Finite Group Databasebeta

← #13147766

Subgroups — #13147766

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#11721716C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic
4#11721116C211yesmaximal, characteristic
5#11721093C211yesmaximal, characteristic
6#11721091C212yesmaximal
7#11721087C212yesmaximal
8#11721082C211yesmaximal, characteristic
9#11721082C211yesmaximal, characteristic
10#11721079C212yesmaximal
11#11721077C212yesmaximal
12#10593853—31nomaximal
13#10593852—31nomaximal
14C2×#634⋊C3C2211yescharacteristic
15C2×#634⋊C3C2211yescharacteristic
16C2×#6196C2212yes—
17C42×H(3,𝔽3)C2211yescharacteristic
18C4⋊#1084C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540C2212yes—
21#88540C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24#3750⋊C3C2211yescharacteristic
25#3750⋊C3C2212yes—
26#3750⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88449C2212yes—
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37C4×#634D311yescharacteristic
38C4×#634D311yescharacteristic
39C3×#870—31no—
40C3×#870—31no—
41C3×#870—61no—
42C3×C4×Dic6—31no—
43C3×C4×Dic6—31no—
44#63919—31no—
45#63918—31no—
46#63872—31no—
47#63870—32no—
48#63869—31no—
49#63867—32no—
50#63847—32no—
51#63845—32no—
52#63844—31no—
53#63844—31no—
54#63843—31no—
55#63843—31no—
56#63840—32no—
57#63839—32no—
58#63838—32no—
59#63837—32no—
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2312yes—
64C2×#1084C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71#634⋊C3—22no—
72#634⋊C3Q814yescomplemented
73#6196—24no—
74#5685—91no—
75#5569—91no—
76#5567—91no—
77C2×#634C2×D311yescharacteristic
78C2×#634C2×D311yescharacteristic
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×#627—32no—
82C2×#627—32no—
83C32×C42—21no—
84C32×C42C2×D311yescharacteristic
85C32×C42C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88#3750C2×D311yescharacteristic
89#3750C2×D312yes—
90#3750C2×D311yescharacteristic
91C4×C3⋊Dic3C2×D312yes—
92C4×C3⋊Dic3C2×D311yescharacteristic
93C4×C3⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—61no—
101C3×C4⋊Dic3—61no—
102C3×Dic3⋊C4—32no—
103C3×Dic3⋊C4—31no—
104C3×Dic3⋊C4—32no—
105C3×Dic3⋊C4—32no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—31no—
109C3×Dic3⋊C4—62no—
110C3×Dic3⋊C4—62no—
111C3×C4×Dic3—32no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—31no—
114#3724—32no—
115#3724—32no—
116#3724—32no—
117#3724—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic32—32no—
125Dic32—32no—
126Dic32—32no—
127Dic32—32no—
128C22×H(3,𝔽3)C2411yescharacteristic
129C4×H(3,𝔽3)—21no—
130C4×H(3,𝔽3)—21no—
131C4×H(3,𝔽3)C2×Q812yes—
132#1084—22no—
133#1084—22no—
134(C3⋊Dic3)⋊C3—22no—
135(C3⋊Dic3)⋊C3—21no—
136(C3⋊Dic3)⋊C3—21no—
137(C3⋊Dic3)⋊C3—21no—
138(C3⋊Dic3)⋊C3C2×Q814yes—
139(C3⋊Dic3)⋊C3#4112yes—
140C3×#125—91no—
141C3×C4×Q8—91no—
142C3×C4×Q8—91no—
143Q8×Dic3—91no—
144Q8×Dic3—91no—
145#944—92no—
146#890—91no—
147#890—92no—
148#890—91no—
149#889—92no—
150#889—92no—
151#889—92no—
152#889—92no—
153#888—91no—
154#888—92no—
155#888—92no—
156#888—91no—
157#887—92no—
158#887—91no—
159#887—91no—
160#887—92no—
161#870—31no—
162#870—31no—
163#870—61no—
164#870—92no—
165#870—91no—
166#869—91no—
167C4×Dic6—31no—
168C4×Dic6—31no—
169C4×Dic6—91no—
170C4×Dic6—91no—
171C4×Dic6—92no—
172C4×Dic6—91no—
173C4×Dic6—91no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3C22×D311yescharacteristic
184C2×C3⋊Dic3C22×D311yescharacteristic
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D311yescharacteristic
189#634—22no—
190#634Dic614yescomplemented
191C2×C3×Dic3—32no—
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—32no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—62no—
202C2×C3×Dic3—62no—
203C3×Dic6—34no—
204C3×Dic6—62no—
205#627—64no—
206#627—64no—
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—91no—
213C2×Dic6—31no—
214C2×Dic6—31no—
215C2×Dic6—91no—
216C2×Dic6—91no—
217C2×Dic6—92no—
218C2×Dic6—92no—
219C2×Dic6—92no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4—92no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4—91no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—92no—
229C3×C42—31no—
230C3×C42—31no—
231C3×C42—61no—
232C3×C42—91no—
233C3×C42—91no—
234C3×C42—92no—
235C3×C42D3211yescharacteristic
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—31no—
241C4⋊Dic3—31no—
242C4⋊Dic3—61no—
243C4⋊Dic3—61no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—92no—
248C4⋊Dic3—92no—
249C4⋊Dic3—92no—
250C4⋊Dic3—92no—
251Dic3⋊C4—32no—
