Finite Group Databasebeta

← #13141166

Subgroups — #13141166

Order 1728.

1237 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1084C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic
3C2×#3757⋊C3C211yesmaximal, characteristic, complemented
4C2×#88450C212yesmaximal, complemented
5#11720334C212yesmaximal, complemented
6#10591711—31nomaximal
7#10591710—31nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C22×#1084—22no—
11C22×#1084—24no—
12C22×#1084C2212yes—
13C2×#638⋊C3—22no—
14C2×#638⋊C3—22no—
15C2×#638⋊C3C414yescomplemented
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#3757⋊C3—22no—
19#88450—24no—
20C22×#638D311yescharacteristic
21C22×C3×#67—31no—
22C23×C3×Dic3—31no—
23C23×C3×Dic3—31no—
24C23×C3×Dic3—61no—
25C2×#3757D311yescharacteristic, complemented
26C2×C3×#217—31no—
27C2×#3722—32no—
28C2×(C2×D3)⋊Dic3—32no—
29#63579—32no—
30#63579—32no—
31C23×H(3,𝔽3)—22no—
32C23×H(3,𝔽3)—22no—
33C23×H(3,𝔽3)C2×C412yes—
34C23×H(3,𝔽3)D412yes—
35C23×H(3,𝔽3)D412yes—
36C23×H(3,𝔽3)C2311yescharacteristic
37C22×(C3⋊D3)⋊C3—22no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—22no—
41C22×(C3⋊D3)⋊C3C2×C412yescomplemented
42C2×#1084—28no—
43C2×#1084—22no—
44C2×#1084—22no—
45C2×#1084D414yescomplemented
46#638⋊C3—28no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—22no—
50C2×(C3⋊Dic3)⋊C3—22no—
51C2×(C3⋊Dic3)⋊C3—22no—
52C2×(C3⋊Dic3)⋊C3C2×C412yescomplemented
53#5220—91no—
54#5213—91no—
55#4958—91no—
56C24×C32—21no—
57C24×C32C2×D311yescharacteristic
58C24×C32C2×D311yescharacteristic
59C23×C3⋊D3C2×D311yescharacteristic
60C23×C3×D3—31no—
61C2×#638—22no—
62C2×#638—22no—
63C2×#638Dic314yescomplemented
64C22×C3⋊Dic3C2×D312yes—
65C22×C3⋊Dic3C2×D311yescharacteristic
66C2×C3×#67—34no—
67C2×C3×#67—62no—
68C2×C3×#67—62no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—32no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—64no—
74C22×C3×Dic3—64no—
75C22×C3×Dic3—62no—
76C22×C3×Dic3—68no—
77C22×C3×Dic3—64no—
78C22×C3×Dic3—62no—
79C22×C3×Dic3—62no—
80#3757—22no—
81C3×#217—62no—
82#3722—64no—
83(C2×D3)⋊Dic3—64no—
84C22×H(3,𝔽3)—24no—
85C22×H(3,𝔽3)—22no—
86C22×H(3,𝔽3)—24no—
87C22×H(3,𝔽3)—22no—
88C22×H(3,𝔽3)C22⋊C414yescomplemented
89C22×H(3,𝔽3)C22×C411yescharacteristic
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C22×H(3,𝔽3)C2×D412yes—
93C22×H(3,𝔽3)#4112yes—
94C2×(C3⋊D3)⋊C3—22no—
95C2×(C3⋊D3)⋊C3—24no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—42no—
98C2×(C3⋊D3)⋊C3—41no—
99#1084—24no—
100(C3⋊Dic3)⋊C3—22no—
101(C3⋊Dic3)⋊C3—42no—
102C22×C3×D4—91no—
103C23×C3×C4—91no—
104C22×#67—31no—
105C22×#67—91no—
106C23×Dic3—31no—
107C23×Dic3—31no—
108C23×Dic3—61no—
109C2×C3×C22⋊C4—91no—
110C2×#217—31no—
111C2×#217—91no—
112C2×#217—92no—
113C2×(C2×D3)⋊C4—92no—
114C2×(C2×D3)⋊C4—92no—
115C3⋊#95—92no—
116C3⋊#95—92no—
117C3⋊#95—92no—
118C23×C32—24no—
119C23×C32—24no—
120C23×C32—22no—
121C23×C32—22no—
