Finite Group Databasebeta

← #13150871

Subgroups — #13150871

Order 1728.

1315 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1315 conjugacy classes of subgroups — too many to draw. Only its 203 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×#88677C211yesmaximal, characteristic
2C2×#88542C211yesmaximal, characteristic, complemented
3C4×#638⋊C3C212yesmaximal, complemented
4#11721320C211yesmaximal, characteristic, complemented
5#11721315C212yesmaximal, complemented
6#11721160C212yesmaximal, complemented
7#11721154C212yesmaximal, complemented
8#11721123C212yesmaximal, complemented
9#11721076C212yesmaximal, complemented
10#10595056—32nomaximal
11C23×#272C2211yescharacteristic
12C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic, complemented
14C22×#1084C2211yescharacteristic, complemented
15C2×#638⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
17#88677C418yescomplemented
18C2×#6200C2212yes—
19C2×#6199C2211yescharacteristic
20C2×#6199C2211yescharacteristic, complemented
21C2×#6199C2211yescharacteristic, complemented
22C2×((C3⋊D3)⋊C3)⋊C4C2212yescomplemented
23#88542C2212yes—
24#88542C2212yescomplemented
25#3757⋊C3C2212yescomplemented
26#3752⋊C3C2212yescomplemented
27#3750⋊C3C2212yescomplemented
28C4×(C3⋊Dic3)⋊C3C2212yescomplemented
29#88451C2212yes—
30#88450C2212yescomplemented
31#88448C2212yescomplemented
32#88447C2212yescomplemented
33C2×D3×#67—32no—
34C4×#638D312yescomplemented
35C3×C4×#67—32no—
36C2×C3×(C2×D3)⋊C4—32no—
37C2×C3×(C2×D3)⋊C4—61no—
38Dic3×#67—32no—
39D3×#217—32no—
40#63952—32no—
41#63921—32no—
42D3×(C2×D3)⋊C4—32no—
43#63902—32no—
44#63881—32no—
45#63880—32no—
46#63879—32no—
47#63878—32no—
48#63877—32no—
49C23×H(3,𝔽3)C2311yescharacteristic
50C22×#272—41no—
51C22×#272—41no—
52C22×#272C2×C414yescomplemented
53C22×#272C2311yescharacteristic
54C22×#272C2311yescharacteristic
55C22×(C3⋊D3)⋊C3C2312yes—
56C2×((C3⋊D3)⋊C3)⋊C2—41no—
57C2×((C3⋊D3)⋊C3)⋊C2—41no—
58C2×((C3⋊D3)⋊C3)⋊C2C2×C414yescomplemented
59C2×C4×H(3,𝔽3)—41no—
60C2×C4×H(3,𝔽3)C2312yes—
61C2×#1084—41no—
62C2×#1084C2312yes—
63#638⋊C3C2×C418yescomplemented
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C4×(C3⋊D3)⋊C3—42no—
68#6200C2×C418yescomplemented
69#6199—41no—
70#6199—41no—
71#6199C2×C414yescomplemented
72((C3⋊D3)⋊C3)⋊C4—42no—
73#5545—92no—
74#5486—91no—
75C23×C3×D3—32no—
76C23×C3×D3—61no—
77C22×D32—32no—
78C22×C32×C4—21no—
79C22×C32×C4C2×D312yescomplemented
80C2×#638C2×D312yes—
81C2×C4×C3⋊D3C2×D312yescomplemented
82C2×C3×#67—32no—
83C22×C3×Dic3—32no—
84C22×C3×Dic3—61no—
85C2×C3×C4×D3—32no—
86D3×#67—316no—
87C2×#626—32no—
88C2×#625—32no—
89C2×#624—32no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92C2×D3×Dic3—32no—
93C2×D3×Dic3—32no—
94#3757C2×D312yescomplemented
95#3752C2×D312yescomplemented
96#3750C2×D312yescomplemented
97C4×C3⋊Dic3C2×D312yescomplemented
98C3×#217—32no—
99C3×(C2×D3)⋊C4—32no—
100C3×(C2×D3)⋊C4—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—62no—
105C3×(C2×D3)⋊C4—62no—
106C3×Dic3⋊C4—32no—
107C3×C4×Dic3—32no—
108#3724—32no—
109Dic3⋊Dic3—32no—
110#3722—32no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114Dic32—32no—
