Finite Group Databasebeta

← #13149547

Subgroups — #13149547

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721756C211yesmaximal, characteristic, complemented
2#64106⋊C3C211yesmaximal, characteristic, complemented
3C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4C4×#6198C211yesmaximal, characteristic
5#11721135C212yesmaximal, complemented
6#11721129C211yesmaximal, characteristic, complemented
7#11721122C212yesmaximal
8#11721107C212yesmaximal, complemented
9#11721106C211yesmaximal, characteristic, complemented
10#11721086C212yesmaximal, complemented
11#11721084C211yesmaximal, characteristic, complemented
12#10594537—31nomaximal
13#10594393—31nomaximal
14C2×#6229C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic
19C2×#6198C2211yescharacteristic
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C4⋊#1084C2211yescharacteristic
23#88540C2212yes—
24C4×#1084C2211yescharacteristic
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28#3750⋊C3C2212yes—
29#88450C2212yes—
30#88450C2211yescharacteristic
31#88450C2211yescharacteristic
32#88449C2211yescharacteristic
33#88449C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64106D311yescharacteristic, complemented
38C4⋊C4×C3⋊D3D311yescharacteristic, complemented
39C3×#897—31no—
40C3×D3×C4⋊C4—31no—
41C3×#888—31no—
42C3×#888—31no—
43C3×#888—61no—
44#63904—32no—
45C4×#626—31no—
46C4×#626—31no—
47#63900—31no—
48D3×C4⋊Dic3—31no—
49#63888—32no—
50#63887—32no—
51#63881—32no—
52#63871—31no—
53#63871—32no—
54#63857—31no—
55#63856—32no—
56#63846—32no—
57#63845—32no—
58#63844—31no—
59Dic3×Dic6—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2312yes—
65C2×#1084C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67#6229—42no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4×(C3⋊D3)⋊C3—22no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—42no—
75#634⋊C3—22no—
76#6198Q814yescomplemented
77((C3⋊D3)⋊C3)⋊C4—42no—
78#5730—91no—
79#5635—91no—
80#5511—91no—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D312yes—
84C2×#634C2×D311yescharacteristic
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—32no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—32no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—62no—
92C2×#626—31no—
93C2×#626—31no—
94C2×#624—32no—
95C2×D3×Dic3—32no—
96C32×C4⋊C4—21no—
97C32×C4⋊C4C2×D311yescharacteristic
98C32×C4⋊C4C2×D311yescharacteristic
99#3752C2×D312yes—
100C4⋊(C3⋊Dic3)C2×D311yescharacteristic
101C4⋊(C3⋊Dic3)C2×D312yes—
102#3750C2×D312yes—
103C3×(C2×D3)⋊C4—32no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—32no—
108C3×C4⋊Dic3—61no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—62no—
113C3×C4×Dic3—31no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—61no—
116#3724—32no—
117#3724—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—31no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—31no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124#3722—32no—
125#3722—31no—
126#3722—31no—
127(C2×D3)⋊Dic3—31no—
128(C2×D3)⋊Dic3—32no—
129(C2×D3)⋊Dic3—31no—
130Dic32—32no—
131Dic32—32no—
132C22×H(3,𝔽3)C2411yescharacteristic
133C2×(C3⋊D3)⋊C3—21no—
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3—21no—
137C2×(C3⋊D3)⋊C3C2×Q812yes—
138C2×(C3⋊D3)⋊C3C2×Q812yes—
139C4×H(3,𝔽3)—22no—
140C4×H(3,𝔽3)#4112yes—
141#1084—21no—
142#1084—22no—
143#1084C2×Q812yes—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—21no—
147(C3⋊Dic3)⋊C3—22no—
148C3×C4⋊Q8—91no—
149C3×C22⋊Q8—91no—
150C2×C3×C4⋊C4—91no—
151#946—92no—
152#946—92no—
153#946—91no—
154Q8×Dic3—91no—
155C4×#67—91no—
156C4×#67—91no—
157C4×#67—92no—
158C2×C4⋊Dic3—91no—
159#924—91no—
160#924—92no—
161#897—31no—
162#897—91no—
163#897—91no—
164#897—92no—
165#896—92no—
166#893—92no—
167D3×C4⋊C4—31no—
168D3×C4⋊C4—91no—
169D3×C4⋊C4—91no—
170D3×C4⋊C4—91no—
171#890—92no—
172#889—92no—
173#889—92no—
174#888—31no—
