Finite Group Databasebeta

← #13154258

Subgroups — #13154258

Order 1728.

1078 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#126C211yesmaximal, characteristic, complemented
2#11721766C211yesmaximal, characteristic, complemented
3#11721765C211yesmaximal, characteristic, complemented
4#11721764C211yesmaximal, characteristic, complemented
5#11721758C211yesmaximal, characteristic, complemented
6#11721757C211yesmaximal, characteristic, complemented
7#11721756C211yesmaximal, characteristic, complemented
8#11721742C211yesmaximal, characteristic
9#11721741C211yesmaximal, characteristic
10#11721739C211yesmaximal, characteristic
11#11721735C211yesmaximal, characteristic, complemented
12#11721734C211yesmaximal, characteristic, complemented
13#11721734C211yesmaximal, characteristic, complemented
14#11721722C211yesmaximal, characteristic, complemented
15#11721715C211yesmaximal, characteristic, complemented
16C3×#5712—34nomaximal
17C2×#6235C2211yescharacteristic
18C22×#1084C2211yescharacteristic
19C2×C4×#272C2211yescharacteristic
20C2×#6229C2211yescharacteristic
21C2×#6229C2211yescharacteristic
22C2×#6229C2211yescharacteristic
23C4⋊C4×H(3,𝔽3)C2211yescharacteristic
24C4⋊C4×H(3,𝔽3)C2211yescharacteristic
25C4⋊C4×H(3,𝔽3)C2211yescharacteristic
26C22⋊C4×H(3,𝔽3)C2211yescharacteristic
27C22⋊C4×H(3,𝔽3)C2211yescharacteristic
28C22⋊C4×H(3,𝔽3)C2211yescharacteristic
29C42×H(3,𝔽3)C2211yescharacteristic
30#88552C2211yescharacteristic
31#88552C2211yescharacteristic
32#88542C2211yescharacteristic
33#88542C2211yescharacteristic
34#88542C2211yescharacteristic
35#88542C2211yescharacteristic
36#88542C2211yescharacteristic
37C4⋊#1084C2211yescharacteristic
38C4⋊#1084C2211yescharacteristic
39C4⋊#1084C2211yescharacteristic
40C4⋊#1084C2211yescharacteristic
41C4⋊#1084C2211yescharacteristic
42C4⋊#1084C2211yescharacteristic
43#88540C2211yescharacteristic
44#88540C2211yescharacteristic
45#88540C2211yescharacteristic
46#88540C2211yescharacteristic
47#88540C2211yescharacteristic
48#88540C2211yescharacteristic
49#88540C2211yescharacteristic
50C4×#1084C2211yescharacteristic
51C4×#1084C2211yescharacteristic
52C32×#126D314yescomplemented
53C3×#898—34no—
54C3×#897—34no—
55C3×#896—34no—
56C3×#890—34no—
57C3×#889—34no—
58C3×#888—34no—
59C3×#886—34no—
60C3×#885—34no—
61C3×#883—34no—
62C3×#879—34no—
63C3×#878—34no—
64C3×#878—34no—
65C3×#876—34no—
66C3×#869—34no—
67C23×H(3,𝔽3)C2311yescharacteristic
68C22×#272C2311yescharacteristic
69C2×C4×H(3,𝔽3)C2311yescharacteristic
70C2×C4×H(3,𝔽3)C2311yescharacteristic
71C2×C4×H(3,𝔽3)C2311yescharacteristic
72C2×C4×H(3,𝔽3)C2311yescharacteristic
73C2×C4×H(3,𝔽3)C2311yescharacteristic
74C2×C4×H(3,𝔽3)C2311yescharacteristic
75#6235—41no—
76C2×#1084—41no—
77C2×#1084C2311yescharacteristic
78C2×#1084C2311yescharacteristic
79C2×#1084C2311yescharacteristic
80C2×#1084C2311yescharacteristic
81C2×#1084C2311yescharacteristic
82C2×#1084C2311yescharacteristic
83C2×#1084C2311yescharacteristic
84C4×#272—41no—
85#6229—41no—
86#6229—41no—
87#6229—41no—
88C3×#563—91no—
89#5712—34no—
90C2×C3×#67—34no—
91C22×C3×Dic3—34no—
92C2×C3×C4×D3—34no—
93C2×C3×Dic6—34no—
94C2×C3×Dic6—34no—
95C2×C3×Dic6—34no—
96C32×C4⋊C4C2×D314yes—
97C32×C4⋊C4C2×D314yes—
98C32×C4⋊C4C2×D314yes—
99C32×C22⋊C4C2×D314yes—
100C32×C22⋊C4C2×D314yes—
101C32×C22⋊C4C2×D314yes—
102C32×C42C2×D314yes—
103C3×#217—34no—
104C3×#217—34no—
105C3×(C2×D3)⋊C4—34no—
106C3×(C2×D3)⋊C4—34no—
107C3×(C2×D3)⋊C4—34no—
108C3×(C2×D3)⋊C4—34no—
109C3×(C2×D3)⋊C4—34no—
110C3×C4⋊Dic3—34no—
111C3×C4⋊Dic3—34no—
112C3×C4⋊Dic3—34no—
113C3×C4⋊Dic3—34no—
114C3×C4⋊Dic3—34no—
115C3×C4⋊Dic3—34no—
116C3×Dic3⋊C4—34no—
117C3×Dic3⋊C4—34no—
118C3×Dic3⋊C4—34no—
119C3×Dic3⋊C4—34no—
120C3×Dic3⋊C4—34no—
121C3×Dic3⋊C4—34no—
