Finite Group Databasebeta

← D4×#157⋊D3 #13181152

Subgroups — D4×#157⋊D3 #13181152

Order 1728.

1436 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1436 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22×#157⋊D3C212yesmaximal, complemented
2D4×#1113C212yesmaximal, complemented
3C3×D4×#157C211yesmaximal, characteristic, complemented
4#11724355C212yesmaximal, complemented
5#11724354C214yesmaximal, complemented
6#11724344C211yesmaximal, characteristic, complemented
7#11724343C212yesmaximal, complemented
8C4×#157⋊D3C211yesmaximal, characteristic, complemented
9D4×GO+(4,𝔽2)—31nomaximal
10C22×#1113C2214yescomplemented
11C22×C3×#157C2212yescomplemented
12C2×#157⋊D3—28no—
13C2×#157⋊D3C2211yescharacteristic
14C2×#157⋊D3C2214yescomplemented
15C2×#157⋊D3C2212yescomplemented
16C2×#157⋊D3C2212yescomplemented
17C3×D4×C3⋊D3—22no—
18C3×D4×C3⋊D3C2211yescharacteristic
19#89049—24no—
20#89049C2214yescomplemented
21C4⋊#1113—22no—
22C4⋊#1113C2212yescomplemented
23C4×#1113C2212yescomplemented
24C3×C22⋊#157C2212yescomplemented
25C3×C4⋊#157C2211yescharacteristic, complemented
26C3×C4×#157C2211yescharacteristic, complemented
27#88940C2211yescharacteristic, complemented
28#88939C2212yescomplemented
29#88939C2214yescomplemented
30C22×GO+(4,𝔽2)—32no—
31D32×D4—32no—
32D32×D4—62no—
33D4×#157D311yescharacteristic, complemented
34#64242—32no—
35#64241—34no—
36#64231—31no—
37#64230—32no—
38C4×GO+(4,𝔽2)—31no—
39C22×C3×C3⋊D3—24no—
40C22×C3×C3⋊D3C2312yes—
41C2×#1113—28no—
42C2×#1113—24no—
43C2×#1113—44no—
44C2×#1113C2314yes—
45C2×#1113C2312yes—
46C2×C3×#157—24no—
47C2×C3×#157C2312yes—
48C2×C3×#157C2311yescharacteristic
49C2×C3×#157C2311yescharacteristic
50#157⋊D3—28no—
51#157⋊D3—24no—
52#157⋊D3—42no—
53#157⋊D3D414yesdirect product factor
54C33×D4D411yescharacteristic, complemented
55C3×#638—24no—
56C3×#638D412yescomplemented
57C3×C4⋊(C3⋊D3)—22no—
58C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
59C3×C4×C3⋊D3—22no—
60C3×C4×C3⋊D3C2311yescharacteristic
61#6295—24no—
62#6295—22no—
63#6294—22no—
64#6294C2312yes—
65D4×D12—91nomaximal
66C22×D32—34no—
67C22×D32—64no—
68C22×#157C2×D312yescomplemented
69C2×GO+(4,𝔽2)—31no—
70C2×GO+(4,𝔽2)—34no—
71C2×GO+(4,𝔽2)—32no—
72C2×GO+(4,𝔽2)—32no—
73C2×GO+(4,𝔽2)—68no—
74D4×C3⋊D3—22no—
75D4×C3⋊D3C2×D311yescharacteristic
76C3×D3×D4—62no—
77C3×D3×D4—62no—
78C3×D3×D4—62no—
79C3×D3×D4—122no—
80#3811—34no—
81#3811—64no—
82D3×#67—64no—
83D3×#67—64no—
84D3×#67—64no—
85#3802—32no—
86#3802—62no—
87D3×D12—62no—
88D3×D12—62no—
89D3×D12—62no—
90C4×D32—32no—
91C4×D32—62no—
92C22⋊#157C2×D312yescomplemented
93C4⋊#157C2×D311yescharacteristic, complemented
94C4×#157C2×D311yescharacteristic, complemented
95#3773—31no—
96D32⋊C4—32no—
97D32⋊C4—34no—
98C22×C33C2×D412yescomplemented
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—24no—
103C2×C3×C3⋊D3—48no—
104C2×C3×C3⋊D3C2×D412yescomplemented
105C2×C3×C3⋊D3C2411yescharacteristic
106#1113—22no—
107#1113—24no—
