Finite Group Databasebeta

← #10592475⋊C3 #13141793

Subgroups — #10592475⋊C3 #13141793

Order 1728.

708 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 104 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
2C2×#3754⋊C3C211yesmaximal, characteristic, complemented
3C2×#3744⋊C3C211yesmaximal, characteristic, complemented
4#64106⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#63636⋊C3C211yesmaximal, characteristic, complemented
8#10592475C311yesmaximal, characteristic, complemented
9C3×#4776—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic, complemented
18C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
20#3754⋊C3—22no—
21#3752⋊C3—21no—
22C4⋊((C3⋊Dic3)⋊C3)—21no—
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
24#3744⋊C3—22no—
25C2×D4×C3⋊D3C2×C311yescharacteristic
26C2×C3×D3×D4—31no—
27C2×#3754C2×C311yescharacteristic, complemented
28C2×#3744C2×C311yescharacteristic, complemented
29#64106C2×C311yescharacteristic, complemented
30C2×C3×#214—31no—
31C2×C3×#204—31no—
32C3×#897—31no—
33C32×#102—21no—
34C32×#102D311yescharacteristic, complemented
35C32×#102C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63636C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—31no—
39C3×#809—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3—42no—
44C22×(C3⋊D3)⋊C3—42no—
45C22×(C3⋊D3)⋊C3D411yescharacteristic
46D4×H(3,𝔽3)—21no—
47D4×H(3,𝔽3)D412yes—
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50#638⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3D411yescharacteristic
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)D412yes—
55C4×(C3⋊D3)⋊C3—21no—
56C4×(C3⋊D3)⋊C3—21no—
57#634⋊C3—21no—
58#634⋊C3—21no—
59C8×H(3,𝔽3)—21no—
60(C3⋊(C3⋊C8))⋊C3—21no—
61C3×#450—91no—
62#4776—31no—
63#4776—91no—
64C23×C3⋊D3—21no—
65C23×C3×D3—61no—
66C2×C32×D4—21no—
67C2×C32×D4C2×D311yescharacteristic
68C2×C32×D4C22×C311yescharacteristic
69C2×#638—21no—
70D4×C3⋊D3—22no—
71D4×C3⋊D3—22no—
72C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
73C2×C4×C3⋊D3C22×C311yescharacteristic
74C2×#634C22×C311yescharacteristic, complemented
75C2×C3×#67—61no—
76C3×D3×D4—62no—
77C3×D3×D4—62no—
78C2×C3×D12—31no—
79C2×C3×C4×D3—31no—
80C2×C3×Dic6—31no—
81C2×C32×C8—21no—
82C2×C32×C8C2×D311yescharacteristic, complemented
83C2×C32×C8C22×C311yescharacteristic, complemented
84C32×C4⋊C4—21no—
85C32×C4⋊C4C2×D311yescharacteristic, complemented
86C32×C4⋊C4C22×C311yescharacteristic, complemented
87#3754—22no—
88#3752—21no—
89C4⋊(C3⋊Dic3)—21no—
90C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
91#3744—22no—
92C3×#214—62no—
93C3×(C2×D3)⋊C4—61no—
94C3×C4⋊Dic3—61no—
95C2×C3×C3⋊C8—31no—
96C3×#204—62no—
97C22×H(3,𝔽3)—22no—
98C22×H(3,𝔽3)—41no—
99C22×H(3,𝔽3)—41no—
100C22×H(3,𝔽3)C2×D411yescharacteristic
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—22no—
104C2×(C3⋊D3)⋊C3—48no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3SD1612yescomplemented
108C4×H(3,𝔽3)—41no—
109C4×H(3,𝔽3)C2×D411yescharacteristic
