Finite Group Databasebeta

← #13140262

Subgroups — #13140262

Order 1728.

1259 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 1259 conjugacy classes of subgroups — too many to draw. Only its 133 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
2C22×#6200C212yesmaximal, complemented
3C2×#88542C211yesmaximal, characteristic, complemented
4C2×#88450C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10591395—31nomaximal
7#10591389—31nomaximal
8C23×#272C2211yescharacteristic
9C23×(C3⋊D3)⋊C3C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)—24no—
14C2×C4⋊((C3⋊D3)⋊C3)—22no—
15C2×#6200—24no—
16C2×#6200—24no—
17C2×#6200—24no—
18#88542—22no—
19#88450—24no—
20C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
21C22×C3×D12—31no—
22C22×#626—32no—
23C22×#625—32no—
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×(C2×D3)⋊C4—31no—
26C2×C3×(C2×D3)⋊C4—61no—
27C2×#3722—32no—
28C2×(C2×D3)⋊Dic3—32no—
29#63658D311yescharacteristic, complemented
30C3×C3⋊#95—31no—
31C23×H(3,𝔽3)C2311yescharacteristic
32C22×#272—21no—
33C22×#272—22no—
34C22×#272—22no—
35C22×#272—21no—
36C22×#272D412yescomplemented
37C22×(C3⋊D3)⋊C3—22no—
38C22×(C3⋊D3)⋊C3—22no—
39C22×(C3⋊D3)⋊C3—22no—
40C22×(C3⋊D3)⋊C3—22no—
41C22×(C3⋊D3)⋊C3—22no—
42C22×(C3⋊D3)⋊C3—22no—
43C22×(C3⋊D3)⋊C3D414yescomplemented
44C2×C4×H(3,𝔽3)—22no—
45C2×C4×H(3,𝔽3)D412yescomplemented
46C2×#1084—21no—
47C2×#1084—22no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3—22no—
50C2×(C3⋊Dic3)⋊C3D414yescomplemented
51C4⋊((C3⋊D3)⋊C3)—28no—
52#6200—216no—
53#5223—91no—
54#4960—91no—
55#4671—91no—
56C23×C3⋊D3C2×D312yes—
57C23×C3×D3—31no—
58C23×C3×D3—31no—
59C23×C3×D3—32no—
60C23×C3×D3—61no—
61C22×C32×C4—21no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D311yescharacteristic
64C22×C3⋊Dic3C2×D312yes—
65C2×C4⋊(C3⋊D3)—24no—
66C2×C4⋊(C3⋊D3)—22no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—61no—
71C2×C3×D12—64no—
72C2×C3×D12—62no—
73C2×#626—64no—
74C2×#626—64no—
75C2×#626—64no—
76C2×#625—64no—
77C2×#625—64no—
78C2×#625—64no—
79C3×(C2×D3)⋊C4—64no—
80C3×(C2×D3)⋊C4—62no—
81C3×(C2×D3)⋊C4—62no—
82#3722—64no—
83(C2×D3)⋊Dic3—64no—
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C22×H(3,𝔽3)C2×D412yes—
86C22×H(3,𝔽3)C2×D412yes—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C22×H(3,𝔽3)#4111yescharacteristic
89C2×#272—24no—
90C2×#272—22no—
91C2×#272—41no—
92C2×#272—42no—
93C2×#272—41no—
94C2×(C3⋊D3)⋊C3—24no—
95C2×(C3⋊D3)⋊C3—24no—
96C2×(C3⋊D3)⋊C3—24no—
97C2×(C3⋊D3)⋊C3—42no—
98C2×(C3⋊D3)⋊C3—42no—
99C2×(C3⋊D3)⋊C3—42no—
100C2×(C3⋊D3)⋊C3—42no—
101C4×H(3,𝔽3)—22no—
102#1084—41no—
103(C3⋊Dic3)⋊C3—24no—
104C22×C3×D4—91no—
105C22×#67—92no—
106C22×#67—92no—
107C22×D12—31no—
108C22×D12—91no—
109C22×D12—92no—
110C2×C3×C22⋊C4—91no—
111C2×#217—92no—
112C2×(C2×D3)⋊C4—31no—
113C2×(C2×D3)⋊C4—31no—
114C2×(C2×D3)⋊C4—61no—
115C2×(C2×D3)⋊C4—92no—
116C2×(C2×D3)⋊C4—92no—
