Finite Group Databasebeta

← #13151326

Subgroups — #13151326

Order 1728.

892 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#1084C211yesmaximal, characteristic
2D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
3D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
4#11721309C212yesmaximal, complemented
5C2×#88448C212yesmaximal, complemented
6#11721299C212yesmaximal, complemented
7#11721299C212yesmaximal, complemented
8#11721116C211yesmaximal, characteristic, complemented
9C4×#6196C211yesmaximal, characteristic, complemented
10#11721095C211yesmaximal, characteristic, complemented
11#11721093C211yesmaximal, characteristic, complemented
12#10595274—31nomaximal
13#10595272—31nomaximal
14C2×D4×H(3,𝔽3)C2211yescharacteristic
15C22×#1084C2212yes—
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C22×(C3⋊Dic3)⋊C3C2212yes—
18C2×#6196C2211yescharacteristic
19C2×#6196C2212yes—
20#88552C2212yes—
21C4⋊#1084C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23#3757⋊C3C2212yes—
24#3757⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29#88449C2211yescharacteristic
30#88449C2212yes—
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88448C2211yescharacteristic
35#88448C2212yes—
36#88448C2211yescharacteristic
37D4×C3⋊Dic3D311yescharacteristic
38D4×C3⋊Dic3D311yescharacteristic
39C3×D4×Dic3—31no—
40C3×D4×Dic3—31no—
41C3×D4×Dic3—31no—
42C3×D4×Dic3—31no—
43C3×D4×Dic3—61no—
44#63982—32no—
45#63982—32no—
46C2×#3724—32no—
47C2×Dic3⋊Dic3—32no—
48#63964—32no—
49#63964—32no—
50#63964—32no—
51#63964—32no—
52#63919—31no—
53#63918—31no—
54C4×#627—31no—
55C4×#627—31no—
56#63872—31no—
57#63869—31no—
58#63869—31no—
59#63869—31no—
60C23×H(3,𝔽3)C2312yes—
61D4×H(3,𝔽3)Q814yescomplemented
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084—42no—
64C2×#1084C2312yes—
65C2×#1084C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3—24no—
68C2×(C3⋊Dic3)⋊C3—42no—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75#6196—22no—
76#6196—42no—
77#5543—91no—
78#5539—91no—
79#5539—91no—
80C2×C32×D4—21no—
81C2×C32×D4C2×D311yescharacteristic
82C2×C32×D4C2×D311yescharacteristic
83C22×C3⋊Dic3C2×D312yes—
84C22×C3⋊Dic3C2×D312yes—
85C22×C3×Dic3—32no—
86C22×C3×Dic3—32no—
87C22×C3×Dic3—32no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—62no—
90C2×#627—31no—
91C2×#627—31no—
92C2×#627—32no—
93C2×#627—32no—
94#3757C2×D312yes—
95#3757C2×D312yes—
96C4⋊(C3⋊Dic3)C2×D311yescharacteristic
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98C4×C3⋊Dic3C2×D311yescharacteristic
99C4×C3⋊Dic3C2×D311yescharacteristic
100C3×#217—32no—
101C3×#217—32no—
102C3×#217—32no—
103C3×#217—32no—
104C3×#217—62no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×C4×Dic3—31no—
111C3×C4×Dic3—31no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—31no—
114C3×C4×Dic3—61no—
115#3724—32no—
116#3724—32no—
117#3724—32no—
118#3724—31no—
119#3724—31no—
120#3724—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—31no—
126Dic3⋊Dic3—31no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—31no—
130Dic3⋊Dic3—31no—
131C22×H(3,𝔽3)—22no—
132C22×H(3,𝔽3)—22no—
133C22×H(3,𝔽3)C2×Q814yes—
134C22×H(3,𝔽3)C2411yescharacteristic
135C4×H(3,𝔽3)C2×Q812yes—
136#1084—21no—
137#1084—21no—
138#1084—22no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3#4112yes—
145C3×C4×D4—91no—
146C3×C4×D4—91no—
147C3×C4×D4—91no—
148D4×Dic3—31no—
149D4×Dic3—31no—
150D4×Dic3—31no—
151D4×Dic3—31no—
152D4×Dic3—61no—
153D4×Dic3—91no—
154D4×Dic3—91no—
155C2×C4⋊Dic3—92no—
156#924—92no—
157#924—92no—
158#924—92no—
159C2×Dic3⋊C4—92no—
160C2×Dic3⋊C4—92no—
161#890—91no—
162#890—91no—
163#890—91no—
164#890—91no—
165#888—91no—
