Finite Group Databasebeta

← #13149415

Subgroups — #13149415

Order 1728.

756 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721758C211yesmaximal, characteristic
2#64097⋊C3C211yesmaximal, characteristic
3#64096⋊C3C211yesmaximal, characteristic
4C4×#6196C211yesmaximal, characteristic
5#11721093C211yesmaximal, characteristic
6#11721091C212yesmaximal
7#11721085C212yesmaximal
8#11721084C211yesmaximal, characteristic
9#11721082C212yesmaximal
10#11721082C211yesmaximal, characteristic
11#11721079C212yesmaximal
12#10594322—31nomaximal
13#10594318—31nomaximal
14C2×#634⋊C3C2212yes—
15C2×#634⋊C3C2211yescharacteristic
16C2×#6196C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18C4⋊#1084C2212yes—
19C4⋊#1084C2211yescharacteristic
20#88540C2212yes—
21C4×#1084C2211yescharacteristic
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23#3750⋊C3C2212yes—
24#3750⋊C3C2212yes—
25C4×(C3⋊Dic3)⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28#88449C2212yes—
29#88449C2211yescharacteristic
30#88448C2212yes—
31#88448C2211yescharacteristic
32#88448C2212yes—
33#88448C2212yes—
34#88448C2211yescharacteristic
35#88447C2212yes—
36#88447C2212yes—
37#64097D311yescharacteristic
38#64096D311yescharacteristic
39C3×#890—31no—
40C3×#890—31no—
41C3×#890—61no—
42C3×#888—31no—
43C3×#887—31no—
44C4×#627—31no—
45C4×#627—31no—
46#63872—31no—
47#63870—32no—
48#63869—31no—
49#63867—32no—
50#63847—32no—
51#63846—32no—
52#63844—31no—
53#63844—31no—
54#63844—32no—
55#63843—32no—
56#63843—31no—
57Dic3×Dic6—31no—
58#63840—32no—
59#63839—32no—
60C2×C4×H(3,𝔽3)C2312yes—
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×#1084C2311yescharacteristic
63C2×#1084C2311yescharacteristic
64C2×#1084C2312yes—
65C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71#634⋊C3—24no—
72#634⋊C3Q814yescomplemented
73#6196—22no—
74#5721—91no—
75#5685—91no—
76#5569—91no—
77C2×#634C2×D312yes—
78C2×#634C2×D311yescharacteristic
79C2×C3×Dic6—31no—
80C2×C3×Dic6—32no—
81C2×#627—31no—
82C2×#627—31no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic
87#3750C2×D312yes—
88#3750C2×D312yes—
89C4×C3⋊Dic3C2×D312yes—
90C4×C3⋊Dic3C2×D311yescharacteristic
91C4×C3⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—32no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—62no—
98C3×C4⋊Dic3—61no—
99C3×Dic3⋊C4—32no—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—32no—
103C3×Dic3⋊C4—62no—
104C3×C4×Dic3—32no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—31no—
107C3×C4×Dic3—31no—
108C3×C4×Dic3—31no—
109C3×C4×Dic3—61no—
110#3724—32no—
111#3724—31no—
112#3724—32no—
113#3724—32no—
114#3724—31no—
115Dic3⋊Dic3—32no—
116Dic3⋊Dic3—32no—
117Dic3⋊Dic3—32no—
118Dic3⋊Dic3—31no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—31no—
122Dic3⋊Dic3—31no—
123Dic3⋊Dic3—31no—
124Dic32—32no—
125Dic32—32no—
126Dic32—32no—
127Dic32—32no—
128C22×H(3,𝔽3)C2411yescharacteristic
129C4×H(3,𝔽3)—22no—
130C4×H(3,𝔽3)C2×Q812yes—
131#1084—21no—
132#1084—21no—
133#1084—22no—
134(C3⋊Dic3)⋊C3—21no—
135(C3⋊Dic3)⋊C3—22no—
136(C3⋊Dic3)⋊C3—21no—
137(C3⋊Dic3)⋊C3—21no—
138(C3⋊Dic3)⋊C3C2×Q814yes—
139(C3⋊Dic3)⋊C3#4112yes—
140C3×C4⋊Q8—91no—
141C3×#125—91no—
142C3×C4×Q8—91no—
143Q8×Dic3—91no—
144Q8×Dic3—92no—
145Q8×Dic3—91no—
146#944—92no—
147#890—31no—
148#890—31no—
