Finite Group Databasebeta

← #13149413

Subgroups — #13149413

Order 1728.

813 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#272C211yesmaximal, characteristic
2#64099⋊C3C211yesmaximal, characteristic, complemented
3#64097⋊C3C211yesmaximal, characteristic, complemented
4#11721116C211yesmaximal, characteristic, complemented
5#11721097C211yesmaximal, characteristic
6#11721096C212yesmaximal
7#11721094C212yesmaximal
8#11721093C211yesmaximal, characteristic, complemented
9#11721089C211yesmaximal, characteristic
10#11721086C212yesmaximal, complemented
11#11721079C212yesmaximal, complemented
12#10594344—31nomaximal
13#10594342—31nomaximal
14C2×C4×#272C2212yes—
15C2×C4×#272C2211yescharacteristic
16C2×#634⋊C3C2212yes—
17C2×#6196C2212yes—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540C2212yes—
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24#3750⋊C3C2212yes—
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28#88451C2212yes—
29#88451C2211yescharacteristic
30#88451C2211yescharacteristic
31#88449C2211yescharacteristic
32#88449C2211yescharacteristic
33#88449C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64099D311yescharacteristic, complemented
38#64097D311yescharacteristic, complemented
39C3×D3×C4⋊C4—31no—
40C3×D3×C4⋊C4—31no—
41C3×D3×C4⋊C4—61no—
42C3×#890—31no—
43C3×#888—31no—
44#63919—31no—
45#63918—31no—
46#63872—31no—
47#63869—31no—
48#63857—32no—
49#63856—32no—
50#63855—31no—
51#63855—31no—
52#63851—32no—
53#63851—32no—
54#63846—32no—
55#63845—32no—
56#63841—31no—
57#63841—31no—
58#63840—32no—
59#63839—32no—
60C22×#272C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2312yes—
64C2×#1084C2311yescharacteristic
65C4×#272—42no—
66C4×#272Q814yescomplemented
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73#634⋊C3—42no—
74#6196—42no—
75#5729—91no—
76#5689—91no—
77#5500—91no—
78C2×#634C2×D312yes—
79C2×C3×C4×D3—31no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—61no—
84C2×C3×C4×D3—62no—
85C2×C3×Dic6—32no—
86C2×#627—32no—
87C2×#627—32no—
88C32×C4⋊C4—21no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C2×D311yescharacteristic
91C4⋊(C3⋊Dic3)C2×D311yescharacteristic
92C4⋊(C3⋊Dic3)C2×D311yescharacteristic
93C4⋊(C3⋊Dic3)C2×D312yes—
94#3750C2×D312yes—
95#3750C2×D312yes—
96C4×C3⋊Dic3C2×D311yescharacteristic
97C4×C3⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—32no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—61no—
104C3×Dic3⋊C4—32no—
105C3×Dic3⋊C4—32no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—62no—
109C3×C4×Dic3—31no—
110C3×C4×Dic3—31no—
111#3724—32no—
112#3724—32no—
113Dic3⋊Dic3—31no—
114Dic3⋊Dic3—31no—
115Dic3⋊Dic3—32no—
116Dic3⋊Dic3—31no—
117Dic3⋊Dic3—32no—
118Dic3⋊Dic3—31no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121(C2×D3)⋊Dic3—32no—
122(C2×D3)⋊Dic3—32no—
123(C2×D3)⋊Dic3—31no—
124(C2×D3)⋊Dic3—31no—
125(C2×D3)⋊Dic3—31no—
126(C2×D3)⋊Dic3—31no—
127Dic32—32no—
128Dic32—32no—
129C22×H(3,𝔽3)C2411yescharacteristic
130C2×#272—21no—
131C2×#272—21no—
132C2×#272C2×Q812yes—
133C4×H(3,𝔽3)—22no—
134C4×H(3,𝔽3)C2×Q812yes—
135#1084—22no—
136#1084C2×Q812yes—
137(C3⋊Dic3)⋊C3—21no—
138(C3⋊Dic3)⋊C3—22no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3—21no—
143C3×C4⋊Q8—91no—
144C3×#125—91no—
145C2×C3×C4⋊C4—91no—
146#946—92no—
147#944—92no—
148#926—91no—
149#926—91no—
150#924—92no—
151#924—92no—
152#897—92no—
153#896—92no—
154#896—92no—
155#892—91no—
156#892—91no—
157D3×C4⋊C4—31no—
158D3×C4⋊C4—31no—