117C3×#95—91no—
118C3⋊#95—31no—
119C3⋊#95—91no—
120C23×C32—21no—
121C23×C32C22×D311yescharacteristic
122C23×C32C22×D311yescharacteristic
123C22×C3⋊D3—22no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3—22no—
126C22×C3⋊D3—22no—
127C22×C3⋊D3—22no—
128C22×C3⋊D3—22no—
129C22×C3⋊D3#6714yescomplemented
130C22×C3×D3—34no—
131C22×C3×D3—32no—
132C22×C3×D3—32no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C22×C3×D3—62no—
136C22×C3×D3—62no—
137C22×C3×D3—62no—
138C22×C3×D3—62no—
139C22×C3×D3—61no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C22×C3×D3—61no—
146C22×C3×D3—62no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C22×C3×D3—62no—
150C22×C3×D3—122no—
151C22×C3×D3—122no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—42no—
156C2×C32×C4D1212yescomplemented
157C2×C32×C4#6712yescomplemented
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3D1214yescomplemented
161C4⋊(C3⋊D3)—28no—
162C2×C3×Dic3—34no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—122no—
171C3×D12—68no—
172#626—616no—
173#625—616no—
174#392—271noSylow
175C2×H(3,𝔽3)#12012yes—
176C2×H(3,𝔽3)#12011yescharacteristic
177C2×H(3,𝔽3)#12111yescharacteristic
178C2×H(3,𝔽3)#12112yes—
179C2×H(3,𝔽3)#12711yescharacteristic
180#272—42no—
181(C3⋊D3)⋊C3—44no—
182C24×C3—92no—
183C24×C3—91no—
184C23×D3—31no—
185C23×D3—31no—
186C23×D3—32no—
187C23×D3—61no—
188C23×D3—92no—
189C2×C3×D4—182no—
190C2×C3×D4—184no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—61no—
194C22×C3×C4—92no—
195C22×C3×C4—91no—
196C22×C3×C4D3211yescharacteristic
197C2×#67—184no—
198C2×#67—184no—
199C2×#67—184no—
200C2×#67—184no—
201C2×#67—184no—
202C2×#67—184no—
203C22×Dic3—32no—
204C22×Dic3—31no—
205C22×Dic3—31no—
206C22×Dic3—61no—
207C22×Dic3—92no—
208C2×D12—64no—
209C2×D12—62no—
210C2×D12—182no—
211C2×D12—184no—
212C2×D12—184no—
213C2×D12—184no—
214C2×D12—184no—
215C3×C22⋊C4—182no—
216#217—184no—
217(C2×D3)⋊C4—62no—
218(C2×D3)⋊C4—62no—
219(C2×D3)⋊C4—64no—
220(C2×D3)⋊C4—184no—
221(C2×D3)⋊C4—184no—
222C22×C32—21no—
223C22×C32—22no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—22no—
227C22×C32C2×D1212yes—
228C22×C32C2×D1211yescharacteristic
229C22×C32D3×D412yes—
230C22×C32D3×D411yescharacteristic
231C22×C32D3×D411yescharacteristic
232C22×C32D3×D412yes—
233C22×C32#23711yescharacteristic
234C22×C32#23911yescharacteristic
235C22×C32C2×#6711yescharacteristic
236C22×C32C2×#6712yes—
237C2×C3⋊D3—24no—
238C2×C3⋊D3—24no—
239C2×C3⋊D3—24no—
240C2×C3⋊D3—42no—
241C2×C3⋊D3—42no—
242C2×C3⋊D3—42no—
243C2×C3⋊D3—42no—
244C2×C3×D3—64no—
245C2×C3×D3—64no—
246C2×C3×D3—64no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—62no—
250C2×C3×D3—64no—
251C2×C3×D3—62no—
252C2×C3×D3—124no—
253C2×C3×D3—124no—
254C2×C3×D3—122no—
255C2×C3×D3—122no—
256C2×C3×D3—122no—
257C2×C3×D3—121no—
258C2×C3×D3—122no—
259C2×C3×D3—121no—