252Dic3⋊C4—32no—
253Dic3⋊C4—32no—
254Dic3⋊C4—32no—
255Dic3⋊C4—31no—
256Dic3⋊C4—32no—
257Dic3⋊C4—31no—
258Dic3⋊C4—62no—
259Dic3⋊C4—62no—
260Dic3⋊C4—91no—
261Dic3⋊C4—91no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273C4×Dic3—31no—
274C4×Dic3—32no—
275C4×Dic3—31no—
276C4×Dic3—92no—
277C4×Dic3—91no—
278C4×Dic3—91no—
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—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—22no—
293C32×C4—41no—
294C32×C4—41no—
295C32×C4C2×Dic612yes—
296C32×C4D3×Q812yes—
297C3⋊Dic3—21no—
298C3⋊Dic3—22no—
299C3⋊Dic3—21no—
300C3⋊Dic3—21no—
301C3⋊Dic3C2×Dic614yes—
302C3⋊Dic3#23512yes—
303C3×Dic3—34no—
304C3×Dic3—32no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—61no—
308C3×Dic3—61no—
309C3×Dic3—62no—
310C3×Dic3—61no—
311C3×Dic3—62no—
312C3×Dic3—62no—
313C3×Dic3—122no—
314C3×Dic3—122no—
315C4⋊Q8—271no—
316C4⋊Q8—272no—
317#125—91no—
318#125—272no—
319#125—271no—
320#125—272no—
321C4×Q8—91no—
322C4×Q8—91no—
323C4×Q8—272no—
324C4×Q8—271no—
325C4×Q8—271no—
326H(3,𝔽3)#55711yescharacteristic, complemented, Sylow
327C3×Q8—94no—
328C3×Q8—182no—
329C2×C3×C4—31no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—92no—
339C2×C3×C4—91no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—91no—
343C2×C3×C4—92no—
344C2×C3×C4—92no—
345C2×C3×C4—92no—
346C2×C3×C4C2×D3211yescharacteristic
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×Dic3—32no—
350C2×Dic3—31no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—32no—
355C2×Dic3—31no—
356C2×Dic3—31no—
357C2×Dic3—32no—
358C2×Dic3—31no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—92no—
362C2×Dic3—91no—
363C2×Dic3—91no—
364C2×Dic3—91no—
365C2×Dic3—92no—
366C2×Dic3—91no—
367Dic6—34no—
368Dic6—62no—
369Dic6—94no—
370Dic6—182no—
371Dic6—184no—
372Dic6—184no—
373Dic6—184no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32C22×Dic611yescharacteristic
378C2×C32C2×#23511yescharacteristic
379C2×C32C2×D3×Q811yescharacteristic
380C2×C32#100711yescharacteristic
381C2×C32D3×#4111yescharacteristic
382C2×C32#101011yescharacteristic
383C2×Q8—91no—
384C2×Q8—91no—
385C2×Q8—272no—
386C4⋊C4—91no—
387C4⋊C4—91no—
388C4⋊C4—91no—
389C4⋊C4—92no—
390C4⋊C4—91no—
391C4⋊C4—92no—
392C4⋊C4—91no—
393C4⋊C4—91no—
394C4⋊C4—92no—
395C4⋊C4—272no—
396C4⋊C4—272no—
397C4⋊C4—272no—
398C4⋊C4—272no—
399C4⋊C4—272no—
400C42—92no—
401C42—91no—
402C42—91no—
403C42—272no—
404C42—272no—
405C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
406C22×C3—31no—
407C22×C3—31no—
408C22×C3—61no—
409C22×C3C22×D3211yescharacteristic
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—32no—
413C3×C4—32no—
414C3×C4—61no—
415C3×C4—61no—
416C3×C4—61no—
417C3×C4—61no—
418C3×C4—62no—
419C3×C4—92no—
420C3×C4—94no—
421C3×C4—121no—
422C3×C4—121no—
423C3×C4—182no—
424C3×C4—181no—
425C3×C4—181no—
426C3×C4—181no—
427C3×C4—182no—
428C3×C4—182no—
429C3×C4D3×Dic612yes—
430Dic3—34no—
431Dic3—32no—
432Dic3—62no—
433Dic3—62no—
434Dic3—62no—
435Dic3—62no—
436Dic3—61no—
437Dic3—61no—
438Dic3—61no—
439Dic3—62no—
440Dic3—94no—
441Dic3—92no—
442Dic3—122no—
443Dic3—122no—
444Dic3—181no—
445Dic3—182no—
446Dic3—181no—
447Dic3—181no—
448C32—21no—
449C32#556711yescharacteristic, complemented
450C32#556911yescharacteristic, complemented
451Q8—94no—
452Q8—182no—
453Q8—544no—
454C2×C4—92no—
455C2×C4—91no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—92no—
459C2×C4—91no—
460C2×C4—92no—
461C2×C4—92no—
462C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
463C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
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×C3#6429711yescharacteristic
476C2×C3D3×#23511yescharacteristic
477C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
478C4—21no—
479C4—21no—
480C4—94no—
481C4—92no—
482C4—182no—
483C4—181no—
484C4—181no—
485C4—181no—
486C4—182no—
487C4—182no—
488C4((C3⋊D3)⋊C3)⋊Q812yes—
489C3—31no—
490C3—31no—
491C3—61no—
492C3#1059385111yescharacteristic
493C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
494C2#1172248211yescharacteristic
495C2#1172249011yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#557
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
72. #634⋊C3Q8semidirect
190. #634Dic6semidirect
326. H(3,𝔽3)#557semidirect
449. C32#5567semidirect
450. C32#5569semidirect

Maximal subgroups

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

subgroupindex
#117217162
C4×#634⋊C32
#117211162
#117210932
#117210912
#117210872
#117210822
#117210792
#117210772
#105938533
#105938523