110C4×H(3,𝔽3)C2×D411yescharacteristic
111(C3⋊Dic3)⋊C3—21no—
112(C3⋊Dic3)⋊C3—41no—
113C22×C3×D4—91no—
114C2×D3×D4—31no—
115C2×D3×D4—91no—
116C2×C3×SD16—91no—
117C2×C3×SD16—91no—
118C3×C22⋊Q8—91no—
119C2×#214—31no—
120C2×#214—91no—
121C2×#204—31no—
122C2×#204—91no—
123#897—31no—
124#897—91no—
125C3×#102—31no—
126C3×#102—31no—
127C3×#102—61no—
128C3×#102—91no—
129C3×#102C3×D311yescharacteristic
130C3×C22⋊C8—91no—
131(C2×D3)⋊C8—31no—
132(C2×D3)⋊C8—91no—
133#809—31no—
134#809—91no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—41no—
138C22×C3⋊D3—24no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3—42no—
141C22×C3⋊D3—42no—
142C22×C3⋊D3C3×D411yescharacteristic
143C22×C3×D3—31no—
144C22×C3×D3—64no—
145C22×C3×D3—61no—
146C22×C3×D3—122no—
147C22×C3×D3—122no—
148C32×D4—21no—
149C32×D4—21no—
150C32×D4—22no—
151C32×D4—41no—
152C32×D4D1212yes—
153C32×D4C3×D412yes—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160#638—24no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3C3×D411yescharacteristic
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)C3×D412yes—
165C4×C3⋊D3—21no—
166C4×C3⋊D3—21no—
167#634—21no—
168#634—21no—
169C3×#67—64no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—61no—
172C3×D12—32no—
173C3×D12—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×Dic6—61no—
177C3×Dic6—61no—
178C32×C8—21no—
179C32×C8—21no—
180C32×C8—41no—
181C3⋊(C3⋊C8)—21no—
182C3×C3⋊C8—61no—
183#450—91noSylow
184C2×H(3,𝔽3)—42no—
185C2×H(3,𝔽3)#12011yescharacteristic
186C2×H(3,𝔽3)C2×SD1611yescharacteristic
187C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
188(C3⋊D3)⋊C3—22no—
189(C3⋊D3)⋊C3—42no—
190C24×C3—181no—
191C23×D3—61no—
192C23×D3—181no—
193C2×C3×Q8—91no—
194C2×C3×D4—31no—
195C2×C3×D4—31no—
196C2×C3×D4—61no—
197C2×C3×D4—91no—
198C2×C3×D4—181no—
199C2×C3×D4—182no—
200C2×C3×D4—182no—
201C2×C3×D4C2×C3×D311yescharacteristic
202C22×C3×C4—91no—
203C2×#67—61no—
204C2×#67—181no—
205D3×D4—62no—
206D3×D4—62no—
207D3×D4—182no—
208D3×D4—182no—
209C2×D12—31no—
210C2×D12—91no—
211C2×C4×D3—31no—
212C2×C4×D3—91no—
213C2×Dic6—31no—
214C2×Dic6—91no—
215C3×SD16—182no—
216C3×SD16—182no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8—61no—
220C2×C3×C8—91no—
221C2×C3×C8C2×C3×D311yescharacteristic
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—61no—
225C3×C4⋊C4—181no—
226C3×C4⋊C4C2×C3×D311yescharacteristic
227C3×C22⋊C4—181no—
228#214—62no—
229#214—182no—
230(C2×D3)⋊C4—61no—
231(C2×D3)⋊C4—181no—
232C4⋊Dic3—61no—
233C4⋊Dic3—181no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—91no—
236#204—62no—
237#204—182no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—21no—
241C22×C32—42no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32—41no—
246C22×C32—81no—
247C22×C32—81no—
248C22×C32D3×D411yescharacteristic
249C22×C32C2×C3×D411yescharacteristic
250C2×C3⋊D3—21no—
251C2×C3⋊D3—21no—
252C2×C3⋊D3—22no—
253C2×C3⋊D3—48no—
254C2×C3⋊D3—41no—
255C2×C3⋊D3—41no—
256C2×C3⋊D3C3×SD1612yescomplemented
