Finite Group Databasebeta

← #13149574

Subgroups — #13149574

Order 1728.

1219 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1219 conjugacy classes of subgroups — too many to draw. Only its 163 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×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2C2×#88651C211yesmaximal, characteristic, complemented
3C4⋊C4×#272C211yesmaximal, characteristic, complemented
4#64105⋊C3C212yesmaximal, complemented
5#11721161C211yesmaximal, characteristic
6#11721161C211yesmaximal, characteristic
7#11721147C211yesmaximal, characteristic, complemented
8#11721147C211yesmaximal, characteristic, complemented
9#11721133C212yesmaximal, complemented
10#11721128C212yesmaximal, complemented
11#11721101C212yesmaximal, complemented
12D3×#896—32nomaximal
13C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2212yescomplemented
19C2×#6199C2211yescharacteristic
20C2×#6199C2211yescharacteristic
21C2×#6199C2211yescharacteristic
22C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
23C2×#6196C2212yescomplemented
24C4×((C3⋊D3)⋊C3)⋊C2—24no—
25#88651—24no—
26C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
27C4⋊#1084C2211yescharacteristic, complemented
28#88540C2211yescharacteristic, complemented
29#88540C2211yescharacteristic, complemented
30#3752⋊C3C2212yes—
31#3752⋊C3C2212yes—
32#3750⋊C3C2212yescomplemented
33#3750⋊C3C2212yescomplemented
34#88450C2212yes—
35#88450C2212yes—
36#88448C2212yescomplemented
37#88448C2212yescomplemented
38C2×C4×D32—32no—
39C2×D3×Dic6—32no—
40#64105D312yescomplemented
41C3×#896—32no—
42C3×D3×C4⋊C4—32no—
43C3×D3×C4⋊C4—61no—
44D3×(C2×D3)⋊C4—32no—
45D3×(C2×D3)⋊C4—32no—
46#63899—32no—
47#63896—32no—
48#63895—32no—
49#63888—32no—
50D3×Dic3⋊C4—32no—
51D3×Dic3⋊C4—32no—
52#63865—32no—
53#63851—32no—
54C22×#272C2311yescharacteristic
55C22×(C3⋊D3)⋊C3C2312yes—
56C2×((C3⋊D3)⋊C3)⋊C2—22no—
57C2×((C3⋊D3)⋊C3)⋊C2—22no—
58C2×((C3⋊D3)⋊C3)⋊C2Q814yescomplemented
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×#1084C2311yescharacteristic
63C2×#1084C2311yescharacteristic
64C2×#1084C2311yescharacteristic
65C4×#272—22no—
66C4×#272—22no—
67C4×#272—22no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C4×(C3⋊D3)⋊C3—24no—
72#634⋊C3—24no—
73#6199—22no—
74#6199—22no—
75#6199D414yes—
76((C3⋊D3)⋊C3)⋊C4—24no—
77#6196—24no—
78D3×C22⋊Q8—92no—
79C2×#896—91no—
80C22×D32—32no—
81C2×C4×C3⋊D3C2×D312yes—
82C2×#634C2×D312yescomplemented
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—61no—
88C2×C3×C4×D3—61no—
89C2×C3×C4×D3—61no—
90C2×C3×Dic6—32no—
91C2×#627—32no—
92C2×#627—32no—
93C2×#624—32no—
94C2×D3×Dic3—32no—
95C2×D3×Dic3—32no—
96C2×D3×Dic3—32no—
97C2×D3×Dic3—32no—
98C4×D32—68no—
99D3×Dic6—68no—
100C32×C4⋊C4—21no—
101C32×C4⋊C4C2×D312yescomplemented
102#3752C2×D312yes—
103#3752C2×D312yes—
104#3750C2×D312yescomplemented