122C23×C32—22no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32—22no—
127C23×C32—21no—
128C23×C32C4×D312yes—
129C23×C32D1212yes—
130C23×C32C2×Dic312yes—
131C23×C32#6712yes—
132C23×C32#6712yes—
133C23×C32#6712yes—
134C23×C32C22×D311yescharacteristic
135C23×C32C22×D311yescharacteristic
136C22×C3⋊D3—22no—
137C22×C3⋊D3—22no—
138C22×C3⋊D3—21no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3C2×Dic312yescomplemented
141C22×C3×D3—32no—
142C22×C3×D3—61no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C22×C3×D3—61no—
146#638—28no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3C2×Dic312yescomplemented
153C3×#67—68no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—616no—
159C2×C3×Dic3—68no—
160C2×C3×Dic3—68no—
161C2×C3×Dic3—64no—
162C2×C3×Dic3—64no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—62no—
172#386—271noSylow
173C2×H(3,𝔽3)—24no—
174C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
175C2×H(3,𝔽3)C4×D412yes—
176C2×H(3,𝔽3)#12011yescharacteristic
177C2×H(3,𝔽3)#12112yes—
178C2×H(3,𝔽3)#12311yescharacteristic
179(C3⋊D3)⋊C3—42no—
180C24×C3—31no—
181C24×C3—31no—
182C24×C3—61no—
183C24×C3—91no—
184C24×C3D3211yescharacteristic
185C23×D3—31no—
186C23×D3—91no—
187C2×C3×D4—94no—
188C2×C3×D4—182no—
189C2×C3×D4—182no—
190C22×C3×C4—91no—
191C22×C3×C4—92no—
192C22×C3×C4—92no—
193C22×C3×C4—184no—
194C22×C3×C4—182no—
195C2×#67—34no—
196C2×#67—62no—
197C2×#67—62no—
198C2×#67—94no—
199C2×#67—182no—
200C2×#67—182no—
201C22×Dic3—32no—
202C22×Dic3—32no—
203C22×Dic3—31no—
204C22×Dic3—32no—
205C22×Dic3—62no—
206C22×Dic3—64no—
207C22×Dic3—64no—
208C22×Dic3—62no—
209C22×Dic3—62no—
210C22×Dic3—68no—
211C22×Dic3—64no—
212C22×Dic3—91no—
213C22×Dic3—92no—
214C3×C22⋊C4—182no—
215#217—62no—
216#217—182no—
217#217—184no—
218(C2×D3)⋊C4—184no—
219(C2×D3)⋊C4—184no—
220C22×C32—24no—
221C22×C32—24no—
222C22×C32—28no—
223C22×C32—28no—
224C22×C32—24no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—24no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—21no—
234C22×C32—22no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—22no—
238C22×C32(C2×D3)⋊C414yescomplemented
239C22×C32#21714yescomplemented
240C22×C32C2×C4×D311yescharacteristic
241C22×C32C2×D1211yescharacteristic
242C22×C32D3×D412yes—
243C22×C32D3×D412yes—
244C22×C32#23712yes—
245C22×C32#23712yes—
246C22×C32C22×Dic311yescharacteristic
247C22×C32C2×#6711yescharacteristic
248C22×C32C2×#6711yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C2×C3⋊D3—24no—
251C2×C3⋊D3—22no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C2×C3⋊D3—42no—
255C2×C3×D3—64no—
256C2×C3×D3—62no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—122no—
260C3⋊Dic3—22no—
261C3⋊Dic3—42no—
262C3×Dic3—68no—
263C3×Dic3—64no—
264C3×Dic3—62no—
265C3×Dic3—64no—
266C3×Dic3—122no—
267C22×D4—91no—
268C23×C4—91no—
269C2×C22⋊C4—91no—
270C2×C22⋊C4—272no—
271#95—272no—
