Finite Group Databasebeta

← #13140530

Subgroups — #13140530

Order 1728.

787 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 101 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×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11721310C212yesmaximal, complemented
3#11720580C211yesmaximal, characteristic, complemented
4#63661⋊C3C211yesmaximal, characteristic, complemented
5#11720333C212yesmaximal, complemented
6#10591527—31nomaximal
7D3×#834—31nomaximal
8C23×(C3⋊D3)⋊C3C411yescharacteristic, complemented
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3C411yescharacteristic, complemented
12C2×#638⋊C3C2211yescharacteristic
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
16C2×C4×(C3⋊D3)⋊C3C411yescharacteristic, complemented
17C2×((C3⋊D3)⋊C3)⋊C4—22no—
18#88552C2212yescomplemented
19#3757⋊C3C2212yescomplemented
20#88450—22no—
21C2×D4×C3⋊D3D311yescharacteristic
22C2×C3×D3×D4—31no—
23#63974—32no—
24D3×#217—32no—
25#63661D311yescharacteristic, complemented
26C3×#834—31no—
27C3×#834—31no—
28C3×#834—31no—
29C3×#834—61no—
30#63581—32no—
31#63580—32no—
32C23×H(3,𝔽3)C2×C411yescharacteristic, complemented
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×(C3⋊D3)⋊C3—21no—
35C22×(C3⋊D3)⋊C3—21no—
36C22×(C3⋊D3)⋊C3—42no—
37C22×(C3⋊D3)⋊C3—42no—
38C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C22×(C3⋊D3)⋊C3C2×C411yescharacteristic, complemented
41C22×(C3⋊D3)⋊C3D412yes—
42C22×(C3⋊D3)⋊C3D412yes—
43D4×H(3,𝔽3)—22no—
44C2×C4×H(3,𝔽3)C2×C411yescharacteristic, complemented
45C2×#1084—22no—
46#638⋊C3—22no—
47#638⋊C3—22no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic, complemented
50C4⋊((C3⋊D3)⋊C3)—22no—
51C4×(C3⋊D3)⋊C3—21no—
52C4×(C3⋊D3)⋊C3—21no—
53((C3⋊D3)⋊C3)⋊C4—44no—
54C2×#834—91no—
55D3×#99—91no—
56D3×#99—91no—
57C23×C3⋊D3Dic311yescharacteristic, complemented
58C23×C3⋊D3C2×D311yescharacteristic
59C23×C3×D3—31no—
60C23×C3×D3—31no—
61C2×C32×D4—21no—
62C2×C32×D4C2×D311yescharacteristic
63C2×C32×D4C2×D311yescharacteristic
64C2×#638Dic311yescharacteristic, complemented
65C2×#638C2×D311yescharacteristic
66D4×C3⋊D3—22no—
67D4×C3⋊D3—22no—
68C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
69C2×C4×C3⋊D3Dic311yescharacteristic, complemented
70C2×C3×#67—31no—
71C2×C3×#67—31no—
72C3×D3×D4—62no—
73C3×D3×D4—62no—
74C2×C3×D12—31no—
75C2×C3×C4×D3—31no—
76C2×#624—62no—
77C2×D3×Dic3—62no—
78#3757C2×D312yescomplemented
79C3×#217—32no—
80C3×#217—32no—
81C3×#217—32no—
82C3×#217—62no—
83#3722—62no—
84(C2×D3)⋊Dic3—62no—
85C22×H(3,𝔽3)—21no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)C22×C411yescharacteristic
89C22×H(3,𝔽3)C2×D412yes—
90C2×(C3⋊D3)⋊C3—24no—
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—42no—
95C2×(C3⋊D3)⋊C3—42no—
96C2×(C3⋊D3)⋊C3—42no—
97C2×(C3⋊D3)⋊C3—41no—
98C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
99C2×(C3⋊D3)⋊C3C22⋊C412yes—
100C2×(C3⋊D3)⋊C3C22⋊C411yescharacteristic
101C4×H(3,𝔽3)—21no—
102#1084—42no—
103(C3⋊Dic3)⋊C3—21no—
104(C3⋊Dic3)⋊C3—42no—
105C22×C3×D4—91no—
106C2×D3×D4—31no—
107C2×D3×D4—91no—
108C2×#217—92no—
109D3×C22⋊C4—92no—
110D3×C22⋊C4—92no—