166#888—91no—
167#878—92no—
168#878—92no—
169#878—92no—
170#878—92no—
171#878—92no—
172#878—92no—
173#870—91no—
174#870—91no—
175#869—91no—
176C4×Dic6—91no—
177C4×Dic6—91no—
178C4×Dic6—91no—
179C23×C32—22no—
180C23×C32C22×D312yes—
181C23×C32C22×D312yes—
182C32×D4—24no—
183C32×D4Dic614yescomplemented
184C32×D4Dic614yescomplemented
185C2×C32×C4—21no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3—24no—
189C2×C3⋊Dic3—42no—
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3×Dic3—31no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—61no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3—64no—
212C2×C3×Dic3—122no—
213C2×C3×Dic3—122no—
214C2×C3×Dic3—122no—
215C2×C3×Dic3—124no—
216#627—62no—
217#627—62no—
218#627—122no—
219#627—122no—
220#554—271noSylow
221C2×H(3,𝔽3)—24no—
222C2×H(3,𝔽3)C22×Q811yescharacteristic
223C2×H(3,𝔽3)C2×#4111yescharacteristic
224C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
225C2×C3×D4—31no—
226C2×C3×D4—31no—
227C2×C3×D4—61no—
228C2×C3×D4D3211yescharacteristic
229C22×C3×C4—92no—
230C22×C3×C4—92no—
231C22×C3×C4—92no—
232C22×Dic3—32no—
233C22×Dic3—32no—
234C22×Dic3—32no—
235C22×Dic3—32no—
236C22×Dic3—62no—
237C22×Dic3—92no—
238C22×Dic3—92no—
239C2×Dic6—91no—
240C2×Dic6—91no—
241C2×Dic6—91no—
242C2×Dic6—92no—
243C2×Dic6—92no—
244C2×Dic6—92no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—91no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4—92no—
251C3×C42—91no—
252C3×C42—91no—
253C3×C42—91no—
254#217—32no—
255#217—32no—
256#217—32no—
257#217—32no—
258#217—62no—
259#217—92no—
260#217—92no—
261C4⋊Dic3—31no—
262C4⋊Dic3—31no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266C4⋊Dic3—91no—
267C4⋊Dic3—91no—
268C4⋊Dic3—91no—
269C4⋊Dic3—92no—
270C4⋊Dic3—91no—
271C4⋊Dic3—91no—
272C4⋊Dic3—92no—
273C4⋊Dic3—92no—
274C4⋊Dic3—92no—
275C4⋊Dic3—92no—
276Dic3⋊C4—91no—
277Dic3⋊C4—91no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—91no—
283Dic3⋊C4—91no—
284Dic3⋊C4—91no—
285Dic3⋊C4—91no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292C4×Dic3—31no—
293C4×Dic3—31no—
294C4×Dic3—31no—
295C4×Dic3—31no—
296C4×Dic3—61no—
297C4×Dic3—91no—
298C4×Dic3—91no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32—22no—
302C22×C32—22no—
303C22×C32—21no—
304C22×C32—24no—
305C22×C32—42no—
306C22×C32—42no—
307C22×C32C2×Dic614yes—
308C22×C32C2×Dic614yes—
309C22×C32C23×D311yescharacteristic
310C22×C32C23×D311yescharacteristic
311C32×C4—22no—
312C32×C4C2×Dic612yes—
313C32×C4C2×Dic612yes—
314C3⋊Dic3—22no—
315C3⋊Dic3—22no—
316C3⋊Dic3—21no—
317C3⋊Dic3—21no—
318C3⋊Dic3—21no—
319C3⋊Dic3#23712yes—
320C3×Dic3—32no—
321C3×Dic3—62no—
322C3×Dic3—61no—
323C3×Dic3—61no—
324C3×Dic3—61no—
325C3×Dic3—62no—
326C3×Dic3—61no—
327C3×Dic3—62no—
328C3×Dic3—61no—
329C3×Dic3—62no—
330C3×Dic3—61no—
331C3×Dic3—62no—
332C3×Dic3—61no—
333C3×Dic3—121no—
334C3×Dic3—122no—
335C4⋊Q8—271no—
336#125—271no—
337#125—271no—
338C22⋊Q8—272no—
339C22⋊Q8—272no—
340C22⋊Q8—272no—
341C4×Q8—271no—
342C4×D4—91no—
343C4×D4—91no—
344C4×D4—91no—
345C2×C4⋊C4—272no—
346H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
347C23×C3—32no—
348C23×C3—32no—
349C23×C3—62no—
350C23×C3C2×D3212yes—
351C3×D4—34no—
352C3×D4—34no—
353C3×D4—64no—
354C3×D4#62714yes—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—61no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—92no—
361C2×C3×C4—91no—
362C2×C3×C4—92no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—92no—
367C2×C3×C4—184no—
368C2×C3×C4—362no—
369C2×C3×C4—362no—
370C2×C3×C4C2×D3211yescharacteristic
371C2×Dic3—31no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—32no—
377C2×Dic3—32no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—31no—
