Finite Group Databasebeta

← #13150750

Subgroups — #13150750

Order 1728.

890 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1084C211yesmaximal, characteristic, complemented
2C4×#638⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721315C212yesmaximal
5#11721309C212yesmaximal, complemented
6#11721133C212yesmaximal, complemented
7#11721129C211yesmaximal, characteristic, complemented
8#11721116C211yesmaximal, characteristic, complemented
9#11721106C211yesmaximal, characteristic, complemented
10#11721085C212yesmaximal, complemented
11#11721082C211yesmaximal, characteristic, complemented
12#10594991—31nomaximal
13#10594988—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C22×#1084C2212yes—
16C2×#638⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
20C2×#6196C2212yes—
21C4⋊#1084C2211yescharacteristic
22C4⋊#1084C2212yes—
23C4⋊#1084C2211yescharacteristic
24#3757⋊C3C2211yescharacteristic
25#3757⋊C3C2212yes—
26#3752⋊C3C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2212yes—
29#3750⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31#88450C2212yes—
32#88449C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37C4×#638D311yescharacteristic
38#64133D311yescharacteristic, complemented
39C3×C4×#67—31no—
40C2×C3×C4⋊Dic3—31no—
41C2×C3×C4⋊Dic3—31no—
42C2×C3×C4⋊Dic3—61no—
43C3×#924—31no—
44Dic3×#67—32no—
45#63964—32no—
46#63964—32no—
47#63952—32no—
48#63919—31no—
49#63918—31no—
50#63900—31no—
51#63899—32no—
52#63895—32no—
53D3×C4⋊Dic3—31no—
54#63871—31no—
55#63857—31no—
56#63847—32no—
57#63846—32no—
58#63844—31no—
59#63843—31no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084—42no—
66C2×#1084C2312yes—
67C2×#1084C2312yes—
68#638⋊C3Q814yescomplemented
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C4×(C3⋊D3)⋊C3—22no—
75#634⋊C3—22no—
76((C3⋊D3)⋊C3)⋊C4—42no—
77#6196—42no—
78#5635—91no—
79#5543—91no—
80#5511—91no—
81C22×C32×C4—21no—
82C22×C32×C4C2×D311yescharacteristic
83C22×C32×C4C2×D311yescharacteristic
84C2×#638C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×#634C2×D311yescharacteristic
87C2×C3×#67—31no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—32no—
90C22×C3×Dic3—62no—
91C2×C3×C4×D3—31no—
92C2×C3×Dic6—31no—
93C2×#627—32no—
94C2×#627—32no—
95C2×#624—32no—
96C2×D3×Dic3—32no—
97#3757C2×D312yes—
98#3757C2×D311yescharacteristic
99#3752C2×D311yescharacteristic
100C4⋊(C3⋊Dic3)C2×D311yescharacteristic
101#3750C2×D312yes—
102#3750C2×D311yescharacteristic
103C4×C3⋊Dic3C2×D311yescharacteristic
104C3×#217—32no—
105C3×#217—31no—
106C3×(C2×D3)⋊C4—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—32no—
111C3×C4⋊Dic3—32no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—61no—
115C3×C4⋊Dic3—61no—
116C3×C4⋊Dic3—62no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—31no—
119C3×C4×Dic3—31no—
120#3724—32no—
121#3724—32no—
122#3724—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128#3722—32no—
129(C2×D3)⋊Dic3—32no—
130Dic32—32no—
131Dic32—32no—
132C22×H(3,𝔽3)—21no—
133C22×H(3,𝔽3)—21no—
134C22×H(3,𝔽3)C2×Q812yes—
135C22×H(3,𝔽3)C2411yescharacteristic
136C2×(C3⋊D3)⋊C3—21no—
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3C2×Q812yes—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)#4112yes—
141#1084—22no—
142#1084—22no—
143(C3⋊Dic3)⋊C3—22no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3C2×Q812yes—
148C3×C22⋊Q8—91no—
149C3×C4×D4—91no—
150C2×C3×C4⋊C4—91no—
151#946—91no—
152Q8×Dic3—91no—
153D4×Dic3—92no—
154C4×#67—31no—
155C4×#67—91no—
156C4×#67—92no—
157C2×C4⋊Dic3—31no—
158C2×C4⋊Dic3—31no—
159C2×C4⋊Dic3—61no—
160C2×C4⋊Dic3—91no—
161#924—31no—
162#924—91no—
163#924—91no—
164#924—92no—
165#897—92no—
166#897—91no—
167#896—92no—
168D3×C4⋊C4—91no—
169D3×C4⋊C4—91no—
170#890—92no—
171#889—92no—
172#889—92no—
173#888—91no—
174#888—91no—
175#887—91no—
176#67⋊C4—92no—
177#878—92no—
178#878—92no—
179#878—92no—
180#869—91no—