272H(3,𝔽3)#38611yescharacteristic, complemented, Sylow
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—32no—
276C23×C3—32no—
277C23×C3—32no—
278C23×C3—32no—
279C23×C3—32no—
280C23×C3—32no—
281C23×C3—31no—
282C23×C3—31no—
283C23×C3—62no—
284C23×C3—62no—
285C23×C3—62no—
286C23×C3—62no—
287C23×C3—64no—
288C23×C3—64no—
289C23×C3—62no—
290C23×C3—62no—
291C23×C3—62no—
292C23×C3—61no—
293C23×C3—92no—
294C23×C3—182no—
295C23×C3—181no—
296C23×C3—181no—
297C23×C3—182no—
298C23×C3D3×Dic312yes—
299C23×C3#62512yes—
300C23×C3#62612yes—
301C23×C3C2×D3211yescharacteristic
302C22×D3—32no—
303C22×D3—62no—
304C22×D3—61no—
305C22×D3—61no—
306C22×D3—62no—
307C22×D3—92no—
308C22×D3—182no—
309C22×D3—181no—
310C22×D3—181no—
311C22×D3—182no—
312C3×D4—188no—
313C2×C3×C4—92no—
314C2×C3×C4—94no—
315C2×C3×C4—188no—
316C2×C3×C4—181no—
317C2×C3×C4—181no—
318C2×C3×C4—182no—
319C2×C3×C4—182no—
320C2×C3×C4—182no—
321C2×C3×C4—182no—
322C2×C3×C4—182no—
323#67—68no—
324#67—188no—
325C2×Dic3—34no—
326C2×Dic3—34no—
327C2×Dic3—32no—
328C2×Dic3—64no—
329C2×Dic3—62no—
330C2×Dic3—62no—
331C2×Dic3—62no—
332C2×Dic3—62no—
333C2×Dic3—68no—
334C2×Dic3—68no—
335C2×Dic3—616no—
336C2×Dic3—64no—
337C2×Dic3—64no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—62no—
341C2×Dic3—62no—
342C2×Dic3—62no—
343C2×Dic3—92no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—182no—
347C2×Dic3—182no—
348C2×Dic3—182no—
349C2×C32—28no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—22no—
355C2×C32—22no—
356C2×C32—21no—
357C2×C32#67⋊C412yes—
358C2×C32#88211yescharacteristic
359C2×C32#88611yescharacteristic
360C2×C32C2×(C2×D3)⋊C411yescharacteristic
361C2×C32D4×Dic312yes—
362C2×C32#93511yescharacteristic
363C2×C32#93711yescharacteristic
364C2×C32#93912yes—
365C2×C32#93912yes—
366C2×C32C2×#21711yescharacteristic
367C3⋊D3—42no—
368C3×D3—122no—
369C24—91no—
370C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
371C2×D4—94no—
372C2×D4—182no—
373C2×D4—182no—
374C22×C4—91no—
375C22×C4—92no—
376C22×C4—92no—
377C22×C4—184no—
378C22×C4—182no—
379C22⋊C4—182no—
380C22⋊C4—544no—
381C22×C3—22no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—22no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—32no—
388C22×C3—32no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—32no—
394C22×C3—32no—
395C22×C3—34no—
396C22×C3—34no—
397C22×C3—62no—
398C22×C3—62no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—64no—
402C22×C3—64no—
403C22×C3—64no—
404C22×C3—64no—
405C22×C3—64no—
406C22×C3—64no—
407C22×C3—68no—
408C22×C3—68no—
409C22×C3—61no—
410C22×C3—62no—
411C22×C3—61no—
412C22×C3—61no—
413C22×C3—62no—
414C22×C3—64no—
415C22×C3—182no—
416C22×C3—184no—
417C22×C3—361no—
418C22×C3—362no—
419C22×C3—361no—
420C22×C3(C2×D3)⋊Dic314yes—
421C22×C3C2×D3×Dic311yescharacteristic
422C22×C3#380512yes—
423C22×C3C2×#62511yescharacteristic
424C22×C3C2×#62611yescharacteristic
425C22×C3#381112yes—