149#890—61no—
150#890—91no—
151#890—91no—
152#890—92no—
153#889—92no—
154#889—92no—
155#889—92no—
156#889—92no—
157#888—31no—
158#888—91no—
159#888—92no—
160#888—92no—
161#887—31no—
162#887—91no—
163#887—91no—
164#887—92no—
165#887—91no—
166#870—92no—
167#870—91no—
168C4×Dic6—91no—
169C4×Dic6—91no—
170C4×Dic6—91no—
171C4×Dic6—91no—
172C4×Dic6—91no—
173C4×Dic6—92no—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3C22×D312yes—
181C2×C3⋊Dic3C22×D311yescharacteristic
182C2×C3⋊Dic3C22×D311yescharacteristic
183C2×C3⋊Dic3C22×D311yescharacteristic
184C2×C3⋊Dic3C22×D312yes—
185C2×C3⋊Dic3C22×D311yescharacteristic
186#634—24no—
187#634Dic614yescomplemented
188C2×C3×Dic3—32no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—62no—
203C3×Dic6—34no—
204C3×Dic6—64no—
205#627—62no—
206#627—62no—
207#557—271noSylow
208C2×H(3,𝔽3)C22×Q811yescharacteristic
209C2×H(3,𝔽3)C2×#4111yescharacteristic
210C2×H(3,𝔽3)2−1+411yescharacteristic
211C2×C3×Q8—91no—
212C2×C3×Q8—92no—
213C2×Dic6—31no—
214C2×Dic6—32no—
215C2×Dic6—91no—
216C2×Dic6—92no—
217C2×Dic6—91no—
218C2×Dic6—91no—
219C2×Dic6—91no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4—91no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—92no—
229C3×C4⋊C4D3211yescharacteristic
230C3×C42—91no—
231C3×C42—91no—
232C3×C42—92no—
233C3×C42—91no—
234C4⋊Dic3—32no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—32no—
238C4⋊Dic3—31no—
239C4⋊Dic3—62no—
240C4⋊Dic3—61no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—92no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—92no—
248C4⋊Dic3—92no—
249Dic3⋊C4—32no—
250Dic3⋊C4—32no—
251Dic3⋊C4—32no—
252Dic3⋊C4—32no—
253Dic3⋊C4—62no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—91no—
257Dic3⋊C4—92no—
258Dic3⋊C4—91no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—91no—
262Dic3⋊C4—91no—
263Dic3⋊C4—92no—
264Dic3⋊C4—91no—
265Dic3⋊C4—91no—
266Dic3⋊C4—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270C4×Dic3—31no—
271C4×Dic3—31no—
272C4×Dic3—31no—
273C4×Dic3—31no—
274C4×Dic3—32no—
275C4×Dic3—61no—
276C4×Dic3—91no—
277C4×Dic3—91no—
278C4×Dic3—92no—
279C4×Dic3—92no—
280C4×Dic3—92no—
281C4×Dic3—92no—
282C4×Dic3—92no—
283C4×Dic3—92no—
284C4×Dic3—92no—
285C22×C32—21no—
286C22×C32C23×D311yescharacteristic
287C22×C32C23×D311yescharacteristic
288C32×C4—22no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4—42no—
292C32×C4C2×Dic612yes—
293C32×C4D3×Q812yes—
294C3⋊Dic3—22no—
295C3⋊Dic3—21no—
296C3⋊Dic3—21no—
297C3⋊Dic3—21no—
298C3⋊Dic3C2×Dic614yes—
299C3⋊Dic3#23712yes—
300C3×Dic3—34no—
301C3×Dic3—32no—
302C3×Dic3—61no—
303C3×Dic3—61no—
304C3×Dic3—61no—
305C3×Dic3—61no—
306C3×Dic3—61no—
307C3×Dic3—62no—
308C3×Dic3—61no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C3×Dic3—61no—
312C3×Dic3—121no—
313C3×Dic3—121no—
314C3×Dic3—122no—
315C4⋊Q8—91no—
316C4⋊Q8—272no—
317#125—91no—
318#125—272no—
319#125—271no—
320#125—272no—
321C4×Q8—91no—
322C4×Q8—271no—
323C4×Q8—272no—
324C4×Q8—271no—
325C4×Q8—271no—
326H(3,𝔽3)#55711yescharacteristic, complemented, Sylow
327C3×Q8—94no—
328C3×Q8—184no—
329C2×C3×C4—31no—
330C2×C3×C4—31no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—61no—
334C2×C3×C4—62no—
335C2×C3×C4—92no—
336C2×C3×C4—91no—
337C2×C3×C4—92no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—92no—