260C2×C3×D3—122no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263C2×C3×D3—122no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C32×C4—22no—
267C32×C4—22no—
268C32×C4—42no—
269C3⋊Dic3—24no—
270C3×Dic3—64no—
271C3×Dic3—122no—
272C3×Dic3—121no—
273C3×Dic3—121no—
274C22×D4—91no—
275C22×D4—272no—
276C2×C22⋊C4—91no—
277C2×C22⋊C4—272no—
278#95—91no—
279H(3,𝔽3)#39211yescharacteristic, complemented, Sylow
280C23×C3—31no—
281C23×C3—31no—
282C23×C3—61no—
283C23×C3—94no—
284C23×C3—92no—
285C23×C3—182no—
286C23×C3—182no—
287C23×C3—182no—
288C23×C3—182no—
289C23×C3—182no—
290C23×C3—182no—
291C23×C3—181no—
292C23×C3—182no—
293C23×C3—181no—
294C23×C3—182no—
295C23×C3C2×D3211yescharacteristic
296C22×D3—32no—
297C22×D3—32no—
298C22×D3—34no—
299C22×D3—62no—
300C22×D3—61no—
301C22×D3—61no—
302C22×D3—62no—
303C22×D3—62no—
304C22×D3—62no—
305C22×D3—62no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—62no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—62no—
312C22×D3—62no—
313C22×D3—62no—
314C22×D3—62no—
315C22×D3—62no—
316C22×D3—94no—
317C22×D3—122no—
318C22×D3—122no—
319C22×D3—182no—
320C22×D3—182no—
321C22×D3—182no—
322C22×D3—182no—
323C22×D3—182no—
324C22×D3—182no—
325C3×D4—188no—
326C2×C3×C4—22no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—94no—
333C2×C3×C4—122no—
334C2×C3×C4—182no—
335C2×C3×C4—182no—
336C2×C3×C4—182no—
337C2×C3×C4—181no—
338C2×C3×C4#62612yes—
339#67—1816no—
340#67—1816no—
341C2×Dic3—34no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—62no—
349C2×Dic3—94no—
350C2×Dic3—122no—
351C2×Dic3—182no—
352C2×Dic3—182no—
353D12—68no—
354D12—188no—
355D12—1816no—
356C2×C32—22no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—22no—
360C2×C32—21no—
361C2×C32#87411yescharacteristic
362C2×C32#88212yes—
363C2×C32#88211yescharacteristic
364C2×C32#89512yes—
365C2×C32#89511yescharacteristic
366C2×C32#93712yes—
367C2×C32#93811yescharacteristic
368C2×C32#93912yes—
369C2×C32#94011yescharacteristic
370C2×C32#95311yescharacteristic
371C3⋊D3—44no—
372C3×D3—124no—
373C3×D3—124no—
374C3×D3—122no—
375C3×D3—122no—
376C24—91no—
377C24—92no—
378C2×D4—184no—
379C2×D4—182no—
380C2×D4—544no—
381C2×D4—544no—
382C2×D4—544no—
383C22×C4—92no—
384C22×C4—91no—
385C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
386C22⋊C4—182no—
387C22⋊C4—544no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—32no—
391C22×C3—32no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—31no—
396C22×C3—32no—
397C22×C3—32no—
398C22×C3—61no—
399C22×C3—62no—
400C22×C3—61no—
401C22×C3—61no—
402C22×C3—62no—
403C22×C3—184no—
404C22×C3—184no—
405C22×C3—184no—
406C22×C3—182no—
407C22×C3—184no—
408C22×C3—362no—
409C22×C3—362no—
410C22×C3—362no—
411C22×C3—362no—
412C22×C3—362no—
413C22×C3—361no—
414C22×C3—361no—
415C22×C3#379611yescharacteristic
416C22×C3D3×D1212yes—
417C22×C3#380211yescharacteristic
418C22×C3C2×#62611yescharacteristic
419C22×C3D3×#6712yes—
420C2×D3—64no—