108#1113—24no—
109#1113—48no—
110#1113C2×D414yescomplemented
111C3×#157—22no—
112C3×#157—21no—
113C3×#157C2×D412yescomplemented
114C33×C4C2×D411yescharacteristic, complemented
115C3×C3⋊Dic3—22no—
116C3×C3⋊Dic3C2×D411yescharacteristic, complemented
117C2×D3×D4—92no—
118C2×D3×D4—182no—
119C22×D12—92no—
120C3×C4×D4—91no—
121#930—92no—
122#895—92no—
123#882—94no—
124#874—91no—
125C4×D12—91no—
126C22×C3⋊D3—24no—
127C22×C3⋊D3C22×D312yes—
128C22×C3×D3—64no—
129C22×C3×D3—64no—
130C22×C3×D3—64no—
131C22×C3×D3—124no—
132C2×D32—32no—
133C2×D32—34no—
134C2×D32—62no—
135C2×D32—64no—
136C2×D32—64no—
137C2×D32—64no—
138C2×D32—64no—
139C2×D32—68no—
140C2×D32—128no—
141C2×D32—124no—
142C2×D32—128no—
143C2×#157—24no—
144C2×#157C22×D312yes—
145C2×#157C22×D311yescharacteristic
146C2×#157C22×D311yescharacteristic
147GO+(4,𝔽2)—34no—
148GO+(4,𝔽2)—68no—
149GO+(4,𝔽2)—64no—
150GO+(4,𝔽2)—122no—
151C32×D4—22no—
152C32×D4—42no—
153C32×D4D1211yescharacteristic, complemented
154#638—24no—
155#638D1212yescomplemented
156C4⋊(C3⋊D3)—22no—
157C4⋊(C3⋊D3)D1211yescharacteristic, complemented
158C4×C3⋊D3—22no—
159C4×C3⋊D3C22×D311yescharacteristic
160C3×#67—64no—
161C3×#67—64no—
162C3×#67—64no—
163C3×#67—124no—
164C3×D12—62no—
165C3×D12—62no—
166C3×D12—62no—
167C3×D12—122no—
168C3×C4×D3—62no—
169C3×C4×D3—62no—
170C3×C4×D3—62no—
171C3×C4×D3—122no—
172#626—64no—
173#626—64no—
174#626—64no—
175#625—64no—
176#625—62no—
177#624—32no—
178#624—62no—
179D3×Dic3—62no—
180D3×Dic3—62no—
181D3×Dic3—62no—
182D42—271noSylow
183C2×C33—22no—
184C2×C33C22×D411yescharacteristic
185C3×C3⋊D3—24no—
186C3×C3⋊D3—22no—
187C3×C3⋊D3—44no—
188C3×C3⋊D3C22×D411yescharacteristic
189C3×C3⋊D3H(5,𝔽2)11yescharacteristic
190C23×D3—94no—
191C23×D3—184no—
192C2×C3×D4—91no—
193C2×C3×D4—182no—
194C22×C3×C4—92no—
195C2×#67—94no—
196C2×#67—184no—
197D3×D4—62no—
198D3×D4—62no—
199D3×D4—62no—
200D3×D4—122no—
201D3×D4—182no—
202D3×D4—184no—
203D3×D4—182no—
204D3×D4—182no—
205D3×D4—184no—
206D3×D4—184no—
207C2×D12—92no—
208C2×D12—91no—
209C2×D12—92no—
210C2×D12—94no—
211C2×D12—92no—
212C2×D12—182no—
213C2×D12—188no—
214C2×C4×D3—92no—
215C2×C4×D3—182no—
216C3×C4⋊C4—91no—
217C3×C22⋊C4—92no—
218C3×C42—91no—
219(C2×D3)⋊C4—92no—
220(C2×D3)⋊C4—94no—
221C4⋊Dic3—91no—
222C22×C32—24no—
223C22×C32—44no—
224C22×C32C2×D1212yescomplemented
225C2×C3⋊D3—22no—
226C2×C3⋊D3—24no—
227C2×C3⋊D3—22no—
228C2×C3⋊D3—22no—
229C2×C3⋊D3—48no—
230C2×C3⋊D3C2×D1212yescomplemented
231C2×C3⋊D3C23×D311yescharacteristic
232C2×C3×D3—62no—
233C2×C3×D3—64no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—124no—
239C2×C3×D3—124no—
240C2×C3×D3—122no—
241C2×C3×D3—124no—
242C2×C3×D3—128no—
243C2×C3×D3—128no—
244C2×C3×D3—248no—
245D32—34no—
246D32—64no—
247D32—64no—
248D32—62no—
249D32—64no—
250D32—64no—
251D32—124no—
252D32—124no—
253D32—128no—
254D32—128no—
255D32—128no—
256#157—22no—
257#157—21no—
258#157D3×D412yescomplemented
259C32×C4—22no—
260C32×C4—42no—