111C3×#99—91no—
112C3×#99—91no—
113#834—31no—
114#834—31no—
115#834—31no—
116#834—61no—
117#834—91no—
118#834—92no—
119#806—92no—
120#806—92no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32C4×D311yescharacteristic, complemented
124C23×C32C2×Dic311yescharacteristic, complemented
125C23×C32C22×D311yescharacteristic
126C23×C32C22×D311yescharacteristic
127C22×C3⋊D3—21no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—42no—
130C22×C3⋊D3—42no—
131C22×C3⋊D3C2×Dic311yescharacteristic
132C22×C3⋊D3C2×Dic311yescharacteristic
133C22×C3⋊D3C2×Dic311yescharacteristic, complemented
134C22×C3⋊D3#6712yes—
135C22×C3⋊D3#6712yes—
136C22×C3×D3—31no—
137C22×C3×D3—32no—
138C22×C3×D3—31no—
139C22×C3×D3—32no—
140C22×C3×D3—31no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C22×C3×D3—122no—
144C22×C3×D3—122no—
145C32×D4—22no—
146C32×D4—22no—
147C32×D4—42no—
148C2×C32×C4—21no—
149C2×C32×C4C4×D311yescharacteristic, complemented
150C2×C32×C4C2×Dic311yescharacteristic, complemented
151#638—22no—
152#638—22no—
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3C2×Dic311yescharacteristic, complemented
155C4⋊(C3⋊D3)—22no—
156C4×C3⋊D3—21no—
157C4×C3⋊D3—21no—
158C3×#67—62no—
159C3×#67—62no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—122no—
165C3×D12—62no—
166C3×C4×D3—61no—
167C3×C4×D3—61no—
168#624—124no—
169D3×Dic3—124no—
170C2×#99—271noSylow
171C2×H(3,𝔽3)—22no—
172C2×H(3,𝔽3)—21no—
173C2×H(3,𝔽3)—41no—
174C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
175(C3⋊D3)⋊C3—21no—
176(C3⋊D3)⋊C3—22no—
177(C3⋊D3)⋊C3—41no—
178(C3⋊D3)⋊C3#9912yescomplemented
179C24×C3—91no—
180C24×C3—91no—
181C23×D3—31no—
182C23×D3—31no—
183C23×D3—91no—
184C23×D3—91no—
185C2×C3×D4—31no—
186C2×C3×D4—31no—
187C2×C3×D4—61no—
188C2×C3×D4—91no—
189C2×C3×D4—91no—
190C2×C3×D4—91no—
191C2×C3×D4—182no—
192C2×C3×D4—182no—
193C2×C3×D4D3211yescharacteristic
194C22×C3×C4—91no—
195C2×#67—31no—
196C2×#67—31no—
197C2×#67—91no—
198C2×#67—91no—
199C22×Dic3—182no—
200D3×D4—62no—
201D3×D4—62no—
202D3×D4—182no—
203D3×D4—182no—
204C2×D12—31no—
205C2×D12—91no—
206C2×C4×D3—31no—
207C2×C4×D3—91no—
208C2×C4×D3—182no—
209C2×C4×D3—182no—
210C3×C22⋊C4—92no—
211C3×C22⋊C4—92no—
212#217—32no—
213#217—32no—
214#217—32no—
215#217—62no—
216#217—92no—
217#217—182no—
218(C2×D3)⋊C4—182no—
219(C2×D3)⋊C4—182no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—22no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32—81no—
230C22×C32—81no—
231C22×C32C2×C4×D311yescharacteristic
232C22×C32D3×D412yes—
233C22×C32C22×Dic311yescharacteristic
234C22×C32C2×#6712yes—
235C2×C3⋊D3—24no—
236C2×C3⋊D3—22no—
237C2×C3⋊D3—21no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3—42no—
240C2×C3⋊D3—42no—
241C2×C3⋊D3—42no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3#21711yescharacteristic
244C2×C3⋊D3#21712yes—
245C2×C3⋊D3#21711yescharacteristic
246C2×C3×D3—31no—
247C2×C3×D3—31no—
248C2×C3×D3—32no—
249C2×C3×D3—64no—
250C2×C3×D3—62no—
251C2×C3×D3—61no—
252C2×C3×D3—122no—
253C2×C3×D3—121no—
254C2×C3×D3—122no—
255C2×C3×D3—122no—
256C2×C3×D3—121no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—41no—