257C2×C3×D3—32no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—62no—
261C2×C3×D3—128no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4—81no—
269C32×C4C2×D1211yescharacteristic
270C32×C4D3×D411yescharacteristic
271C32×C4C2×C3×D411yescharacteristic
272C32×C4C2×C3×D411yescharacteristic
273C3⋊Dic3—21no—
274C3⋊Dic3—41no—
275C3×Dic3—61no—
276C3×Dic3—121no—
277C22×D4—91no—
278C2×SD16—91no—
279C2×SD16—91no—
280C22⋊Q8—91no—
281#102—91no—
282#102(C3⋊D3)⋊C311yescharacteristic, complemented
283C22⋊C8—91no—
284H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
285C23×C3—21no—
286C23×C3—61no—
287C23×C3—61no—
288C23×C3—91no—
289C23×C3—121no—
290C23×C3—181no—
291C23×C3—184no—
292C23×C3—362no—
293C23×C3—362no—
294C22×D3—31no—
295C22×D3—64no—
296C22×D3—61no—
297C22×D3—91no—
298C22×D3—122no—
299C22×D3—122no—
300C22×D3—181no—
301C22×D3—184no—
302C22×D3—362no—
303C22×D3—362no—
304C3×Q8—181no—
305C3×Q8—181no—
306C3×D4—21no—
307C3×D4—32no—
308C3×D4—32no—
309C3×D4—61no—
310C3×D4—61no—
311C3×D4—62no—
312C3×D4—92no—
313C3×D4—121no—
314C3×D4—181no—
315C3×D4—184no—
316C3×D4C3×D1212yes—
317C2×C3×C4—21no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—91no—
324C2×C3×C4—121no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4—181no—
328C2×C3×C4C22×C3×D311yescharacteristic
329#67—64no—
330#67—184no—
331C2×Dic3—31no—
332C2×Dic3—61no—
333C2×Dic3—91no—
334C2×Dic3—181no—
335D12—32no—
336D12—61no—
337D12—92no—
338D12—181no—
339C4×D3—61no—
340C4×D3—61no—
341C4×D3—181no—
342C4×D3—181no—
343Dic6—61no—
344Dic6—61no—
345Dic6—181no—
346Dic6—181no—
347C3×C8—21no—
348C3×C8—61no—
349C3×C8—61no—
350C3×C8—121no—
351C3×C8—181no—
352C3⋊C8—61no—
353C3⋊C8—181no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—42no—
358C2×C32—42no—
359C2×C32—82no—
360C2×C32#88211yescharacteristic
361C2×C32#91111yescharacteristic
362C2×C32D3×SD1611yescharacteristic
363C2×C32C3×#12011yescharacteristic
364C2×C32C2×C3×SD1611yescharacteristic
365C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
366C3⋊D3—22no—
367C3⋊D3—42no—
368C3×D3—62no—
369C3×D3—122no—
370C24—181no—
371C2×Q8—91no—
372C2×D4—91no—
373C2×D4—181no—
374C2×D4—182no—
375C2×D4—182no—
376C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
377C22×C4—91no—
378SD16—182no—
379SD16—182no—
380C2×C8—91no—
381C2×C8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
382C4⋊C4—181no—
383C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
384C22⋊C4—181no—
385C22×C3—22no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—62no—
391C22×C3—62no—
392C22×C3—61no—
393C22×C3—92no—
394C22×C3—122no—
395C22×C3—121no—
396C22×C3—121no—
397C22×C3—121no—
398C22×C3—121no—
399C22×C3—181no—
400C22×C3—181no—
401C22×C3—182no—
402C22×C3—241no—
403C22×C3—241no—
404C22×C3—361no—
405C22×C3—361no—
406C22×C3—368no—
407C22×C3C3×D3×D411yescharacteristic
408C2×D3—32no—
409C2×D3—61no—
410C2×D3—61no—
411C2×D3—62no—
412C2×D3—92no—
413C2×D3—128no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—181no—