105#3750C2×D312yescomplemented
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×C4⋊Dic3—32no—
109C3×C4⋊Dic3—61no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—61no—
115C3×Dic3⋊C4—61no—
116#3724—32no—
117#3724—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120#3722—32no—
121#3722—32no—
122(C2×D3)⋊Dic3—32no—
123(C2×D3)⋊Dic3—32no—
124C22×H(3,𝔽3)C2411yescharacteristic
125C2×#272C2×D412yes—
126C2×#272C2×Q812yes—
127C2×#272#4112yes—
128C2×(C3⋊D3)⋊C3—22no—
129C2×(C3⋊D3)⋊C3—22no—
130C2×(C3⋊D3)⋊C3C2×Q814yescomplemented
131((C3⋊D3)⋊C3)⋊C2—28no—
132C4×H(3,𝔽3)—21no—
133C4×H(3,𝔽3)—21no—
134C4×H(3,𝔽3)—21no—
135#1084—21no—
136#1084—21no—
137#1084—21no—
138(C3⋊Dic3)⋊C3—22no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3C2×D414yes—
141C2×D3×Q8—92no—
142C22×C4×D3—91no—
143C22×C4×D3—92no—
144C22×Dic6—91no—
145C3×C22⋊Q8—92no—
146C2×C3×C4⋊C4—91no—
147#946—92no—
148C2×(C2×D3)⋊C4—91no—
149C2×(C2×D3)⋊C4—91no—
150#924—92no—
151C2×Dic3⋊C4—91no—
152C2×Dic3⋊C4—91no—
153#897—92no—
154#896—32no—
155#896—92no—
156#896—92no—
157#896—92no—
158#896—92no—
159#896—92no—
160D3×C4⋊C4—32no—
161D3×C4⋊C4—61no—
162D3×C4⋊C4—92no—
163D3×C4⋊C4—92no—
164D3×C22⋊C4—92no—
165D3×C22⋊C4—92no—
166#878—92no—
167#878—92no—
168C22×C3⋊D3C22×D312yes—
169C22×C3×D3—32no—
170C22×C3×D3—32no—
171C22×C3×D3—61no—
172C2×D32—38no—
173C2×D32—64no—
174C2×D32—64no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D312yes—
181C2×C3⋊Dic3C22×D312yes—
182C2×C3⋊Dic3C22×D312yes—
183C2×C3⋊Dic3C22×D312yes—
184C4×C3⋊D3—24no—
185#634—24no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C3×C4×D3—64no—
196C3×C4×D3—64no—
197C3×C4×D3—64no—
198C3×C4×D3—64no—
199C3×C4×D3—64no—
200C3×C4×D3—122no—
201C3×C4×D3—122no—
202C3×Dic6—64no—
203#627—64no—
204#627—64no—
205#624—64no—
206D3×Dic3—38no—
207D3×Dic3—64no—
208D3×Dic3—64no—
209D3×Dic3—64no—
210C2×C22⋊Q8—271noSylow
211C2×H(3,𝔽3)C22×D411yescharacteristic
212C2×H(3,𝔽3)C22×Q811yescharacteristic
213C2×H(3,𝔽3)C2×#4111yescharacteristic
214#272C22⋊Q814yescomplemented
215(C3⋊D3)⋊C3—24no—
216C23×D3—91no—
217C23×D3—92no—
218C2×C3×Q8—92no—
219C22×C3×C4—92no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C22×Dic3—91no—
224C22×Dic3—91no—
225C22×Dic3—91no—
226C22×Dic3—92no—
227D3×Q8—188no—
228C2×C4×D3—32no—
229C2×C4×D3—32no—
230C2×C4×D3—32no—
231C2×C4×D3—32no—
232C2×C4×D3—61no—
233C2×C4×D3—61no—
234C2×C4×D3—61no—
235C2×C4×D3—92no—
236C2×C4×D3—92no—
237C2×C4×D3—92no—
238C2×C4×D3—92no—
239C2×C4×D3—92no—
240C2×C4×D3—92no—
241C2×C4×D3—184no—
242C2×C4×D3—188no—
243C2×Dic6—32no—
244C2×Dic6—92no—
245C2×Dic6—92no—
246C2×Dic6—92no—
247C2×Dic6—92no—
248C2×Dic6—184no—
249C3×C4⋊C4—32no—
250C3×C4⋊C4—61no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—91no—
255C3×C4⋊C4—91no—
256C3×C4⋊C4D3211yescharacteristic
257C3×C22⋊C4—92no—
258C3×C22⋊C4—92no—
259#217—92no—
260#217—92no—