115Dic32—32no—
116C22×H(3,𝔽3)—21no—
117C22×H(3,𝔽3)—21no—
118C22×H(3,𝔽3)C22×C412yes—
119C22×H(3,𝔽3)C2411yescharacteristic
120C2×#272—21no—
121C2×#272—21no—
122C2×#272—21no—
123C2×#272—21no—
124C2×#272—44no—
125C2×#272C22×C412yes—
126C2×#272C22×C412yes—
127C2×(C3⋊D3)⋊C3—22no—
128C2×(C3⋊D3)⋊C3—22no—
129C2×(C3⋊D3)⋊C3C22×C414yes—
130((C3⋊D3)⋊C3)⋊C2—44no—
131C4×H(3,𝔽3)—22no—
132#1084—22no—
133(C3⋊Dic3)⋊C3—22no—
134(C3⋊Dic3)⋊C3—22no—
135(C3⋊Dic3)⋊C3C22×C414yes—
136C22×#67—91no—
137C2×D3×D4—92no—
138C3×C4×D4—92no—
139C2×C3×C22⋊C4—91no—
140C2×#217—91no—
141D4×Dic3—92no—
142C4×#67—32no—
143C4×#67—92no—
144C4×#67—92no—
145C4×#67—92no—
146C2×(C2×D3)⋊C4—32no—
147C2×(C2×D3)⋊C4—61no—
148C2×(C2×D3)⋊C4—92no—
149#893—92no—
150#892—92no—
151#67⋊C4—92no—
152#67⋊C4—92no—
153#67⋊C4—92no—
154#67⋊C4—92no—
155D3×C22⋊C4—92no—
156D3×C22⋊C4—92no—
157D3×C22⋊C4—92no—
158#877—92no—
159C4×D12—92no—
160#872—92no—
161C23×C32—21no—
162C23×C32C22×D312yes—
163C22×C3⋊D3C22×D312yes—
164C22×C3×D3—32no—
165C22×C3×D3—32no—
166C22×C3×D3—32no—
167C22×C3×D3—38no—
168C22×C3×D3—64no—
169C22×C3×D3—61no—
170C22×C3×D3—61no—
171C22×C3×D3—122no—
172C22×C3×D3—122no—
173C22×C3×D3—122no—
174C22×C3×D3—122no—
175C2×D32—38no—
176C2×D32—122no—
177C2×D32—122no—
178C2×C32×C4—22no—
179C2×C32×C4—42no—
180C2×C32×C4—42no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D312yes—
183#638C4×D318yescomplemented
184C2×C3⋊Dic3C22×D312yes—
185C2×C3⋊Dic3C22×D312yes—
186C2×C3⋊Dic3C22×D312yes—
187C4×C3⋊D3—42no—
188C3×#67—38no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—122no—
196C2×C3×Dic3—122no—
197C3×C4×D3—122no—
198#626—38no—
199#625—38no—
200#624—122no—
201D3×Dic3—38no—
202D3×Dic3—122no—
203D3×Dic3—122no—
204D3×Dic3—122no—
205#518—271noSylow
206C2×H(3,𝔽3)—22no—
207C2×H(3,𝔽3)C23×C411yescharacteristic
208C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
209C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
210#272—22no—
211#272—22no—
212(C3⋊D3)⋊C3—24no—
213C24×C3—91no—
214C23×D3—32no—
215C23×D3—61no—
216C23×D3—91no—
217C23×D3—92no—
218C2×C3×D4—92no—
219C22×C3×C4—32no—
220C22×C3×C4—61no—
221C22×C3×C4—92no—
222C22×C3×C4—91no—
223C22×C3×C4D3211yescharacteristic
224C2×#67—32no—
225C2×#67—92no—
226C2×#67—92no—
227C2×#67—98no—
228C2×#67—92no—
229C22×Dic3—32no—
230C22×Dic3—61no—
231C22×Dic3—91no—
232C22×Dic3—91no—
233C22×Dic3—91no—
234C22×Dic3—92no—
235D3×D4—916no—
236C2×D12—92no—
237C2×C4×D3—32no—
238C2×C4×D3—92no—
239C2×C4×D3—92no—
240C2×C4×D3—92no—
241C2×C4×D3—92no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C3×C4⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C42—92no—
250#217—32no—
251#217—92no—
252#217—92no—
253#217—92no—
254(C2×D3)⋊C4—32no—
255(C2×D3)⋊C4—32no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—62no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266C4⋊Dic3—92no—
267Dic3⋊C4—32no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271C4×Dic3—32no—
272C4×Dic3—92no—
273C4×Dic3—92no—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—21no—
281C22×C32—22no—
282C22×C32C2×C4×D314yes—
283C22×C32C23×D312yes—
284C2×C3⋊D3—22no—
285C2×C3⋊D3—22no—