260C3⋊Dic3—21no—
261C3⋊Dic3—42no—
262C3×Dic3—61no—
263C3×Dic3—122no—
264C3×Dic3—122no—
265C3×Dic3—122no—
266C3×Dic3—242no—
267C22×D4—91no—
268C2×C22⋊C4—272no—
269#99—91no—
270#99—91no—
271#99—272no—
272H(3,𝔽3)C2×#9911yescharacteristic, complemented, Sylow
273C23×C3—31no—
274C23×C3—31no—
275C23×C3—31no—
276C23×C3—31no—
277C23×C3—61no—
278C23×C3—61no—
279C23×C3—92no—
280C23×C3—91no—
281C23×C3—92no—
282C23×C3—91no—
283C23×C3—91no—
284C23×C3—181no—
285C23×C3—181no—
286C23×C3—362no—
287C23×C3—362no—
288C23×C3D3×Dic311yescharacteristic
289C23×C3C2×D3211yescharacteristic
290C22×D3—31no—
291C22×D3—32no—
292C22×D3—31no—
293C22×D3—32no—
294C22×D3—31no—
295C22×D3—61no—
296C22×D3—61no—
297C22×D3—91no—
298C22×D3—92no—
299C22×D3—91no—
300C22×D3—92no—
301C22×D3—91no—
302C22×D3—122no—
303C22×D3—122no—
304C22×D3—181no—
305C22×D3—181no—
306C22×D3—362no—
307C22×D3—362no—
308C3×D4—22no—
309C3×D4—62no—
310C3×D4—62no—
311C3×D4—122no—
312C3×D4—182no—
313C3×D4—182no—
314C3×D4—182no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—61no—
318C2×C3×C4—91no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4—182no—
322C2×C3×C4—182no—
323C2×C3×C4D3×Dic311yescharacteristic
324#67—62no—
325#67—62no—
326#67—182no—
327#67—182no—
328C2×Dic3—31no—
329C2×Dic3—62no—
330C2×Dic3—62no—
331C2×Dic3—62no—
332C2×Dic3—91no—
333C2×Dic3—122no—
334C2×Dic3—182no—
335C2×Dic3—364no—
336D12—62no—
337D12—182no—
338C4×D3—61no—
339C4×D3—61no—
340C4×D3—181no—
341C4×D3—181no—
342C4×D3—364no—
343C4×D3—364no—
344C2×C32—22no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—42no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32—81no—
354C2×C32D3×C22⋊C411yescharacteristic
355C2×C32C2×#21711yescharacteristic
356C3⋊D3—21no—
357C3⋊D3—22no—
358C3⋊D3—41no—
359C3⋊D3#83412yescomplemented
360C3×D3—32no—
361C3×D3—62no—
362C3×D3—61no—
363C3×D3—121no—
364C24—91no—
365C24—91no—
366C2×D4—91no—
367C2×D4—91no—
368C2×D4—91no—
369C2×D4—182no—
370C2×D4—182no—
371C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C22×C4—91no—
373C22×C4—542no—
374C22⋊C4—92no—
375C22⋊C4—92no—
376C22⋊C4—542no—
377C22×C3—21no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—32no—
381C22×C3—32no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—61no—
385C22×C3—61no—
386C22×C3—61no—
387C22×C3—62no—
388C22×C3—91no—
389C22×C3—91no—
390C22×C3—92no—
391C22×C3—121no—
392C22×C3—121no—
393C22×C3—121no—
394C22×C3—121no—
395C22×C3—121no—
396C22×C3—181no—
397C22×C3—184no—
398C22×C3—182no—
399C22×C3—241no—
400C22×C3—241no—
401C22×C3—362no—
402C22×C3—361no—
403C22×C3—361no—
404C22×C3—362no—
405C22×C3—362no—
406C22×C3C2×D3×Dic311yescharacteristic
407C22×C3D3×#6712yes—
408C2×D3—31no—
409C2×D3—31no—
410C2×D3—32no—
411C2×D3—64no—
412C2×D3—62no—
413C2×D3—61no—
414C2×D3—91no—
415C2×D3—91no—
416C2×D3—92no—
417C2×D3—122no—
418C2×D3—122no—
419C2×D3—121no—
420C2×D3—122no—
421C2×D3—121no—
422C2×D3—184no—
423C2×D3—182no—
424C2×D3—181no—
425C2×D3—362no—
426C2×D3—362no—
427C2×D3—361no—
428C2×D3—362no—
429C2×D3—361no—
430C3×C4—21no—
431C3×C4—61no—
432C3×C4—61no—
433C3×C4—121no—
434C3×C4—181no—