181C4×Dic6—91no—
182C23×C32—21no—
183C23×C32C22×D311yescharacteristic
184C23×C32C22×D311yescharacteristic
185C22×C3⋊D3C22×D311yescharacteristic
186C22×C3×D3—31no—
187C2×C32×C4—24no—
188C2×C32×C4—22no—
189C2×C32×C4—22no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196#638Dic614yescomplemented
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D311yescharacteristic
202C4×C3⋊D3—22no—
203#634—22no—
204C3×#67—34no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3—122no—
217C2×C3×Dic3—122no—
218C2×C3×Dic3—124no—
219C3×C4×D3—62no—
220C3×Dic6—62no—
221#627—122no—
222#627—122no—
223#624—122no—
224D3×Dic3—122no—
225#554—271noSylow
226C2×H(3,𝔽3)—22no—
227C2×H(3,𝔽3)C22×Q811yescharacteristic
228C2×H(3,𝔽3)C2×#4111yescharacteristic
229C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
230(C3⋊D3)⋊C3—22no—
231C2×C3×Q8—91no—
232C2×C3×D4—91no—
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—61no—
236C22×C3×C4—91no—
237C22×C3×C4—92no—
238C22×C3×C4D3211yescharacteristic
239C2×#67—31no—
240C2×#67—91no—
241C22×Dic3—32no—
242C22×Dic3—32no—
243C22×Dic3—62no—
244C22×Dic3—92no—
245C2×C4×D3—31no—
246C2×C4×D3—91no—
247C2×C4×D3—92no—
248C2×C4×D3—92no—
249C2×Dic6—31no—
250C2×Dic6—91no—
251C2×Dic6—92no—
252C2×Dic6—92no—
253C2×Dic6—92no—
254C3×C4⋊C4—91no—
255C3×C4⋊C4—92no—
256C3×C4⋊C4—91no—
257C3×C4⋊C4—91no—
258C3×C4⋊C4—91no—
259C3×C4⋊C4—92no—
260C3×C22⋊C4—91no—
261C3×C22⋊C4—91no—
262C3×C22⋊C4—92no—
263C3×C42—91no—
264#217—32no—
265#217—31no—
266#217—91no—
267#217—92no—
268#217—92no—
269(C2×D3)⋊C4—31no—
270(C2×D3)⋊C4—91no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—92no—
273C4⋊Dic3—31no—
274C4⋊Dic3—31no—
275C4⋊Dic3—32no—
276C4⋊Dic3—31no—
277C4⋊Dic3—32no—
278C4⋊Dic3—31no—
279C4⋊Dic3—31no—
280C4⋊Dic3—61no—
281C4⋊Dic3—62no—
282C4⋊Dic3—61no—
283C4⋊Dic3—91no—
284C4⋊Dic3—92no—
285C4⋊Dic3—92no—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288Dic3⋊C4—31no—
289Dic3⋊C4—32no—
290Dic3⋊C4—92no—
291Dic3⋊C4—91no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300C4×Dic3—31no—
301C4×Dic3—91no—
302C4×Dic3—92no—
303C4×Dic3—92no—
304C4×Dic3—92no—
305C22×C32—22no—
306C22×C32—22no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—22no—
313C22×C32C2×Dic612yes—
314C22×C32D3×Q812yes—
315C22×C32C23×D311yescharacteristic
316C22×C32C23×D311yescharacteristic
317C2×C3⋊D3—21no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3C2×Dic612yes—
320C2×C3×D3—32no—
321C2×C3×D3—61no—
322C2×C3×D3—61no—
323C32×C4—22no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—22no—
327C32×C4#23712yes—
328C32×C4#23712yes—
329C3⋊Dic3—22no—
330C3⋊Dic3—21no—
331C3⋊Dic3—22no—
332C3⋊Dic3—21no—
333C3⋊Dic3C2×Dic612yes—
334C3×Dic3—32no—
335C3×Dic3—62no—
336C3×Dic3—61no—
337C3×Dic3—62no—
338C3×Dic3—61no—
339C3×Dic3—62no—
340C3×Dic3—62no—
341C3×Dic3—62no—
342C3×Dic3—62no—
343C3×Dic3—122no—
344C3×Dic3—122no—
345C4⋊Q8—271no—
346#125—272no—
347C22⋊Q8—91no—
348C22⋊Q8—272no—
349C22⋊Q8—272no—
350C22⋊Q8—271no—
351C4×Q8—271no—
352C4×D4—91no—
353C4×D4—272no—
354C2×C4⋊C4—91no—
355C2×C4⋊C4—271no—
356H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
357C23×C3—31no—
358C23×C3—31no—
359C23×C3—61no—
360C23×C3—91no—
361C23×C3C2×D3211yescharacteristic
362C22×D3—31no—
363C22×D3—91no—
364C3×Q8—182no—
365C3×D4—94no—
366C2×C3×C4—22no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—64no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—91no—
377C2×C3×C4—92no—
378C2×C3×C4—91no—
379C2×C3×C4—91no—
380C2×C3×C4—92no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—182no—
384C2×C3×C4—362no—
385C2×C3×C4C2×D3211yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387#67—34no—
388#67—94no—
389C2×Dic3—31no—
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—32no—
393C2×Dic3—32no—
394C2×Dic3—31no—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—31no—
398C2×Dic3—62no—
399C2×Dic3—62no—
400C2×Dic3—91no—
401C2×Dic3—92no—
402C2×Dic3—91no—