286C2×C3⋊D3C2×C4×D314yes—
287C2×C3×D3—34no—
288C2×C3×D3—34no—
289C2×C3×D3—34no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—121no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
301C2×C3×D3—121no—
302C2×C3×D3—128no—
303C2×C3×D3—128no—
304D32—128no—
305C32×C4—22no—
306C32×C4—22no—
307C32×C4—42no—
308C3⋊Dic3—22no—
309C3⋊Dic3—22no—
310C3⋊Dic3C2×C4×D314yes—
311C3×Dic3—34no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—122no—
317C22×D4—271no—
318C4×D4—92no—
319C4×D4—272no—
320C4×D4—272no—
321C4×D4—272no—
322#117—272no—
323C2×C22⋊C4—91no—
324C2×C22⋊C4—272no—
325C2×C22⋊C4—271no—
326H(3,𝔽3)#51811yescharacteristic, complemented, Sylow
327C23×C3—32no—
328C23×C3—61no—
329C23×C3—92no—
330C23×C3—91no—
331C23×C3—91no—
332C23×C3—94no—
333C23×C3—361no—
334C23×C3—361no—
335C23×C3C2×D3211yescharacteristic
336C22×D3—32no—
337C22×D3—32no—
338C22×D3—32no—
339C22×D3—38no—
340C22×D3—61no—
341C22×D3—61no—
342C22×D3—64no—
343C22×D3—92no—
344C22×D3—94no—
345C22×D3—98no—
346C22×D3—122no—
347C22×D3—122no—
348C22×D3—122no—
349C22×D3—122no—
350C22×D3—361no—
351C22×D3—361no—
352C22×D3—362no—
353C22×D3—362no—
354C3×D4—98no—
355C2×C3×C4—32no—
356C2×C3×C4—32no—
357C2×C3×C4—41no—
358C2×C3×C4—62no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4—92no—
362C2×C3×C4—92no—
363C2×C3×C4—122no—
364C2×C3×C4—122no—
365C2×C3×C4—362no—
366C2×C3×C4—361no—
367C2×C3×C4C2×D3212yes—
368#67—38no—
369#67—98no—
370#67—98no—
371#67—98no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—32no—
377C2×Dic3—62no—
378C2×Dic3—92no—
379C2×Dic3—92no—
380C2×Dic3—92no—
381C2×Dic3—94no—
382C2×Dic3—122no—
383C2×Dic3—122no—
384C2×Dic3—361no—
385C2×Dic3—361no—
386C2×Dic3—362no—
387D12—98no—
388C4×D3—98no—
389C4×D3—122no—
390C4×D3—362no—
391C4×D3—362no—
392C4×D3—362no—
393C4×D3—362no—
394C4×D3—362no—
395C2×C32—24no—
396C2×C32—24no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32C22×C4×D312yes—
401C2×C32#100412yes—
402C2×C32#100912yes—
403C3⋊D3—24no—
404C3×D3—64no—
405C3×D3—64no—
406C3×D3—64no—
407C3×D3—122no—
408C3×D3—122no—
409C24—91no—
410C24—271no—
411C2×D4—92no—
412C2×D4—272no—
413C2×D4—278no—
414C22×C4—91no—
415C22×C4—92no—
416C22×C4—271no—
417C22×C4—271no—
418C22×C4—272no—
419C22×C4—271no—
420C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
421C4⋊C4—92no—
422C4⋊C4—272no—
423C22⋊C4—92no—
424C22⋊C4—92no—
425C22⋊C4—92no—
426C22⋊C4—92no—
427C22⋊C4—272no—
428C22⋊C4—272no—
429C42—92no—
430C42—272no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—32no—
434C22×C3—34no—
435C22×C3—62no—
436C22×C3—62no—
437C22×C3—62no—
438C22×C3—62no—
439C22×C3—61no—
440C22×C3—62no—
441C22×C3—94no—
442C22×C3—92no—
443C22×C3—92no—
444C22×C3—182no—
445C22×C3—182no—
446C22×C3—181no—
447C22×C3—181no—
448C22×C3—181no—
449C22×C3—181no—
450C22×C3—364no—
451C22×C3C4×D3212yes—
452C22×C3C22×D3211yescharacteristic
453C2×D3—34no—
454C2×D3—34no—
455C2×D3—34no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—94no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—121no—
468C2×D3—121no—
469C2×D3—128no—
470C2×D3—128no—
471C2×D3—182no—
472C2×D3—182no—
473C2×D3—364no—
474C2×D3—368no—