343C2×C3×C4—91no—
344C2×C3×C4C2×D3212yes—
345C2×C3×C4C2×D3211yescharacteristic
346C2×Dic3—32no—
347C2×Dic3—31no—
348C2×Dic3—31no—
349C2×Dic3—31no—
350C2×Dic3—31no—
351C2×Dic3—31no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—31no—
355C2×Dic3—31no—
356C2×Dic3—31no—
357C2×Dic3—32no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—62no—
361C2×Dic3—92no—
362C2×Dic3—91no—
363C2×Dic3—91no—
364C2×Dic3—91no—
365C2×Dic3—91no—
366C2×Dic3—92no—
367Dic6—34no—
368Dic6—64no—
369Dic6—94no—
370Dic6—184no—
371Dic6—182no—
372Dic6—182no—
373Dic6—182no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32C22×Dic611yescharacteristic
378C2×C32C2×#23711yescharacteristic
379C2×C32C2×D3×Q811yescharacteristic
380C2×C32#100711yescharacteristic
381C2×C32#100711yescharacteristic
382C2×C32D3×#4111yescharacteristic
383C2×Q8—91no—
384C2×Q8—92no—
385C2×Q8—271no—
386C4⋊C4—92no—
387C4⋊C4—92no—
388C4⋊C4—91no—
389C4⋊C4—91no—
390C4⋊C4—92no—
391C4⋊C4—92no—
392C4⋊C4—271no—
393C4⋊C4—272no—
394C4⋊C4—271no—
395C4⋊C4—272no—
396C4⋊C4—271no—
397C4⋊C4—272no—
398C4⋊C4—272no—
399C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
400C42—92no—
401C42—91no—
402C42—91no—
403C42—91no—
404C42—272no—
405C42—272no—
406C22×C3—31no—
407C22×C3—31no—
408C22×C3—61no—
409C22×C3C22×D3211yescharacteristic
410C3×C4—22no—
411C3×C4—32no—
412C3×C4—32no—
413C3×C4—62no—
414C3×C4—62no—
415C3×C4—62no—
416C3×C4—92no—
417C3×C4—94no—
418C3×C4—122no—
419C3×C4—181no—
420C3×C4—182no—
421C3×C4—181no—
422C3×C4—181no—
423C3×C4—181no—
424C3×C4—181no—
425C3×C4—182no—
426C3×C4D3×Dic612yes—
427Dic3—34no—
428Dic3—32no—
429Dic3—61no—
430Dic3—61no—
431Dic3—61no—
432Dic3—61no—
433Dic3—62no—
434Dic3—62no—
435Dic3—62no—
436Dic3—61no—
437Dic3—61no—
438Dic3—61no—
439Dic3—94no—
440Dic3—92no—
441Dic3—121no—
442Dic3—121no—
443Dic3—122no—
444Dic3—182no—
445Dic3—181no—
446Dic3—181no—
447Dic3—181no—
448C32—21no—
449C32#556911yescharacteristic, complemented
450C32#572111yescharacteristic, complemented
451Q8—94no—
452Q8—184no—
453Q8—542no—
454C2×C4—92no—
455C2×C4—91no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—92no—
462C2×C4—92no—
463C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
464C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—31no—
471C2×C3—61no—
472C2×C3—61no—
473C2×C3—61no—
474C2×C3C2×D3×Dic611yescharacteristic
475C2×C3D3×#23711yescharacteristic
476C2×C3#6431611yescharacteristic
477C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
478C4—22no—
479C4—92no—
480C4—94no—
481C4—182no—
482C4—181no—
483C4—181no—
484C4—181no—
485C4—182no—
486C4—181no—
487C4—181no—
488C4((C3⋊D3)⋊C3)⋊Q812yes—
489C3—31no—
490C3—31no—
491C3—61no—
492C3#1059431711yescharacteristic
493C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
494C2#1172250411yescharacteristic
495C2#1172251711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#557
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
72. #634⋊C3Q8semidirect
187. #634Dic6semidirect
326. H(3,𝔽3)#557semidirect
449. C32#5569semidirect
450. C32#5721semidirect

Maximal subgroups

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

subgroupindex
#117217582
#64097⋊C32
#64096⋊C32
C4×#61962
#117210932
#117210912
#117210852
#117210842
#117210822
#117210792
#105943223
#105943183