261C32×C4C2×D1211yescharacteristic, complemented
262C3⋊Dic3—22no—
263C3⋊Dic3C2×D1211yescharacteristic, complemented
264C3×Dic3—62no—
265C3×Dic3—62no—
266C3×Dic3—62no—
267C3×Dic3—122no—
268C22×D4—272no—
269C22×D4—272no—
270#127—271no—
271#121—272no—
272#121—272no—
273#120—274no—
274C4×D4—91no—
275C4×D4—271no—
276C33D4211yescharacteristic, complemented, Sylow
277C23×C3—92no—
278C23×C3—184no—
279C22×D3—64no—
280C22×D3—64no—
281C22×D3—64no—
282C22×D3—92no—
283C22×D3—94no—
284C22×D3—124no—
285C22×D3—182no—
286C22×D3—184no—
287C22×D3—184no—
288C22×D3—184no—
289C22×D3—184no—
290C22×D3—188no—
291C22×D3—368no—
292C22×D3—364no—
293C22×D3—368no—
294C3×D4—22no—
295C3×D4—42no—
296C3×D4—92no—
297C3×D4—91no—
298C3×D4—182no—
299C3×D4—184no—
300C3×D4GO+(4,𝔽2)11yescharacteristic, complemented
301C2×C3×C4—91no—
302C2×C3×C4—91no—
303C2×C3×C4—91no—
304C2×C3×C4—92no—
305C2×C3×C4—182no—
306C2×C3×C4—184no—
307#67—64no—
308#67—64no—
309#67—64no—
310#67—124no—
311#67—184no—
312#67—184no—
313#67—184no—
314C2×Dic3—92no—
315C2×Dic3—182no—
316D12—62no—
317D12—62no—
318D12—62no—
319D12—94no—
320D12—122no—
321D12—182no—
322D12—184no—
323D12—184no—
324D12—188no—
325D12—362no—
326C4×D3—62no—
327C4×D3—62no—
328C4×D3—62no—
329C4×D3—122no—
330C4×D3—182no—
331C4×D3—182no—
332C4×D3—182no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—42no—
336C2×C32—44no—
337C2×C32—84no—
338C2×C32C22×D1211yescharacteristic
339C3⋊D3—24no—
340C3⋊D3—22no—
341C3⋊D3—44no—
342C3⋊D3C2×D3×D411yescharacteristic
343C3⋊D3#100911yescharacteristic
344C3×D3—62no—
345C3×D3—62no—
346C3×D3—64no—
347C3×D3—64no—
348C3×D3—124no—
349C3×D3—124no—
350C3×D3—124no—
351C3×D3—124no—
352C3×D3—244no—
353C24—274no—
354C2×D4—91no—
355C2×D4—182no—
356C2×D4—272no—
357C2×D4—271no—
358C2×D4—274no—
359C2×D4—272no—
360C2×D4—272no—
361C2×D4—274no—
362C2×D4—542no—
363C2×D4—548no—
364C2×D4—544no—
365C22×C4—92no—
366C22×C4—272no—
367C4⋊C4—91no—
368C4⋊C4—271no—
369C22⋊C4—92no—
370C22⋊C4—272no—
371C22⋊C4—274no—
372C42—91no—
373C22×C3—24no—
374C22×C3—44no—
375C22×C3—91no—
376C22×C3—92no—
377C22×C3—182no—
378C22×C3—182no—
379C22×C3—182no—
380C22×C3—184no—
381C22×C3—368no—
382C22×C3C2×GO+(4,𝔽2)12yescomplemented
383C2×D3—62no—
384C2×D3—64no—
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—64no—
388C2×D3—64no—
389C2×D3—94no—
390C2×D3—124no—
391C2×D3—124no—
392C2×D3—122no—
393C2×D3—124no—
394C2×D3—128no—
395C2×D3—128no—
396C2×D3—184no—
397C2×D3—182no—
398C2×D3—184no—
399C2×D3—184no—
400C2×D3—184no—
401C2×D3—248no—
402C2×D3—364no—
403C2×D3—364no—
404C2×D3—368no—
405C2×D3—368no—
406C2×D3—368no—
407C3×C4—22no—
408C3×C4—42no—
409C3×C4—92no—
410C3×C4—91no—
411C3×C4—182no—
412C3×C4—182no—
413C3×C4—181no—
414C3×C4C2×GO+(4,𝔽2)11yescharacteristic, complemented
415Dic3—62no—
416Dic3—62no—
417Dic3—62no—
418Dic3—122no—
419C32—22no—
420C32—42no—
421C32D4×D1211yescharacteristic, complemented
422C23—92no—
423C23—184no—
424C23—272no—
425C23—274no—
426C23—544no—