381C2×Dic3—32no—
382C2×Dic3—32no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—62no—
386C2×Dic3—64no—
387C2×Dic3—91no—
388C2×Dic3—91no—
389C2×Dic3—91no—
390C2×Dic3—91no—
391C2×Dic3—92no—
392C2×Dic3—92no—
393C2×Dic3—122no—
394C2×Dic3—122no—
395C2×Dic3—122no—
396C2×Dic3—124no—
397C2×Dic3—184no—
398C2×Dic3—362no—
399Dic6—182no—
400Dic6—182no—
401Dic6—182no—
402Dic6—362no—
403Dic6—362no—
404Dic6—362no—
405C2×C32—24no—
406C2×C32—24no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—44no—
411C2×C32C22×Dic611yescharacteristic
412C2×C32C22×Dic611yescharacteristic
413C2×C32C2×#23711yescharacteristic
414C2×C32#100411yescharacteristic
415C2×C32D3×#4111yescharacteristic
416C2×C32#100911yescharacteristic
417C2×Q8—271no—
418C2×Q8—272no—
419C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
420C22×C4—92no—
421C22×C4—92no—
422C22×C4—92no—
423C4⋊C4—91no—
424C4⋊C4—91no—
425C4⋊C4—91no—
426C4⋊C4—271no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C4⋊C4—271no—
431C4⋊C4—271no—
432C4⋊C4—272no—
433C4⋊C4—272no—
434C22⋊C4—92no—
435C22⋊C4—92no—
436C22⋊C4—92no—
437C42—91no—
438C42—91no—
439C42—91no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—34no—
445C22×C3—34no—
446C22×C3—62no—
447C22×C3—62no—
448C22×C3—62no—
449C22×C3—62no—
450C22×C3—61no—
451C22×C3—64no—
452C22×C3—122no—
453C22×C3—122no—
454C22×C3C2×#62714yes—
455C22×C3C22×D3211yescharacteristic
456C3×C4—32no—
457C3×C4—32no—
458C3×C4—62no—
459C3×C4—92no—
460C3×C4—181no—
461C3×C4—181no—
462C3×C4—182no—
463C3×C4—182no—
464C3×C4—181no—
465C3×C4—181no—
466C3×C4—182no—
467C3×C4—181no—
468C3×C4C2×#62712yes—
469Dic3—32no—
470Dic3—61no—
471Dic3—61no—
472Dic3—62no—
473Dic3—62no—
474Dic3—61no—
475Dic3—61no—
476Dic3—62no—
477Dic3—61no—
478Dic3—62no—
479Dic3—61no—
480Dic3—62no—
481Dic3—61no—
482Dic3—92no—
483Dic3—122no—
484Dic3—121no—
485Dic3—181no—
486Dic3—181no—
487Dic3—182no—
488Dic3—182no—
489Dic3—181no—
490C32—21no—
491C32#553911yescharacteristic, complemented
492C32#554311yescharacteristic, complemented
493C23C2×((C3⋊D3)⋊C3)⋊C212yes—
494Q8—542no—
495Q8—1082no—
496D4#619614yescomplemented
497C2×C4—91no—
498C2×C4—91no—
499C2×C4—91no—
500C2×C4—91no—
501C2×C4—91no—
502C2×C4—91no—
503C2×C4—92no—
504C2×C4—92no—
505C2×C4—92no—
506C2×C4—184no—
507C2×C4—362no—
508C2×C4—362no—
509C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
510C2×C3—24no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—31no—
517C2×C3—64no—
518C2×C3—64no—
519C2×C3—61no—
520C2×C3—61no—
521C2×C3—61no—
522C2×C3—124no—
523C2×C3D3×#23711yescharacteristic
524C2×C3#6431411yescharacteristic
525C2×C3C22×#62711yescharacteristic
526C22—22no—
527C22—22no—
528C22C2×#619614yes—
529C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530C4—92no—
531C4—182no—
532C4—181no—
533C4—181no—
534C4—181no—
535C4—181no—
536C4—182no—
537C4—182no—
538C4—181no—
539C4C2×#619612yes—
540C3—31no—
541C3—31no—
542C3—61no—
543C3#1059527211yescharacteristic
544C2—24no—
545C2#1172250411yescharacteristic
546C2#1172250711yescharacteristic
547C2C22×#619611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11721309C2semidirect
5. C2×#88448C2semidirect
6. #11721299C2semidirect
7. #11721299C2semidirect
8. #11721116C2semidirect
9. C4×#6196C2semidirect
10. #11721095C2semidirect
11. #11721093C2semidirect
61. D4×H(3,𝔽3)Q8semidirect
183. C32×D4Dic6semidirect
184. C32×D4Dic6semidirect
346. H(3,𝔽3)#554semidirect
491. C32#5539semidirect
492. C32#5543semidirect
496. D4#6196semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
D4×(C3⋊Dic3)⋊C32
#117213092
C2×#884482
#117212992
#117211162
C4×#61962
#117210952
#117210932
#105952743
#105952723