175#888—31no—
176#888—61no—
177#887—91no—
178#67⋊C4—92no—
179#878—92no—
180C4×D12—91no—
181C4×Dic6—91no—
182C22×C3⋊D3C22×D311yescharacteristic
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3×D3—31no—
185C22×C3×D3—31no—
186C2×C32×C4—21no—
187C2×C32×C4—22no—
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D311yescharacteristic
196C4×C3⋊D3—22no—
197C4×C3⋊D3—22no—
198C4×C3⋊D3—42no—
199#634—22no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C2×C3×Dic3—62no—
213C3×C4×D3—62no—
214C3×C4×D3—62no—
215C3×C4×D3—122no—
216C3×Dic6—62no—
217C3×Dic6—122no—
218C3×Dic6—122no—
219C3×Dic6—124no—
220#626—34no—
221#626—34no—
222#624—122no—
223D3×Dic3—122no—
224#554—271noSylow
225C2×H(3,𝔽3)C22×Q811yescharacteristic
226C2×H(3,𝔽3)C2×#4111yescharacteristic
227C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
228(C3⋊D3)⋊C3—22no—
229(C3⋊D3)⋊C3—22no—
230C2×C3×Q8—91no—
231C2×C3×Q8—92no—
232C22×C3×C4—91no—
233C22×C3×C4—91no—
234C22×C3×C4—92no—
235C2×#67—91no—
236C2×#67—91no—
237C22×Dic3—92no—
238C2×D12—91no—
239C2×C4×D3—31no—
240C2×C4×D3—31no—
241C2×C4×D3—32no—
242C2×C4×D3—91no—
243C2×C4×D3—91no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×C4×D3—92no—
247C2×Dic6—31no—
248C2×Dic6—32no—
249C2×Dic6—32no—
250C2×Dic6—62no—
251C2×Dic6—91no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—31no—
254C3×C4⋊C4—61no—
255C3×C4⋊C4—91no—
256C3×C4⋊C4—92no—
257C3×C4⋊C4—92no—
258C3×C4⋊C4—92no—
259C3×C4⋊C4—91no—
260C3×C4⋊C4D3211yescharacteristic
261C3×C22⋊C4—92no—
262C3×C42—91no—
263#217—91no—
264#217—91no—
265#217—92no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—92no—
268(C2×D3)⋊C4—91no—
269(C2×D3)⋊C4—91no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—91no—
273(C2×D3)⋊C4—91no—
274C4⋊Dic3—31no—
275C4⋊Dic3—31no—
276C4⋊Dic3—31no—
277C4⋊Dic3—32no—
278C4⋊Dic3—61no—
279C4⋊Dic3—92no—
280C4⋊Dic3—91no—
281C4⋊Dic3—91no—
282C4⋊Dic3—92no—
283C4⋊Dic3—92no—
284C4⋊Dic3—92no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—62no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—91no—
295Dic3⋊C4—91no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—61no—
301C4×Dic3—92no—
302C4×Dic3—92no—
303C4×Dic3—92no—
304C22×C32—21no—
305C22×C32C23×D311yescharacteristic
306C22×C32C23×D311yescharacteristic
307C2×C3⋊D3—21no—
308C2×C3⋊D3—21no—
309C2×C3⋊D3—21no—
310C2×C3⋊D3—21no—
311C2×C3⋊D3D3×Q812yes—
312C2×C3⋊D3D3×Q812yes—
313C2×C3×D3—32no—
314C2×C3×D3—32no—
315C2×C3×D3—61no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C32×C4—22no—
320C32×C4—22no—
321C32×C4—22no—
322C32×C4—42no—
323C32×C4#23512yes—
324C32×C4#23712yes—
325C3⋊Dic3—21no—
326C3⋊Dic3—22no—
327C3⋊Dic3—21no—
328C3⋊Dic3—22no—
329C3×Dic3—32no—
330C3×Dic3—32no—
331C3×Dic3—62no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—61no—
335C3×Dic3—61no—
336C3×Dic3—62no—
337C3×Dic3—61no—
338C3×Dic3—62no—
339C3×Dic3—62no—
340C3×Dic3—61no—
341C3×Dic3—122no—
342C4⋊Q8—91no—
343#125—272no—
344C22⋊Q8—91no—
345C22⋊Q8—272no—
346C22⋊Q8—271no—
347C22⋊Q8—272no—
348C4×Q8—271no—
349C4×D4—271no—
350C4×D4—272no—
351C2×C4⋊C4—91no—
352C2×C4⋊C4—271no—
353H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
354C23×C3—91no—
355C23×C3—91no—
356C22×D3—31no—
357C22×D3—31no—
358C22×D3—91no—
359C22×D3—91no—
360C3×Q8—182no—
361C3×Q8—362no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—32no—
365C2×C3×C4—32no—
366C2×C3×C4—61no—
367C2×C3×C4—62no—
368C2×C3×C4—91no—
369C2×C3×C4—92no—
370C2×C3×C4—92no—
371C2×C3×C4—91no—
372C2×C3×C4—91no—
373C2×C3×C4—91no—
374C2×C3×C4—92no—
375C2×C3×C4—182no—
376C2×C3×C4—182no—
377C2×C3×C4—362no—
378C2×C3×C4C2×D3212yes—
379C2×C3×C4C2×D3211yescharacteristic
380#67—94no—
381#67—94no—
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—31no—
385C2×Dic3—31no—
386C2×Dic3—32no—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—32no—
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3—62no—
395C2×Dic3—92no—