435C3×C4—362no—
436C3×C4—362no—
437Dic3—61no—
438Dic3—122no—
439Dic3—122no—
440Dic3—122no—
441Dic3—181no—
442Dic3—242no—
443Dic3—362no—
444C32—21no—
445C32D3×#9911yescharacteristic, complemented
446C32C2×#83411yescharacteristic, complemented
447C23—91no—
448C23—92no—
449C23—91no—
450C23—92no—
451C23—91no—
452C23—181no—
453C23—181no—
454C23—362no—
455C23—362no—
456C23((C3⋊D3)⋊C3)⋊C411yescharacteristic, complemented
457C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
458D4—22no—
459D4—182no—
460D4—182no—
461D4—182no—
462C2×C4—91no—
463C2×C4—181no—
464C2×C4—181no—
465C2×C4—182no—
466C2×C4—182no—
467C2×C4—1084no—
468C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic, complemented
469C2×C3—22no—
470C2×C3—21no—
471C2×C3—31no—
472C2×C3—31no—
473C2×C3—41no—
474C2×C3—62no—
475C2×C3—62no—
476C2×C3—61no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—92no—
480C2×C3—122no—
481C2×C3—121no—
482C2×C3—121no—
483C2×C3—121no—
484C2×C3—181no—
485C2×C3—182no—
486C2×C3—241no—
487C2×C3—361no—
488C2×C3D3×#21711yescharacteristic
489D3—32no—
490D3—61no—
491D3—62no—
492D3—92no—
493D3—121no—
494D3—181no—
495D3—182no—
496D3—361no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—91no—
501C22—91no—
502C22—92no—
503C22—184no—
504C22—182no—
505C22—181no—
506C22—362no—
507C22—361no—
508C22—361no—
509C22—362no—
510C22—362no—
511C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
512C22#8867612yes—
513C4—21no—
514C4—181no—
515C4—362no—
516C4—362no—
517C3—31no—
518C3—31no—
519C3—61no—
520C3D3×#83411yescharacteristic
521C2—22no—
522C2—21no—
523C2—41no—
524C2—92no—
525C2—181no—
526C2—182no—
527C2—361no—
528C2#1172131011yescharacteristic
subgroupG/H
centerC2#11721310
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC23×C3C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC2×C3D3×#217
prime radicalC2×D4×(C3⋊D3)⋊C3C2
cosocleC23×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#99
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721310C2semidirect
3. #11720580C2semidirect
4. #63661⋊C3C2semidirect
5. #11720333C2semidirect
8. C23×(C3⋊D3)⋊C3C4semidirect
11. C2×#638⋊C3C4semidirect
15. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
16. C2×C4×(C3⋊D3)⋊C3C4semidirect
18. #88552C22semidirect
19. #3757⋊C3C22semidirect
25. #63661D3semidirect
32. C23×H(3,𝔽3)C2×C4semidirect
40. C22×(C3⋊D3)⋊C3C2×C4semidirect
44. C2×C4×H(3,𝔽3)C2×C4semidirect
49. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
57. C23×C3⋊D3Dic3semidirect
64. C2×#638Dic3semidirect
68. C2×C4⋊(C3⋊D3)Dic3semidirect
69. C2×C4×C3⋊D3Dic3semidirect
78. #3757C2×D3semidirect
123. C23×C32C4×D3semidirect
124. C23×C32C2×Dic3semidirect
133. C22×C3⋊D3C2×Dic3semidirect
149. C2×C32×C4C4×D3semidirect
150. C2×C32×C4C2×Dic3semidirect
154. C2×C3⋊Dic3C2×Dic3semidirect
178. (C3⋊D3)⋊C3#99semidirect
272. H(3,𝔽3)C2×#99semidirect
359. C3⋊D3#834semidirect
445. C32D3×#99semidirect
446. C32C2×#834semidirect
456. C23((C3⋊D3)⋊C3)⋊C4semidirect
468. C2×C4((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
#117213102
#117205802
#63661⋊C32
#117203332
#105915273
D3×#8343