426C2×D3—62no—
427C2×D3—64no—
428C2×D3—121no—
429C2×D3—122no—
430C2×D3—121no—
431C2×D3—182no—
432C2×D3—184no—
433C2×D3—361no—
434C2×D3—362no—
435C2×D3—361no—
436C3×C4—182no—
437C3×C4—184no—
438C3×C4—362no—
439Dic3—64no—
440Dic3—64no—
441Dic3—68no—
442Dic3—62no—
443Dic3—122no—
444Dic3—182no—
445Dic3—362no—
446C32—21no—
447C32#495811yescharacteristic, complemented
448C32#522011yescharacteristic, complemented
449C23—22no—
450C23—22no—
451C23—92no—
452C23—182no—
453C23—181no—
454C23—181no—
455C23—182no—
456C23((C3⋊D3)⋊C3)⋊C412yes—
457C23#619812yes—
458C23#620012yes—
459C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
460D4—188no—
461C2×C4—92no—
462C2×C4—94no—
463C2×C4—188no—
464C2×C4—181no—
465C2×C4—182no—
466C2×C4—181no—
467C2×C4—182no—
468C2×C4—182no—
469C2×C4—182no—
470C2×C4—182no—
471C2×C3—24no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—32no—
477C2×C3—32no—
478C2×C3—32no—
479C2×C3—32no—
480C2×C3—31no—
481C2×C3—31no—
482C2×C3—64no—
483C2×C3—64no—
484C2×C3—68no—
485C2×C3—61no—
486C2×C3—61no—
487C2×C3—62no—
488C2×C3—62no—
489C2×C3—61no—
490C2×C3—362no—
491C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
492C2×C3#6396211yescharacteristic
493C2×C3#67⋊Dic312yes—
494C2×C3#6398012yes—
495C2×C3#6398111yescharacteristic
496D3—122no—
497D3—362no—
498C22—22no—
499C22—24no—
500C22—24no—
501C22—22no—
502C22—182no—
503C22—184no—
504C22—361no—
505C22—362no—
506C22—361no—
507C22#8845014yescomplemented
508C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
509C22#8867212yes—
510C22C2×#619811yescharacteristic
511C22C2×#620011yescharacteristic
512C22#8867712yes—
513C4—182no—
514C4—184no—
515C4—362no—
516C3—31no—
517C3—31no—
518C3—61no—
519C3#1059171311yescharacteristic
520C2—24no—
521C2—362no—
522C2C2×#8845011yescharacteristic
523C2#1172130711yescharacteristic
524C2#1172131512yes—
525C2#1172131812yes—
526C2#1172132311yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC24×H(3,𝔽3)C22
prime radicalC22×#638⋊C3C2
cosocleC23×H(3,𝔽3)C23
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#386
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×#1084C2semidirect
3. C2×#3757⋊C3C2semidirect
4. C2×#88450C2semidirect
5. #11720334C2semidirect
15. C2×#638⋊C3C4semidirect
25. C2×#3757D3semidirect
41. C22×(C3⋊D3)⋊C3C2×C4semidirect
45. C2×#1084D4semidirect
52. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
63. C2×#638Dic3semidirect
88. C22×H(3,𝔽3)C22⋊C4semidirect
140. C22×C3⋊D3C2×Dic3semidirect
152. C2×C3⋊Dic3C2×Dic3semidirect
238. C22×C32(C2×D3)⋊C4semidirect
239. C22×C32#217semidirect
272. H(3,𝔽3)#386semidirect
447. C32#4958semidirect
448. C32#5220semidirect
507. C22#88450semidirect

Maximal subgroups

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

subgroupindex
C23×#10842
C22×#638⋊C32
C2×#3757⋊C32
C2×#884502
#117203342
#105917113
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