Finite Group Databasebeta

← #13150698

Subgroups — #13150698

Order 1728.

1123 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88671C212yesmaximal, complemented
2#11721830C211yesmaximal, characteristic, complemented
3#64139⋊C3C211yesmaximal, characteristic, complemented
4C2×#3750⋊C3C211yesmaximal, characteristic, complemented
5#11721322C212yesmaximal, complemented
6#11721157C212yesmaximal, complemented
7#11721150C212yesmaximal, complemented
8#11721133C212yesmaximal, complemented
9#11721127C212yesmaximal, complemented
10#10594930—31nomaximal
11#10594911—31nomaximal
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C2×#6235C2212yes—
14C2×C4⋊#272C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×#6200C2212yes—
20C2×#6200C2212yes—
21C2×#6199C2212yes—
22#88671—28no—
23C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
24C2×#6196C2212yes—
25#88542C2212yes—
26C4⋊#1084C2211yescharacteristic
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31#88451C2212yes—
32#88450C2212yes—
33#88448C2212yes—
34C2×#3805—32no—
35C2×#3804—32no—
36#64139D311yescharacteristic, complemented
37C2×#3750D311yescharacteristic, complemented
38C3×#930—31no—
39C3×#930—31no—
40C3×#930—31no—
41C3×#930—61no—
42C2×C3×Dic3⋊C4—31no—
43#63980—32no—
44#63978—32no—
45#63910—32no—
46#63899—32no—
47#63899—32no—
48#63895—32no—
49#63893—32no—
50#63886—32no—
51#63885—32no—
52#63885—32no—
53C23×H(3,𝔽3)C2311yescharacteristic
54C22×#272C2312yes—
55C22×(C3⋊D3)⋊C3C2312yes—
56C2×C4×H(3,𝔽3)—41no—
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C2×C4×H(3,𝔽3)C2311yescharacteristic
59#6235—24no—
60C2×#1084C2312yes—
61C4⋊#272—41no—
62#638⋊C3—24no—
63C2×(C3⋊Dic3)⋊C3—41no—
64C2×(C3⋊Dic3)⋊C3D414yescomplemented
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C4⋊((C3⋊D3)⋊C3)—41no—
69#6200—24no—
70#6200—42no—
71#6199—24no—
72((C3⋊D3)⋊C3)⋊C4—24no—
73#6196—24no—
74#5597—91no—
75#5597—91no—
76#5510—91no—
77C22×C32×C4—21no—
78C22×C32×C4C2×D311yescharacteristic
79C22×C32×C4C2×D311yescharacteristic
80C2×#638C2×D312yes—
81C22×C3⋊Dic3C2×D311yescharacteristic
82C22×C3⋊Dic3C2×D311yescharacteristic
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C3×#67—32no—
85C2×C3×#67—32no—
86C2×C3×#67—32no—
87C2×C3×#67—62no—
88C22×C3×Dic3—31no—
89C22×C3×Dic3—31no—
90C2×C3×D12—31no—
91C2×C3×D12—31no—
92C2×C3×D12—31no—
93C2×C3×D12—61no—
94C2×#627—32no—
95C2×#627—32no—
96C2×#626—32no—
97C2×#626—32no—
98C2×#625—32no—
99C2×#625—32no—
100C2×#624—32no—
101#3805—68no—
102#3804—68no—
103C2×D3×Dic3—32no—
104C2×D3×Dic3—32no—
105C2×D3×Dic3—32no—
106#3752C2×D312yes—
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108#3750C2×D312yes—
109#3750C2×D312yes—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—62no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—61no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120#3724—32no—
121Dic3⋊Dic3—32no—
122#3722—32no—
123(C2×D3)⋊Dic3—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126C22×H(3,𝔽3)—21no—
127C22×H(3,𝔽3)—21no—
128C22×H(3,𝔽3)C2×D412yes—
129C22×H(3,𝔽3)C2411yescharacteristic
130C2×#272—22no—
131C2×#272—22no—
132C2×#272—22no—
133C2×(C3⋊D3)⋊C3—22no—
134C2×(C3⋊D3)⋊C3—22no—
135C2×(C3⋊D3)⋊C3—22no—
136C4×H(3,𝔽3)—21no—
137C4×H(3,𝔽3)—21no—
138#1084—22no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3C2×D414yes—
142C2×#237—92no—
143C2×#237—92no—
144C2×#235—92no—
145C3×#121—91no—
146C3×#121—91no—
147C2×C3×C4⋊C4—91no—
148#939—92no—
149#939—92no—
150#939—92no—
151#939—92no—
152#930—31no—
153#930—31no—
154#930—31no—
155#930—61no—
156#930—91no—
157#930—92no—
158C2×Dic3⋊C4—31no—
159C2×Dic3⋊C4—91no—
160#897—92no—
161#897—92no—
162#896—92no—
163#896—92no—
164#895—92no—
165#895—92no—
166#884—92no—
167#884—92no—
168#878—92no—
169#878—92no—
170C23×C32—21no—
171C23×C32C22×D311yescharacteristic
172C23×C32C22×D311yescharacteristic
173C22×C3⋊D3C22×D312yes—
174C22×C3×D3—32no—
175C22×C3×D3—32no—