159D3×C4⋊C4—61no—
160#890—31no—
161#890—91no—
162#890—91no—
163#890—92no—
164#890—91no—
165#889—92no—
166#889—92no—
167#888—31no—
168#888—91no—
169#888—91no—
170#888—92no—
171#888—92no—
172#888—91no—
173#872—91no—
174#872—91no—
175#870—91no—
176#869—91no—
177C22×C3×D3—31no—
178C22×C3×D3—31no—
179C22×C3×D3—61no—
180C2×C32×C4—21no—
181C2×C32×C4—22no—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D311yescharacteristic
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192#634—42no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—62no—
205C3×C4×D3—34no—
206C3×C4×D3—34no—
207C3×C4×D3—64no—
208C3×C4×D3—122no—
209C3×C4×D3—122no—
210C3×C4×D3—124no—
211C3×Dic6—122no—
212#627—122no—
213#627—122no—
214#544—271noSylow
215C2×H(3,𝔽3)C22×Q811yescharacteristic
216C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
217C2×H(3,𝔽3)2−1+411yescharacteristic
218#272—22no—
219C2×C3×Q8—92no—
220C22×C3×C4—91no—
221C22×C3×C4—92no—
222C2×C4×D3—31no—
223C2×C4×D3—31no—
224C2×C4×D3—32no—
225C2×C4×D3—32no—
226C2×C4×D3—61no—
227C2×C4×D3—62no—
228C2×Dic6—32no—
229C2×Dic6—92no—
230C2×Dic6—92no—
231C2×Dic6—92no—
232C2×Dic6—92no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4D3211yescharacteristic
244C3×C42—91no—
245C3×C42—91no—
246#217—91no—
247#217—91no—
248#217—92no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—91no—
252(C2×D3)⋊C4—91no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255C4⋊Dic3—31no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258C4⋊Dic3—32no—
259C4⋊Dic3—31no—
260C4⋊Dic3—61no—
261C4⋊Dic3—91no—
262C4⋊Dic3—91no—
263C4⋊Dic3—92no—
264C4⋊Dic3—91no—
265C4⋊Dic3—92no—
266C4⋊Dic3—91no—
267C4⋊Dic3—92no—
268C4⋊Dic3—92no—
269Dic3⋊C4—32no—
270Dic3⋊C4—32no—
271Dic3⋊C4—32no—
272Dic3⋊C4—32no—
273Dic3⋊C4—62no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—91no—
281Dic3⋊C4—91no—
282Dic3⋊C4—91no—
283Dic3⋊C4—91no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286C4×Dic3—31no—
287C4×Dic3—31no—
288C4×Dic3—91no—
289C4×Dic3—91no—
290C4×Dic3—92no—
291C4×Dic3—92no—
292C4×Dic3—92no—
293C22×C32—21no—
294C22×C32C23×D311yescharacteristic
295C22×C32C23×D311yescharacteristic
296C2×C3×D3—32no—
297C2×C3×D3—32no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C32×C4—22no—
306C32×C4—22no—
307C32×C4—22no—
308C32×C4—42no—
309C32×C4D3×Q812yes—
310C32×C4D3×Q812yes—
311C3⋊Dic3—22no—
312C3⋊Dic3—21no—
313C3⋊Dic3—21no—
314C3⋊Dic3—21no—
315C3⋊Dic3—22no—
316C3⋊Dic3—21no—
317C3×Dic3—32no—
318C3×Dic3—32no—
319C3×Dic3—62no—
320C3×Dic3—61no—
321C3×Dic3—62no—
322C3×Dic3—62no—
323C3×Dic3—61no—
324C3×Dic3—62no—
325C3×Dic3—61no—
326C3×Dic3—62no—
327C3×Dic3—61no—
328C3×Dic3—122no—
329C4⋊Q8—91no—
330C4⋊Q8—271no—
331C4⋊Q8—272no—
332#125—91no—
333#125—271no—
334#125—272no—
335C22⋊Q8—272no—
336C22⋊Q8—272no—
337#117—271no—
338#117—271no—
339C2×C4⋊C4—91no—
340H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
341C23×C3—91no—
342C22×D3—31no—
343C22×D3—31no—
344C22×D3—61no—
345C3×Q8—362no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—32no—
349C2×C3×C4—32no—
350C2×C3×C4—61no—
351C2×C3×C4—62no—
352C2×C3×C4—91no—
353C2×C3×C4—94no—
354C2×C3×C4—92no—
355C2×C3×C4—92no—
356C2×C3×C4—91no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—92no—
361C2×C3×C4—362no—
362C2×C3×C4C2×D3212yes—
363C2×C3×C4C2×D3211yescharacteristic
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—31no—
371C2×Dic3—31no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—61no—
375C2×Dic3—62no—