475C3×C4—22no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—94no—
479C3×C4—122no—
480C3×C4—182no—
481C3×C4—182no—
482C3×C4—182no—
483Dic3—34no—
484Dic3—62no—
485Dic3—62no—
486Dic3—62no—
487Dic3—62no—
488Dic3—94no—
489Dic3—122no—
490Dic3—182no—
491Dic3—182no—
492C32—21no—
493C32#554512yescomplemented
494C23—91no—
495C23—91no—
496C23—92no—
497C23—94no—
498C23—274no—
499C23—361no—
500C23—361no—
501C23—1081no—
502C23—1081no—
503C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
504D4—98no—
505D4—278no—
506C2×C4—41no—
507C2×C4—92no—
508C2×C4—92no—
509C2×C4—92no—
510C2×C4—92no—
511C2×C4—274no—
512C2×C4—362no—
513C2×C4—361no—
514C2×C4—1081no—
515C2×C4—1082no—
516C2×C4—1081no—
517C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
518C2×C3—22no—
519C2×C3—32no—
520C2×C3—32no—
521C2×C3—32no—
522C2×C3—64no—
523C2×C3—64no—
524C2×C3—61no—
525C2×C3—61no—
526C2×C3—61no—
527C2×C3—184no—
528C2×C3—182no—
529C2×C3—182no—
530C2×C3C2×C4×D3211yescharacteristic
531C2×C3#6430811yescharacteristic
532C2×C3#6433011yescharacteristic
533D3—64no—
534D3—64no—
535D3—64no—
536D3—122no—
537D3—122no—
538D3—184no—
539C22—21no—
540C22—21no—
541C22—92no—
542C22—94no—
543C22—92no—
544C22—182no—
545C22—181no—
546C22—181no—
547C22—181no—
548C22—182no—
549C22—181no—
550C22—364no—
551C22—1084no—
552C22C4×((C3⋊D3)⋊C3)⋊C212yes—
553C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
554C4—22no—
555C4—94no—
556C4—182no—
557C4—182no—
558C4—182no—
559C3—32no—
560C3—61no—
561C3#1059505811yescharacteristic
562C2—22no—
563C2—182no—
564C2—182no—
565C2—184no—
566C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
567C2#1172249311yescharacteristic
568C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC2×#88677C2
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#518
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88542C2semidirect
3. C4×#638⋊C3C2semidirect
4. #11721320C2semidirect
5. #11721315C2semidirect
6. #11721160C2semidirect
7. #11721154C2semidirect
8. #11721123C2semidirect
9. #11721076C2semidirect
13. C22×C4×H(3,𝔽3)C22semidirect
14. C22×#1084C22semidirect
16. C2×C4×(C3⋊D3)⋊C3C22semidirect
17. #88677C4semidirect
20. C2×#6199C22semidirect
21. C2×#6199C22semidirect
22. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
24. #88542C22semidirect
25. #3757⋊C3C22semidirect
26. #3752⋊C3C22semidirect
27. #3750⋊C3C22semidirect
28. C4×(C3⋊Dic3)⋊C3C22semidirect
30. #88450C22semidirect
31. #88448C22semidirect
32. #88447C22semidirect
34. C4×#638D3semidirect
52. C22×#272C2×C4semidirect
58. C2×((C3⋊D3)⋊C3)⋊C2C2×C4semidirect
63. #638⋊C3C2×C4semidirect
68. #6200C2×C4semidirect
71. #6199C2×C4semidirect
79. C22×C32×C4C2×D3semidirect
81. C2×C4×C3⋊D3C2×D3semidirect
94. #3757C2×D3semidirect
95. #3752C2×D3semidirect
96. #3750C2×D3semidirect
97. C4×C3⋊Dic3C2×D3semidirect
183. #638C4×D3semidirect
326. H(3,𝔽3)#518semidirect
420. C22×C4((C3⋊D3)⋊C3)⋊C2semidirect
493. C32#5545semidirect

Maximal subgroups

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

subgroupindex
C2×#886772
C2×#885422
C4×#638⋊C32
#117213202
#117213152
#117211602
#117211542
#117211232
#117210762
#105950563