261(C2×D3)⋊C4—32no—
262(C2×D3)⋊C4—32no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4—92no—
268(C2×D3)⋊C4—92no—
269C4⋊Dic3—32no—
270C4⋊Dic3—61no—
271C4⋊Dic3—92no—
272C4⋊Dic3—92no—
273Dic3⋊C4—32no—
274Dic3⋊C4—32no—
275Dic3⋊C4—32no—
276Dic3⋊C4—32no—
277Dic3⋊C4—61no—
278Dic3⋊C4—61no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285C22×C32—21no—
286C22×C32C23×D312yes—
287C2×C3⋊D3—22no—
288C2×C3⋊D3—22no—
289C2×C3⋊D3D3×Q814yescomplemented
290C2×C3×D3—34no—
291C2×C3×D3—34no—
292C2×C3×D3—34no—
293C2×C3×D3—34no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299D32—616no—
300C32×C4—22no—
301C32×C4—22no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—41no—
305C32×C4—41no—
306C3⋊Dic3—22no—
307C3⋊Dic3—22no—
308C3⋊Dic3D3×D414yes—
309C3×Dic3—34no—
310C3×Dic3—62no—
311C3×Dic3—62no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—121no—
317C3×Dic3—121no—
318C22×Q8—271no—
319C23×C4—271no—
320C22⋊Q8—92no—
321C22⋊Q8—272no—
322C22⋊Q8—272no—
323C22⋊Q8—272no—
324C2×C4⋊C4—91no—
325C2×C4⋊C4—271no—
326C2×C4⋊C4—271no—
327C2×C22⋊C4—271no—
328C2×C22⋊C4—271no—
329H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
330C23×C3—92no—
331C23×C3—91no—
332C22×D3—32no—
333C22×D3—32no—
334C22×D3—61no—
335C22×D3—92no—
336C22×D3—94no—
337C22×D3—98no—
338C22×D3—184no—
339C22×D3—184no—
340C22×D3—182no—
341C22×D3—182no—
342C3×Q8—184no—
343C2×C3×C4—32no—
344C2×C3×C4—32no—
345C2×C3×C4—32no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4—92no—
352C2×C3×C4—91no—
353C2×C3×C4—91no—
354C2×C3×C4—91no—
355C2×C3×C4—182no—
356C2×C3×C4—182no—
357C2×C3×C4—182no—
358C2×C3×C4—184no—
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—92no—
374C2×Dic3—94no—
375C2×Dic3—182no—
376C2×Dic3—182no—
377C2×Dic3—184no—
378C4×D3—64no—
379C4×D3—64no—
380C4×D3—64no—
381C4×D3—64no—
382C4×D3—64no—
383C4×D3—98no—
384C4×D3—122no—
385C4×D3—122no—
386C4×D3—184no—
387C4×D3—184no—
388C4×D3—184no—
389C4×D3—184no—
390C4×D3—184no—
391Dic6—64no—
392Dic6—184no—
393Dic6—184no—
394Dic6—184no—
395Dic6—184no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32C2×D3×D412yes—
400C2×C32C2×D3×Q812yes—
401C2×C32D3×#4112yes—
402C3⋊D3—24no—
403C3×D3—38no—
404C3×D3—64no—
405C3×D3—64no—
406C24—271no—
407C2×Q8—92no—
408C2×Q8—272no—
409C2×Q8—544no—
410C22×C4—91no—
411C22×C4—91no—
412C22×C4—91no—
413C22×C4—92no—
414C22×C4—271no—
415C22×C4—271no—
416C22×C4—271no—
417C22×C4—272no—
418C22×C4—544no—
419C4⋊C4—91no—
420C4⋊C4—91no—
421C4⋊C4—91no—
422C4⋊C4—92no—
423C4⋊C4—92no—
424C4⋊C4—272no—
425C4⋊C4—272no—
426C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
427C22⋊C4—92no—
428C22⋊C4—92no—
429C22⋊C4—272no—
430C22⋊C4—272no—
431C22×C3—32no—
432C22×C3—61no—
433C22×C3—94no—
434C22×C3—92no—
435C22×C3—92no—
436C22×C3—92no—
437C22×C3—182no—
438C22×C3—182no—
439C22×C3C22×D3211yescharacteristic
440C2×D3—34no—
441C2×D3—34no—