176C22×C3×D3—32no—
177C22×C3×D3—62no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—41no—
181C2×C32×C4—41no—
182C2×C32×C4—42no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187#638—24no—
188C2×C3⋊Dic3—41no—
189C2×C3⋊Dic3D1214yescomplemented
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D312yes—
193C4⋊(C3⋊D3)—41no—
194C3×#67—64no—
195C3×#67—64no—
196C3×#67—64no—
197C3×#67—124no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—121no—
206C3×D12—121no—
207C3×D12—121no—
208C3×D12—121no—
209C3×D12—122no—
210#627—64no—
211#627—64no—
212#626—64no—
213#626—122no—
214#625—64no—
215#625—122no—
216#624—64no—
217D3×Dic3—64no—
218D3×Dic3—64no—
219D3×Dic3—64no—
220#537—271noSylow
221C2×H(3,𝔽3)—22no—
222C2×H(3,𝔽3)C22×D411yescharacteristic
223C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
224C2×H(3,𝔽3)2−1+411yescharacteristic
225#272—42no—
226(C3⋊D3)⋊C3—42no—
227C2×C3×D4—91no—
228C2×C3×D4—92no—
229C2×C3×D4—92no—
230C2×C3×D4—91no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—61no—
234C22×C3×C4—91no—
235C22×C3×C4—91no—
236C22×C3×C4D3211yescharacteristic
237C2×#67—32no—
238C2×#67—32no—
239C2×#67—32no—
240C2×#67—62no—
241C2×#67—92no—
242C2×#67—92no—
243C2×#67—92no—
244C2×#67—92no—
245C2×#67—92no—
246C22×Dic3—31no—
247C22×Dic3—31no—
248C22×Dic3—91no—
249C22×Dic3—91no—
250C22×Dic3—92no—
251C22×Dic3—92no—
252#237—188no—
253#237—188no—
254#235—188no—
255C2×D12—31no—
256C2×D12—31no—
257C2×D12—31no—
258C2×D12—61no—
259C2×D12—91no—
260C2×D12—92no—
261C2×D12—92no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×C4×D3—92no—
265C2×C4×D3—92no—
266C2×Dic6—92no—
267C2×Dic6—92no—
268C2×Dic6—92no—
269C3×C4⋊C4—92no—
270C3×C4⋊C4—92no—
271C3×C4⋊C4—91no—
272C3×C4⋊C4—91no—
273C3×C22⋊C4—92no—
274C3×C22⋊C4—92no—
275#217—92no—
276#217—92no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—62no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286C4⋊Dic3—31no—
287C4⋊Dic3—31no—
288C4⋊Dic3—31no—
289C4⋊Dic3—61no—
290C4⋊Dic3—91no—
291C4⋊Dic3—92no—
292Dic3⋊C4—32no—
293Dic3⋊C4—32no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—22no—
304C22×C32—41no—
305C22×C32—41no—
306C22×C32C2×D1212yes—
307C22×C32D3×D412yes—
308C22×C32C23×D311yescharacteristic
309C22×C32C23×D311yescharacteristic
310C2×C3⋊D3—22no—
311C2×C3⋊D3—22no—
312C2×C3⋊D3—22no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—122no—
323C2×C3×D3—122no—
324C2×C3×D3—122no—
325C32×C4—22no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—41no—
331C3⋊Dic3—22no—
332C3⋊Dic3—22no—
333C3⋊Dic3C2×D1214yes—
334C3×Dic3—34no—
335C3×Dic3—62no—
336C3×Dic3—62no—
337C3×Dic3—62no—
338C3×Dic3—62no—
339C3×Dic3—122no—
340C2×#41—272no—
341C22⋊Q8—272no—
342C22⋊Q8—272no—
343#121—91no—
344#121—91no—
345#121—272no—
346#121—272no—
347#121—272no—
348C2×C4⋊C4—91no—
349H(3,𝔽3)#53711yescharacteristic, complemented, Sylow
350C23×C3—31no—
351C23×C3—31no—
352C23×C3—61no—
353C23×C3—92no—
354C23×C3—92no—
355C23×C3C2×D3211yescharacteristic
356C22×D3—32no—
357C22×D3—32no—
358C22×D3—32no—
359C22×D3—62no—
360C22×D3—92no—
361C3×D4—184no—
362C3×D4—184no—
363C3×D4—361no—
364C3×D4—361no—
365C2×C3×C4—31no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—41no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—92no—
373C2×C3×C4—94no—
374C2×C3×C4—92no—
375C2×C3×C4—92no—
376C2×C3×C4—92no—
377C2×C3×C4—121no—
378C2×C3×C4—121no—
379C2×C3×C4—122no—
380C2×C3×C4—361no—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383#67—64no—
384#67—64no—
385#67—64no—
386#67—124no—
387#67—184no—
388#67—184no—
389#67—184no—
390#67—362no—
391#67—362no—
392C2×Dic3—32no—
393C2×Dic3—34no—
394C2×Dic3—32no—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—62no—
399C2×Dic3—94no—
400C2×Dic3—92no—
401C2×Dic3—92no—
402C2×Dic3—92no—
403C2×Dic3—121no—
404C2×Dic3—184no—
405C2×Dic3—184no—
406C2×Dic3—361no—
407D12—121no—
408D12—121no—
409D12—121no—
410D12—122no—
411D12—184no—