376C2×Dic3—91no—
377C2×Dic3—91no—
378C2×Dic3—92no—
379C2×Dic3—91no—
380C2×Dic3—91no—
381C2×Dic3—92no—
382C4×D3—34no—
383C4×D3—34no—
384C4×D3—64no—
385C4×D3—122no—
386C4×D3—122no—
387C4×D3—124no—
388Dic6—122no—
389Dic6—362no—
390Dic6—362no—
391Dic6—362no—
392Dic6—362no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32#100411yescharacteristic
397C2×C32C2×D3×Q811yescharacteristic
398C2×C32C2×D3×Q811yescharacteristic
399C2×C32#100711yescharacteristic
400C2×C32#100911yescharacteristic
401C2×C32#101011yescharacteristic
402C3×D3—64no—
403C3×D3—62no—
404C3×D3—62no—
405C2×Q8—92no—
406C2×Q8—272no—
407C22×C4—91no—
408C22×C4—92no—
409C4⋊C4—92no—
410C4⋊C4—91no—
411C4⋊C4—91no—
412C4⋊C4—91no—
413C4⋊C4—92no—
414C4⋊C4—92no—
415C4⋊C4—92no—
416C4⋊C4—271no—
417C4⋊C4—271no—
418C4⋊C4—272no—
419C4⋊C4—272no—
420C4⋊C4—272no—
421C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
422C22⋊C4—271no—
423C22⋊C4—271no—
424C22⋊C4—272no—
425C42—91no—
426C42—91no—
427C42—272no—
428C22×C3—31no—
429C22×C3—31no—
430C22×C3—61no—
431C22×C3—92no—
432C22×C3—181no—
433C22×C3—181no—
434C22×C3C22×D3211yescharacteristic
435C2×D3—32no—
436C2×D3—32no—
437C2×D3—62no—
438C2×D3—61no—
439C2×D3—61no—
440C2×D3—61no—
441C2×D3—61no—
442C2×D3—62no—
443C2×D3—62no—
444C3×C4—22no—
445C3×C4—32no—
446C3×C4—32no—
447C3×C4—62no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—92no—
451C3×C4—122no—
452C3×C4—181no—
453C3×C4—182no—
454C3×C4—181no—
455C3×C4—181no—
456C3×C4—182no—
457C3×C4—181no—
458C3×C4—182no—
459C3×C4#379712yes—
460Dic3—32no—
461Dic3—32no—
462Dic3—62no—
463Dic3—62no—
464Dic3—62no—
465Dic3—61no—
466Dic3—62no—
467Dic3—61no—
468Dic3—61no—
469Dic3—61no—
470Dic3—62no—
471Dic3—122no—
472Dic3—181no—
473Dic3—182no—
474Dic3—181no—
475Dic3—181no—
476Dic3—182no—
477Dic3—181no—
478C32—21no—
479C32#568911yescharacteristic, complemented
480C32#572911yescharacteristic, complemented
481C23—91no—
482Q8—362no—
483Q8—1082no—
484C2×C4—91no—
485C2×C4—94no—
486C2×C4—91no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—92no—
491C2×C4—92no—
492C2×C4—92no—
493C2×C4—362no—
494C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
495C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
496C2×C3—31no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—31no—
502C2×C3—61no—
503C2×C3—61no—
504C2×C3—61no—
505C2×C3—182no—
506C2×C3C2×#379711yescharacteristic
507C2×C3#6431411yescharacteristic
508C2×C3#6431511yescharacteristic
509D3—62no—
510D3—62no—
511D3—64no—
512C22—92no—
513C22—181no—
514C22—181no—
515C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
516C4—22no—
517C4—92no—
518C4—182no—
519C4—181no—
520C4—182no—
521C4—181no—
522C4—181no—
523C4—181no—
524C4—182no—
525C4#8865112yes—
526C3—31no—
527C3—31no—
528C3—61no—
529C3#1059432511yescharacteristic
530C2—182no—
531C2C2×#8865111yescharacteristic
532C2#1172250711yescharacteristic
533C2#1172251611yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
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
2. #64099⋊C3C2semidirect
3. #64097⋊C3C2semidirect
4. #11721116C2semidirect
8. #11721093C2semidirect
10. #11721086C2semidirect
11. #11721079C2semidirect
37. #64099D3semidirect
38. #64097D3semidirect
66. C4×#272Q8semidirect
340. H(3,𝔽3)#544semidirect
479. C32#5689semidirect
480. C32#5729semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#2722
#64099⋊C32
#64097⋊C32
#117211162
#117210972
#117210962
#117210942
#117210932
#117210892
#117210862
#117210792
#105943443
#105943423