427C23—548no—
428C23—1084no—
429D4—92no—
430D4—91no—
431D4—184no—
432D4—182no—
433D4—274no—
434D4—542no—
435D4—544no—
436D4—544no—
437D4—548no—
438D4—1082no—
439D4#157⋊D311yescharacteristic, direct product factor
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—92no—
444C2×C4—184no—
445C2×C4—182no—
446C2×C4—272no—
447C2×C4—542no—
448C2×C3—22no—
449C2×C3—22no—
450C2×C3—44no—
451C2×C3—42no—
452C2×C3—84no—
453C2×C3—91no—
454C2×C3—91no—
455C2×C3—184no—
456C2×C3—182no—
457C2×C3—364no—
458C2×C3C22×GO+(4,𝔽2)11yescharacteristic
459D3—62no—
460D3—62no—
461D3—64no—
462D3—64no—
463D3—124no—
464D3—124no—
465D3—124no—
466D3—124no—
467D3—244no—
468C22—91no—
469C22—92no—
470C22—182no—
471C22—182no—
472C22—182no—
473C22—184no—
474C22—274no—
475C22—368no—
476C22—542no—
477C22—544no—
478C22—544no—
479C22—1088no—
480C22C2×#157⋊D312yescomplemented
481C4—92no—
482C4—91no—
483C4—182no—
484C4—181no—
485C4—182no—
486C4C2×#157⋊D311yescharacteristic, complemented
487C3—22no—
488C3—42no—
489C3D4×GO+(4,𝔽2)11yescharacteristic, complemented
490C2—22no—
491C2—91no—
492C2—91no—
493C2—182no—
494C2—184no—
495C2—364no—
496C2C22×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FittingC33×D4D4
socleC2×C33C22×D4
hypercenterD4#157⋊D3
hypocenterC33D42
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C22×#157⋊D3C2semidirect
2. D4×#1113C2semidirect
3. C3×D4×#157C2semidirect
4. #11724355C2semidirect
5. #11724354C2semidirect
6. #11724344C2semidirect
7. #11724343C2semidirect
8. C4×#157⋊D3C2semidirect
10. C22×#1113C22semidirect
11. C22×C3×#157C22semidirect
14. C2×#157⋊D3C22semidirect
15. C2×#157⋊D3C22semidirect
16. C2×#157⋊D3C22semidirect
20. #89049C22semidirect
22. C4⋊#1113C22semidirect
23. C4×#1113C22semidirect
24. C3×C22⋊#157C22semidirect
25. C3×C4⋊#157C22semidirect
26. C3×C4×#157C22semidirect
27. #88940C22semidirect
28. #88939C22semidirect
29. #88939C22semidirect
33. D4×#157D3semidirect
53. #157⋊D3D4direct (N × H)
54. C33×D4D4semidirect
56. C3×#638D4semidirect
58. C3×C4⋊(C3⋊D3)D4semidirect
68. C22×#157C2×D3semidirect
92. C22⋊#157C2×D3semidirect
93. C4⋊#157C2×D3semidirect
94. C4×#157C2×D3semidirect
98. C22×C33C2×D4semidirect
104. C2×C3×C3⋊D3C2×D4semidirect
110. #1113C2×D4semidirect
113. C3×#157C2×D4semidirect
114. C33×C4C2×D4semidirect
116. C3×C3⋊Dic3C2×D4semidirect
153. C32×D4D12semidirect
155. #638D12semidirect
157. C4⋊(C3⋊D3)D12semidirect
224. C22×C32C2×D12semidirect
230. C2×C3⋊D3C2×D12semidirect
258. #157D3×D4semidirect
261. C32×C4C2×D12semidirect
263. C3⋊Dic3C2×D12semidirect
276. C33D42semidirect
300. C3×D4GO+(4,𝔽2)semidirect
382. C22×C3C2×GO+(4,𝔽2)semidirect
414. C3×C4C2×GO+(4,𝔽2)semidirect
421. C32D4×D12semidirect
439. D4#157⋊D3direct (N × H)
480. C22C2×#157⋊D3semidirect
486. C4C2×#157⋊D3semidirect
489. C3D4×GO+(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C22×#157⋊D32
D4×#11132
C3×D4×#1572
#117243552
#117243542
#117243442
#117243432
C4×#157⋊D32
D4×GO+(4,𝔽2)3
D4×D129