412D12—361no—
413D12—362no—
414C4×D3—184no—
415C4×D3—184no—
416C4×D3—184no—
417C4×D3—184no—
418Dic6—184no—
419Dic6—184no—
420Dic6—184no—
421C2×C32—22no—
422C2×C32—22no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—42no—
427C2×C32C22×D1211yescharacteristic
428C2×C32C2×D3×D411yescharacteristic
429C2×C32#100411yescharacteristic
430C2×C32#100411yescharacteristic
431C2×C32#100711yescharacteristic
432C2×C32#101011yescharacteristic
433C3⋊D3—42no—
434C3×D3—122no—
435C3×D3—122no—
436C3×D3—122no—
437C3×D3—242no—
438#41—548no—
439C2×Q8—272no—
440C2×D4—92no—
441C2×D4—92no—
442C2×D4—91no—
443C2×D4—91no—
444C2×D4—272no—
445C2×D4—272no—
446C22×C4—91no—
447C22×C4—91no—
448C22×C4—272no—
449C22×C4—272no—
450C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
451C4⋊C4—92no—
452C4⋊C4—92no—
453C4⋊C4—91no—
454C4⋊C4—91no—
455C4⋊C4—272no—
456C22⋊C4—92no—
457C22⋊C4—92no—
458C22⋊C4—272no—
459C22⋊C4—272no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—31no—
463C22×C3—31no—
464C22×C3—32no—
465C22×C3—32no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—62no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—182no—
475C22×C3—182no—
476C22×C3—182no—
477C22×C3—182no—
478C22×C3—182no—
479C22×C3—182no—
480C22×C3D3×D1212yes—
481C22×C3C22×D3211yescharacteristic
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—62no—
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—122no—
492C2×D3—122no—
493C2×D3—122no—
494C2×D3—182no—
495C2×D3—182no—
496C2×D3—182no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—61no—
500C3×C4—61no—
501C3×C4—61no—
502C3×C4—61no—
503C3×C4—62no—
504C3×C4—94no—
505C3×C4—121no—
506C3×C4—182no—
507C3×C4—182no—
508C3×C4—182no—
509Dic3—34no—
510Dic3—62no—
511Dic3—62no—
512Dic3—62no—
513Dic3—62no—
514Dic3—94no—
515Dic3—122no—
516Dic3—182no—
517Dic3—182no—
518C32—21no—
519C32#551011yescharacteristic, complemented
520C32#559711yescharacteristic, complemented
521C23—92no—
522C23—92no—
523C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
524Q8—544no—
525D4—184no—
526D4—184no—
527D4—361no—
528D4—361no—
529D4—544no—
530D4—1082no—
531C2×C4—41no—
532C2×C4—92no—
533C2×C4—94no—
534C2×C4—92no—
535C2×C4—92no—
536C2×C4—92no—
537C2×C4—361no—
538C2×C4—544no—
539C2×C4—544no—
540C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
541C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
542C2×C3—22no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—62no—
550C2×C3—62no—
551C2×C3—61no—
552C2×C3—61no—
553C2×C3—61no—
554C2×C3—122no—
555C2×C3—362no—
556C2×C3—362no—
557C2×C3#6429711yescharacteristic
558C2×C3C2×D3×D1211yescharacteristic
559C2×C3#6433011yescharacteristic
560D3—122no—
561D3—122no—
562D3—122no—
563D3—242no—
564D3—362no—
565C22—21no—
566C22—21no—
567C22—182no—
568C22—182no—
569C22—182no—
570C22—182no—
571C22—182no—
572C22—182no—
573C22#8865412yes—
574C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
575C4—21no—
576C4—21no—
577C4—94no—
578C4—182no—
579C4—182no—
580C4—182no—
581C3—31no—
582C3—31no—
583C3—61no—
584C3#1059491111yescharacteristic
585C2—22no—
586C2—362no—
587C2—362no—
588C2#1172248211yescharacteristic
589C2C2×#8865411yescharacteristic
590C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#537
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88671C2semidirect
2. #11721830C2semidirect
3. #64139⋊C3C2semidirect
4. C2×#3750⋊C3C2semidirect
5. #11721322C2semidirect
6. #11721157C2semidirect
7. #11721150C2semidirect
8. #11721133C2semidirect
9. #11721127C2semidirect
36. #64139D3semidirect
37. C2×#3750D3semidirect
64. C2×(C3⋊Dic3)⋊C3D4semidirect
189. C2×C3⋊Dic3D12semidirect
349. H(3,𝔽3)#537semidirect
519. C32#5510semidirect
520. C32#5597semidirect

Maximal subgroups

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

subgroupindex
C2×#886712
#117218302
#64139⋊C32
C2×#3750⋊C32
#117213222
#117211572
#117211502
#117211332
#117211272
#105949303
#105949113