442C2×D3—34no—
443C2×D3—34no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—94no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—1816no—
453C2×D3—188no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—62no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—62no—
461C3×C4—94no—
462C3×C4—121no—
463C3×C4—121no—
464C3×C4—182no—
465C3×C4—182no—
466C3×C4—181no—
467C3×C4—181no—
468C3×C4—181no—
469Dic3—34no—
470Dic3—62no—
471Dic3—62no—
472Dic3—62no—
473Dic3—62no—
474Dic3—62no—
475Dic3—62no—
476Dic3—94no—
477Dic3—121no—
478Dic3—121no—
479Dic3—182no—
480Dic3—182no—
481C32—21no—
482C32D3×C22⋊Q812yescomplemented
483C23—91no—
484C23—92no—
485C23—274no—
486C23—542no—
487C23—542no—
488Q8—184no—
489Q8—544no—
490C2×C4—92no—
491C2×C4—92no—
492C2×C4—92no—
493C2×C4—91no—
494C2×C4—91no—
495C2×C4—91no—
496C2×C4—184no—
497C2×C4—182no—
498C2×C4—182no—
499C2×C4—182no—
500C2×C4—274no—
501C2×C4—544no—
502C2×C4—542no—
503C2×C4—542no—
504C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
505C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
506C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
507C2×C3—32no—
508C2×C3—32no—
509C2×C3—32no—
510C2×C3—61no—
511C2×C3—61no—
512C2×C3—61no—
513C2×C3—94no—
514C2×C3—184no—
515C2×C3#6430611yescharacteristic
516C2×C3D32×D411yescharacteristic
517C2×C3D32×Q811yescharacteristic
518D3—38no—
519D3—64no—
520D3—64no—
521D3—184no—
522C22—92no—
523C22—92no—
524C22—92no—
525C22—94no—
526C22—182no—
527C22—182no—
528C22—548no—
529C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—94no—
534C4—181no—
535C4—181no—
536C4—182no—
537C4—181no—
538C4—182no—
539C3—32no—
540C3—61no—
541C3#1059462411yescharacteristic
542C2—94no—
543C2—184no—
544C2#1172249111yescharacteristic
545C2D4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
546C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#88651C2semidirect
3. C4⋊C4×#272C2semidirect
4. #64105⋊C3C2semidirect
7. #11721147C2semidirect
8. #11721147C2semidirect
9. #11721133C2semidirect
10. #11721128C2semidirect
11. #11721101C2semidirect
18. C2×#634⋊C3C22semidirect
23. C2×#6196C22semidirect
26. C4⋊C4×H(3,𝔽3)C22semidirect
27. C4⋊#1084C22semidirect
28. #88540C22semidirect
29. #88540C22semidirect
32. #3750⋊C3C22semidirect
33. #3750⋊C3C22semidirect
36. #88448C22semidirect
37. #88448C22semidirect
40. #64105D3semidirect
58. C2×((C3⋊D3)⋊C3)⋊C2Q8semidirect
82. C2×#634C2×D3semidirect
101. C32×C4⋊C4C2×D3semidirect
104. #3750C2×D3semidirect
105. #3750C2×D3semidirect
130. C2×(C3⋊D3)⋊C3C2×Q8semidirect
214. #272C22⋊Q8semidirect
289. C2×C3⋊D3D3×Q8semidirect
329. H(3,𝔽3)C2×C22⋊Q8semidirect
426. C4⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
482. C32D3×C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×#886512
C4⋊C4×#2722
#64105⋊C32
#117211612
#117211472
#117211332
#117211282
#117211012
D3×#8963