421C2×D3—64no—
422C2×D3—62no—
423C2×D3—62no—
424C2×D3—64no—
425C2×D3—64no—
426C2×D3—64no—
427C2×D3—64no—
428C2×D3—124no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—122no—
432C2×D3—122no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—122no—
436C2×D3—124no—
437C2×D3—122no—
438C2×D3—122no—
439C2×D3—122no—
440C2×D3—122no—
441C2×D3—122no—
442C2×D3—184no—
443C2×D3—184no—
444C2×D3—184no—
445C2×D3—362no—
446C2×D3—362no—
447C2×D3—362no—
448C2×D3—362no—
449C3×C4—22no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—122no—
453C3×C4—184no—
454C3×C4—361no—
455Dic3—64no—
456Dic3—121no—
457Dic3—121no—
458Dic3—122no—
459Dic3—184no—
460C32—21no—
461C32#467111yescharacteristic, complemented
462C32#522311yescharacteristic, complemented
463C23—92no—
464C23—94no—
465C23—181no—
466C23—182no—
467C23—182no—
468C23—182no—
469C23—182no—
470C23—182no—
471C23—181no—
472C23—182no—
473C23—182no—
474C23—182no—
475C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
476D4—188no—
477D4—5416no—
478C2×C4—22no—
479C2×C4—94no—
480C2×C4—182no—
481C2×C4—182no—
482C2×C4—182no—
483C2×C4—181no—
484C2×C4#620012yescomplemented
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—32no—
488C2×C3—32no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—32no—
494C2×C3—32no—
495C2×C3—61no—
496C2×C3—62no—
497C2×C3—61no—
498C2×C3—61no—
499C2×C3—62no—
500C2×C3—364no—
501C2×C3—362no—
502C2×C3#6388612yes—
503C2×C3#6391111yescharacteristic
504C2×C3#6391611yescharacteristic
505C2×C3#6392212yes—
506C2×C3#6397711yescharacteristic
507D3—122no—
508D3—122no—
509D3—124no—
510D3—124no—
511D3—364no—
512C22—184no—
513C22—184no—
514C22—182no—
515C22—184no—
516C22—184no—
517C22—361no—
518C22—362no—
519C22—362no—
520C22—361no—
521C22—362no—
522C22—362no—
523C22—362no—
524C22#8864911yescharacteristic
525C22#8865412yes—
526C22#8865511yescharacteristic
527C22C2×#620011yescharacteristic
528C22#8867612yes—
529C4—22no—
530C4—184no—
531C4—361no—
532C3—31no—
533C3—31no—
534C3—61no—
535C3#1059139111yescharacteristic
536C2—362no—
537C2—364no—
538C2#1172112712yes—
539C2#1172115811yescharacteristic
540C2#1172115911yescharacteristic
541C2#1172116212yes—
542C2#1172132111yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#392
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
2. C22×#6200C2semidirect
3. C2×#88542C2semidirect
4. C2×#88450C2semidirect
5. #63658⋊C3C2semidirect
20. C22×C4⋊(C3⋊D3)D3semidirect
29. #63658D3semidirect
36. C22×#272D4semidirect
43. C22×(C3⋊D3)⋊C3D4semidirect
45. C2×C4×H(3,𝔽3)D4semidirect
50. C2×(C3⋊Dic3)⋊C3D4semidirect
129. C22×C3⋊D3#67semidirect
156. C2×C32×C4D12semidirect
157. C2×C32×C4#67semidirect
160. C2×C3⋊Dic3D12semidirect
279. H(3,𝔽3)#392semidirect
461. C32#4671semidirect
462. C32#5223semidirect
484. C2×C4#6200semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C22×#62002
C2×#885422
C2×#884502
#63658⋊C32
#105913953
#105913893