—
181C32×D3#10412yescomplemented
182C3×#41—91no—
183C2×C3×Q8—62no—
184#239—62no—
185#239—91no—
186#239—182no—
187#239—182no—
188#239—182no—
189#239—184no—
190#239—184no—
191#239—362no—
192D3×Q8#15711yescharacteristic, complemented
193#237—91no—
194#237—181no—
195D3×D4—91no—
196D3×D4—181no—
197#235—91no—
198#235—181no—
199C2×D12—182no—
200C2×D12—184no—
201C2×C4×D3—91no—
202C2×C4×D3—182no—
203C2×C4×D3—181no—
204C2×C4×D3—184no—
205C2×C4×D3—181no—
206C3×M16—91no—
207C3×C42—91no—
208C4×Dic3—91no—
209C4×Dic3—181no—
210#208—91no—
211ΓL(1,𝔽25)—181no—
212D3×C8—181no—
213C22×C32—31no—
214C2×C3⋊D3—21no—
215C2×C3⋊D3—31no—
216C2×C3⋊D3—62no—
217C2×C3⋊D3—61no—
218C2×C3⋊D3—64no—
219C2×C3⋊D3—62no—
220C2×C3⋊D3—124no—
221C2×C3⋊D3—122no—
222C2×C3⋊D3—122no—
223C2×C3⋊D3D3×D411yescharacteristic
224C2×C3×D3—22no—
225C2×C3×D3—62no—
226C2×C3×D3—64no—
227D32—124no—
228D32—128no—
229#157—22no—
230#157—62no—
231C32×C4—21no—
232C32×C4—24no—
233C32×C4—22no—
234C32×C4—31no—
235C32×C4—44no—
236C32×C4—42no—
237C32×C4—61no—
238C32×C4C2×C4×D311yescharacteristic
239C3⋊Dic3—31no—
240C3⋊Dic3—62no—
241C3⋊Dic3—122no—
242C3⋊Dic3D3×D411yescharacteristic
243C3×Dic3—24no—
244C3×Dic3—22no—
245C3×Dic3—62no—
246C2×#41—271no—
247C2×M16—271no—
248C2×C42—271no—
249#104—91no—
250#104—271no—
251#104—271no—
252#104—271no—
253C33C2×#10411yescharacteristic, complemented, Sylow
254C22×D3—91no—
255C22×D3—182no—
256C22×D3—184no—
257C22×D3—181no—
258C3×Q8—22no—
259C3×Q8—42no—
260C3×Q8—62no—
261C3×Q8—64no—
262C3×Q8C2×#15711yescharacteristic, complemented
263C3×D4—91no—
264C3×D4—181no—
265C2×C3×C4—62no—
266C2×C3×C4—64no—
267C2×C3×C4—91no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270#67—181no—
271#67—181no—
272#67—181no—
273C2×Dic3—91no—
274C2×Dic3—182no—
275C2×Dic3—181no—
276C2×Dic3—181no—
277D12—62no—
278D12—64no—
279D12—91no—
280D12—182no—
281D12—181no—
282D12—184no—
283D12—184no—
284D12—184no—
285D12—184no—
286D12—362no—
287D12—364no—
288C4×D3—21no—
289C4×D3—62no—
290C4×D3—64no—
291C4×D3—91no—
292C4×D3—91no—
293C4×D3—91no—
294C4×D3—182no—
295C4×D3—182no—
296C4×D3—182no—
297C4×D3—181no—
298C4×D3—184no—
299C4×D3—181no—
300C4×D3—184no—
301C4×D3—184no—
302C4×D3—184no—
303C4×D3—362no—
304C4×D3—364no—
305C4×D3C2×#15711yescharacteristic
306Dic6—21no—
307Dic6C2×#15711yescharacteristic, complemented
308C3×C8—181no—
309C3⋊C8—181no—
310C2×C32—22no—
311C2×C32—32no—
312C2×C32—42no—
313C2×C32D3×C22⋊C411yescharacteristic
314C3⋊D3—21no—
315C3⋊D3—41no—
316C3⋊D3—61no—
317C3⋊D3—122no—
318C3⋊D3—121no—
319C3⋊D3—124no—
320C3⋊D3—244no—
321C3⋊D3—242no—
322C3×D3—24no—
323C3×D3—122no—
324C3×D3—124no—
325#41—91no—
326#41—271no—
327#41—271no—
328#41—271no—
329#41—541no—
330#41—541no—
331C2×Q8—31no—
332C2×D4—271no—
333C2×D4—541no—
334C22×C4—271no—
335C22×C4—541no—
336C22×C4—541no—
337M16—91no—
338M16—271no—
339M16—541no—
340C2×C8—541no—
341C42—91no—
342C42—271no—
343C42—541no—
344C22×C3—62no—
345C22×C3—91no—
346C22×C3—181no—
347C2×D3—62no—
348C2×D3—64no—
349C2×D3—91no—