403C2×Dic3—91no—
404C2×Dic3—92no—
405C2×Dic3—122no—
406C2×Dic3—122no—
407C2×Dic3—124no—
408C2×Dic3—362no—
409C4×D3—62no—
410C4×D3—182no—
411C4×D3—362no—
412C4×D3—362no—
413Dic6—62no—
414Dic6—182no—
415Dic6—362no—
416Dic6—362no—
417Dic6—362no—
418C2×C32—24no—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32C22×Dic611yescharacteristic
425C2×C32C2×#23711yescharacteristic
426C2×C32C2×#23711yescharacteristic
427C2×C32C2×D3×Q811yescharacteristic
428C2×C32#100911yescharacteristic
429C2×C32#100911yescharacteristic
430C3⋊D3—22no—
431C3×D3—62no—
432C2×Q8—91no—
433C2×Q8—272no—
434C2×D4—91no—
435C22×C4—92no—
436C22×C4—91no—
437C22×C4—272no—
438C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
439C4⋊C4—91no—
440C4⋊C4—92no—
441C4⋊C4—91no—
442C4⋊C4—91no—
443C4⋊C4—91no—
444C4⋊C4—92no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C4⋊C4—272no—
448C4⋊C4—272no—
449C22⋊C4—91no—
450C22⋊C4—92no—
451C22⋊C4—91no—
452C22⋊C4—272no—
453C42—91no—
454C42—272no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—31no—
458C22×C3—31no—
459C22×C3—32no—
460C22×C3—32no—
461C22×C3—61no—
462C22×C3—61no—
463C22×C3—61no—
464C22×C3—61no—
465C22×C3—62no—
466C22×C3—62no—
467C22×C3—61no—
468C22×C3—62no—
469C22×C3—92no—
470C22×C3—181no—
471C22×C3—181no—
472C22×C3D3×Dic612yes—
473C22×C3C22×D3211yescharacteristic
474C2×D3—32no—
475C2×D3—61no—
476C2×D3—61no—
477C2×D3—92no—
478C2×D3—181no—
479C2×D3—181no—
480C3×C4—21no—
481C3×C4—32no—
482C3×C4—32no—
483C3×C4—61no—
484C3×C4—61no—
485C3×C4—62no—
486C3×C4—62no—
487C3×C4—92no—
488C3×C4—181no—
489C3×C4—182no—
490C3×C4—181no—
491C3×C4—182no—
492C3×C4—182no—
493C3×C4—182no—
494C3×C4#380512yes—
495Dic3—32no—
496Dic3—62no—
497Dic3—62no—
498Dic3—62no—
499Dic3—62no—
500Dic3—61no—
501Dic3—62no—
502Dic3—62no—
503Dic3—61no—
504Dic3—92no—
505Dic3—122no—
506Dic3—122no—
507Dic3—182no—
508Dic3—181no—
509Dic3—182no—
510Dic3—181no—
511C32—21no—
512C32#554311yescharacteristic, complemented
513C32#563511yescharacteristic, complemented
514C23—91no—
515C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
516Q8—182no—
517Q8—1082no—
518D4—94no—
519C2×C4—22no—
520C2×C4—91no—
521C2×C4—91no—
522C2×C4—92no—
523C2×C4—92no—
524C2×C4—92no—
525C2×C4—91no—
526C2×C4—92no—
527C2×C4—182no—
528C2×C4—362no—
529C2×C4—1082no—
530C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
531C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
532C2×C3—22no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—62no—
540C2×C3—62no—
541C2×C3—64no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—182no—
546C2×C3C2×D3×Dic611yescharacteristic
547C2×C3#6430811yescharacteristic
548C2×C3C2×#380511yescharacteristic
549D3—62no—
550D3—182no—
551C22—21no—
552C22—21no—
553C22—92no—
554C22—181no—
555C22—181no—
556C22((C3⋊D3)⋊C3)⋊Q812yes—
557C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
558C4—21no—
559C4—92no—
560C4—181no—
561C4—182no—
562C4—182no—
563C4—182no—
564C4—182no—
565C4—181no—
566C4#8867212yes—
567C3—31no—
568C3—31no—
569C3—61no—
570C3#1059499311yescharacteristic
571C2—22no—
572C2—182no—
573C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
574C2#1172249311yescharacteristic
575C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC2×#638⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
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×C4⋊#1084C2semidirect
3. #64133⋊C3C2semidirect
5. #11721309C2semidirect
6. #11721133C2semidirect
7. #11721129C2semidirect
8. #11721116C2semidirect
9. #11721106C2semidirect
10. #11721085C2semidirect
11. #11721082C2semidirect
38. #64133D3semidirect
68. #638⋊C3Q8semidirect
196. #638Dic6semidirect
356. H(3,𝔽3)#554semidirect
512. C32#5543semidirect
513. C32#5635semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C4×#638⋊C32
#64133⋊C32
#117213152
#117213092
#117211332
#117211292
#117211162
#117211062
#117210852
#117210822
#105949913
#105949883