417C2×D3—181no—
418C2×D3—182no—
419C2×D3—368no—
420C2×D3—361no—
421C2×D3—361no—
422C3×C4—31no—
423C3×C4—31no—
424C3×C4—31no—
425C3×C4—31no—
426C3×C4—41no—
427C3×C4—61no—
428C3×C4—61no—
429C3×C4—121no—
430C3×C4—121no—
431C3×C4—181no—
432C3×C4—241no—
433C3×C4—361no—
434C3×C4C2×C3×D1211yescharacteristic
435C3×C4C3×D3×D411yescharacteristic
436Dic3—61no—
437Dic3—121no—
438Dic3—181no—
439Dic3—361no—
440C32—21no—
441C32#477611yescharacteristic, complemented
442C32C3×#45011yescharacteristic, complemented
443C23—21no—
444C23—91no—
445C23—184no—
446C23—181no—
447C23—362no—
448C23—362no—
449Q8—181no—
450Q8—181no—
451D4—21no—
452D4—92no—
453D4—184no—
454D4—181no—
455D4C4⋊((C3⋊D3)⋊C3)12yes—
456C2×C4—21no—
457C2×C4—91no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
462C8—21no—
463C8—181no—
464C2×C3—31no—
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—42no—
471C2×C3—61no—
472C2×C3—61no—
473C2×C3—61no—
474C2×C3—122no—
475C2×C3—122no—
476C2×C3—182no—
477C2×C3—242no—
478C2×C3—362no—
479C2×C3C3×#88211yescharacteristic
480C2×C3C3×#91111yescharacteristic
481C2×C3C3×D3×SD1611yescharacteristic
482D3—62no—
483D3—122no—
484D3—182no—
485D3—362no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—92no—
490C22—181no—
491C22—181no—
492C22—182no—
493C22—368no—
494C22—361no—
495C22—361no—
496C22D4×(C3⋊D3)⋊C311yescharacteristic
497C4—41no—
498C4—181no—
499C4—361no—
500C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
501C4D4×(C3⋊D3)⋊C311yescharacteristic
502C3—31no—
503C3—31no—
504C3—61no—
505C3C3×#477611yescharacteristic
506C2—42no—
507C2—182no—
508C2—362no—
509C2#64091⋊C311yescharacteristic
510C2#64120⋊C311yescharacteristic
511C2SD16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×(C3⋊D3)⋊C3C2
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#450
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#3754⋊C3C2semidirect
3. C2×#3744⋊C3C2semidirect
4. #64106⋊C3C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #63647⋊C3C2semidirect
7. #63636⋊C3C2semidirect
8. #10592475C3semidirect
17. C2×#634⋊C3C22semidirect
18. C2×C8×H(3,𝔽3)C22semidirect
19. C4⋊C4×H(3,𝔽3)C22semidirect
23. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
27. C2×#3754C2×C3semidirect
28. C2×#3744C2×C3semidirect
29. #64106C2×C3semidirect
34. C32×#102D3semidirect
35. C32×#102C2×C3semidirect
36. #63647C2×C3semidirect
37. #63636C2×C3semidirect
74. C2×#634C22×C3semidirect
82. C2×C32×C8C2×D3semidirect
83. C2×C32×C8C22×C3semidirect
85. C32×C4⋊C4C2×D3semidirect
86. C32×C4⋊C4C22×C3semidirect
90. C2×C3⋊(C3⋊C8)C22×C3semidirect
107. C2×(C3⋊D3)⋊C3SD16semidirect
256. C2×C3⋊D3C3×SD16semidirect
282. #102(C3⋊D3)⋊C3semidirect
284. H(3,𝔽3)#450semidirect
381. C2×C8C2×(C3⋊D3)⋊C3semidirect
383. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
441. C32#4776semidirect
442. C32C3×#450semidirect

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
C2×#3754⋊C32
C2×#3744⋊C32
#64106⋊C32
H(3,𝔽3)×#1022
#63647⋊C32
#63636⋊C32
#105924753
C3×#47763