350C2×D3—182no—
351C2×D3—182no—
352C2×D3—184no—
353C2×D3—181no—
354C2×D3—362no—
355C2×D3—364no—
356C2×D3—364no—
357C2×D3—368no—
358C2×D3—362no—
359C2×D3C22⋊#15711yescharacteristic
360C3×C4—24no—
361C3×C4—21no—
362C3×C4—22no—
363C3×C4—44no—
364C3×C4—42no—
365C3×C4—62no—
366C3×C4—64no—
367C3×C4—91no—
368C3×C4—182no—
369C3×C4C22×#15711yescharacteristic
370Dic3—21no—
371Dic3—62no—
372Dic3—91no—
373Dic3—182no—
374Dic3—182no—
375Dic3—362no—
376Dic3C22⋊#15711yescharacteristic
377C32—22no—
378C32—42no—
379C32D3×#10411yescharacteristic, complemented
380C23—271no—
381C23—541no—
382Q8—31no—
383Q8—61no—
384Q8D3×#15711yescharacteristic, complemented
385D4—91no—
386D4—181no—
387D4—271no—
388D4—541no—
389D4—541no—
390D4—541no—
391D4—541no—
392C2×C4—31no—
393C2×C4—61no—
394C2×C4—91no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4—271no—
398C2×C4—271no—
399C2×C4—271no—
400C2×C4—271no—
401C2×C4—541no—
402C2×C4—541no—
403C2×C4—541no—
404C2×C4—544no—
405C2×C4—541no—
406C8—181no—
407C8—541no—
408C2×C3—22no—
409C2×C3—42no—
410C2×C3—64no—
411C2×C3—181no—
412C2×C3—361no—
413C2×C3C2×C22⋊#15711yescharacteristic
414D3—122no—
415D3—124no—
416D3—181no—
417D3—362no—
418D3—364no—
419D3—361no—
420D3—722no—
421D3—724no—
422D3#6378512yesdirect product factor
423C22—31no—
424C22—91no—
425C22—181no—
426C22—271no—
427C22—542no—
428C22—541no—
429C22—1082no—
430C4—21no—
431C4—31no—
432C4—61no—
433C4—91no—
434C4—182no—
435C4—271no—
436C4—542no—
437C4C2×D3×#15711yescharacteristic
438C3—22no—
439C3—42no—
440C3C2×#6378511yescharacteristic, complemented
441C2—32no—
442C2—181no—
443C2—361no—
444C2—541no—
445C2—1081no—
446C2D3×C22⋊#15711yescharacteristic
subgroupG/H
centerC2D3×C22⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×Q8C2×C4
socleC2×C33C2×C22⋊C4
prime radicalD3×#3832C2
cosocleC3×C4×C3⋊D3C23
hypercenterQ8D3×#157
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
2. C4×D3×#157C2semidirect
3. D3×#3788C2semidirect
4. #11723110C2semidirect
5. C3×#63785C2semidirect
6. #11722713C2semidirect
7. #11722712C2semidirect
10. #89080C4semidirect
12. C32×D3×Q8C4semidirect
14. D3×C4⋊(C3⋊D3)C4semidirect
20. #89017C4semidirect
21. C4×C3⋊#157C22semidirect
22. #88989C22semidirect
23. C3×C4×#157C22semidirect
24. C3×#3788C22semidirect
35. #63785D3direct (N × H)
41. C33×Q8C2×C4semidirect
43. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
47. C3×C4⋊(C3⋊D3)C2×C4semidirect
53. C32×Dic6C2×C4semidirect
92. C4×#157C2×D3semidirect
95. #3788C2×D3semidirect
130. C32×Q8C4×D3semidirect
144. C4⋊(C3⋊D3)C4×D3semidirect
181. C32×D3#104semidirect
192. D3×Q8#157semidirect
253. C33C2×#104semidirect
262. C3×Q8C2×#157semidirect
307. Dic6C2×#157semidirect
379. C32D3×#104semidirect
384. Q8D3×#157semidirect
422. D3#63785direct (N × H)
440. C3C2×#63785semidirect

Maximal subgroups

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

subgroupindex
D3×#38322
C4×D3×#1572
D3×#37882
#117231102
C3×#637852
#117227132
#117227122
C2×#637853
D3×#1049