396C2×Dic3—92no—
397C2×Dic3—91no—
398C2×Dic3—91no—
399C2×Dic3—362no—
400D12—94no—
401C4×D3—62no—
402C4×D3—62no—
403C4×D3—122no—
404C4×D3—182no—
405C4×D3—182no—
406C4×D3—362no—
407C4×D3—362no—
408C4×D3—362no—
409Dic6—62no—
410Dic6—122no—
411Dic6—122no—
412Dic6—124no—
413Dic6—182no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32C2×#23511yescharacteristic
418C2×C32C2×#23711yescharacteristic
419C2×C32#100411yescharacteristic
420C2×C32C2×D3×Q811yescharacteristic
421C2×C32C2×D3×Q811yescharacteristic
422C2×C32#100911yescharacteristic
423C3⋊D3—22no—
424C3⋊D3—22no—
425C3×D3—62no—
426C3×D3—62no—
427C2×Q8—91no—
428C2×Q8—92no—
429C2×D4—271no—
430C22×C4—91no—
431C22×C4—91no—
432C22×C4—92no—
433C22×C4—272no—
434C4⋊C4—91no—
435C4⋊C4—92no—
436C4⋊C4—91no—
437C4⋊C4—92no—
438C4⋊C4—92no—
439C4⋊C4—272no—
440C4⋊C4—271no—
441C4⋊C4—272no—
442C4⋊C4—272no—
443C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
444C22⋊C4—92no—
445C22⋊C4—271no—
446C22⋊C4—271no—
447C22⋊C4—272no—
448C42—91no—
449C42—272no—
450C22×C3—31no—
451C22×C3—31no—
452C22×C3—61no—
453C22×C3—92no—
454C22×C3—92no—
455C22×C3—181no—
456C22×C3—181no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3C22×D3211yescharacteristic
460C2×D3—32no—
461C2×D3—32no—
462C2×D3—61no—
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—92no—
467C2×D3—92no—
468C2×D3—181no—
469C2×D3—181no—
470C2×D3—181no—
471C2×D3—181no—
472C3×C4—22no—
473C3×C4—32no—
474C3×C4—32no—
475C3×C4—62no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—92no—
479C3×C4—122no—
480C3×C4—182no—
481C3×C4—181no—
482C3×C4—181no—
483C3×C4—182no—
484C3×C4—182no—
485C3×C4—181no—
486C3×C4#379512yes—
487Dic3—32no—
488Dic3—32no—
489Dic3—62no—
490Dic3—61no—
491Dic3—61no—
492Dic3—62no—
493Dic3—62no—
494Dic3—62no—
495Dic3—62no—
496Dic3—61no—
497Dic3—61no—
498Dic3—62no—
499Dic3—122no—
500Dic3—182no—
501Dic3—181no—
502Dic3—181no—
503Dic3—182no—
504C32—21no—
505C32#551111yescharacteristic, complemented
506C32#563511yescharacteristic, complemented
507C23—91no—
508C23—91no—
509Q8—182no—
510Q8—362no—
511D4—274no—
512C2×C4—91no—
513C2×C4—91no—
514C2×C4—92no—
515C2×C4—92no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—92no—
519C2×C4—182no—
520C2×C4—182no—
521C2×C4—362no—
522C2×C4—1082no—
523C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
524C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—61no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—182no—
535C2×C3—182no—
536C2×C3C2×#379511yescharacteristic
537C2×C3#6431411yescharacteristic
538C2×C3D32×Q811yescharacteristic
539D3—62no—
540D3—62no—
541D3—182no—
542D3—182no—
543C22—92no—
544C22—92no—
545C22—181no—
546C22—181no—
547C22—181no—
548C22—181no—
549C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
550C4—22no—
551C4—92no—
552C4—182no—
553C4—182no—
554C4—181no—
555C4—181no—
556C4—181no—
557C4—182no—
558C4#8864812yes—
559C3—31no—
560C3—31no—
561C3—61no—
562C3#1059453211yescharacteristic
563C2—182no—
564C2—182no—
565C2C2×#8864811yescharacteristic
566C2#1172250711yescharacteristic
567C2Q8×((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 radicalC2×#6198C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
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
1. #11721756C2semidirect
2. #64106⋊C3C2semidirect
3. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
5. #11721135C2semidirect
6. #11721129C2semidirect
8. #11721107C2semidirect
9. #11721106C2semidirect
10. #11721086C2semidirect
11. #11721084C2semidirect
37. #64106D3semidirect
38. C4⋊C4×C3⋊D3D3semidirect
76. #6198Q8semidirect
353. H(3,𝔽3)#554semidirect
505. C32#5511semidirect
506. C32#5635semidirect

Maximal subgroups

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

subgroupindex
#117217562
#64106⋊C32
C4⋊C4×(C3⋊D3)⋊C32
C4×#61982
#117211352
#117211292
#117211222
#117211072
#117211062
#117210862
#117210842
#105945373
#105943933