122C3×Dic3⋊C4—34no—
123C3×C4×Dic3—34no—
124C3×C4×Dic3—34no—
125C22×H(3,𝔽3)—21no—
126C22×H(3,𝔽3)—21no—
127C22×H(3,𝔽3)—21no—
128C22×H(3,𝔽3)C2411yescharacteristic
129C2×#272—21no—
130C2×#272—21no—
131C2×#272—21no—
132C4×H(3,𝔽3)—21no—
133C4×H(3,𝔽3)—21no—
134C4×H(3,𝔽3)—21no—
135C4×H(3,𝔽3)—21no—
136C4×H(3,𝔽3)—21no—
137C4×H(3,𝔽3)—21no—
138#1084—21no—
139#1084—21no—
140#1084—21no—
141#1084—21no—
142#1084—21no—
143#1084—21no—
144#1084—21no—
145C3×C4⋊Q8—91no—
146C3×C4⋊Q8—91no—
147C3×#126—34no—
148C3×#126—91no—
149C3×#126—91no—
150C3×#126—91no—
151C3×#126C3⋊D311yescharacteristic
152C3×#125—91no—
153C3×#125—91no—
154C3×#124—91no—
155C3×#123—91no—
156C3×#123—91no—
157C3×C22⋊Q8—91no—
158C3×C22⋊Q8—91no—
159C3×C22⋊Q8—91no—
160C3×C22⋊Q8—91no—
161#898—34no—
162#897—34no—
163#896—34no—
164#890—34no—
165#889—34no—
166#888—34no—
167#886—34no—
168#885—34no—
169#883—34no—
170#879—34no—
171#878—34no—
172#878—34no—
173#876—34no—
174#869—34no—
175C23×C32C22×D314yes—
176C22×C3×D3—34no—
177C2×C32×C4C22×D314yes—
178C2×C32×C4C22×D314yes—
179C2×C32×C4C22×D314yes—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D314yes—
183C3×#67—124no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—34no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—124no—
192C3×C4×D3—124no—
193C3×Dic6—124no—
194C3×Dic6—124no—
195C3×Dic6—124no—
196#563—91noSylow
197C2×H(3,𝔽3)—41no—
198C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
199C2×H(3,𝔽3)2−1+411yescharacteristic
200C2×H(3,𝔽3)2−1+411yescharacteristic
201#272—41no—
202C2×C3×Q8—91no—
203C2×C3×Q8—91no—
204C2×C3×Q8—91no—
205C2×C3×D4—91no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C2×#67—34no—
209C22×Dic3—34no—
210C2×C4×D3—34no—
211C2×Dic6—34no—
212C2×Dic6—34no—
213C2×Dic6—34no—
214C3×C4⋊C4—34no—
215C3×C4⋊C4—34no—
216C3×C4⋊C4—34no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—91no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4—91no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4—91no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—91no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4C2×C3⋊D311yescharacteristic
231C3×C4⋊C4C2×C3⋊D311yescharacteristic
232C3×C4⋊C4C2×C3⋊D311yescharacteristic
233C3×C22⋊C4—34no—
234C3×C22⋊C4—34no—
235C3×C22⋊C4—34no—
236C3×C22⋊C4—91no—
237C3×C22⋊C4—91no—
238C3×C22⋊C4—91no—
239C3×C22⋊C4—91no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—91no—
242C3×C22⋊C4—91no—
243C3×C22⋊C4C2×C3⋊D311yescharacteristic
244C3×C22⋊C4C2×C3⋊D311yescharacteristic
245C3×C22⋊C4C2×C3⋊D311yescharacteristic
246C3×C42—34no—
247C3×C42—91no—
248C3×C42—91no—
249C3×C42C2×C3⋊D311yescharacteristic
250#217—34no—
251#217—34no—
252(C2×D3)⋊C4—34no—
253(C2×D3)⋊C4—34no—
254(C2×D3)⋊C4—34no—
255(C2×D3)⋊C4—34no—
256(C2×D3)⋊C4—34no—
257C4⋊Dic3—34no—
258C4⋊Dic3—34no—
259C4⋊Dic3—34no—
260C4⋊Dic3—34no—
261C4⋊Dic3—34no—
262C4⋊Dic3—34no—
263Dic3⋊C4—34no—
264Dic3⋊C4—34no—
265Dic3⋊C4—34no—
266Dic3⋊C4—34no—
267Dic3⋊C4—34no—
268Dic3⋊C4—34no—
269Dic3⋊C4—34no—
270C4×Dic3—34no—
271C4×Dic3—34no—
272C22×C32—24no—
273C22×C32—24no—
274C22×C32—24no—
275C22×C32C23×D314yes—
276C2×C3×D3—64no—
277C2×C3×D3—64no—
278C2×C3×D3—64no—
279C32×C4—24no—
280C32×C4—24no—
281C32×C4—24no—
282C32×C4—24no—
283C32×C4—24no—
284C32×C4—24no—
285C3×Dic3—64no—
286C3×Dic3—64no—
287C3×Dic3—64no—
288C3×Dic3—64no—
289C3×Dic3—64no—
290C3×Dic3—64no—
291C3×Dic3—64no—
292C4⋊Q8—91no—
293C4⋊Q8—91no—
294#126—91no—
295#126—91no—
296#126—91no—
297#126#27211yescharacteristic, complemented
298#125—91no—
299#125—91no—
300#124—91no—
301#123—91no—
302#123—91no—
303C22⋊Q8—91no—
304C22⋊Q8—91no—
305C22⋊Q8—91no—
306C22⋊Q8—91no—
307H(3,𝔽3)#56311yescharacteristic, complemented, Sylow
308C23×C3—34no—
309C23×C3—91no—
310C23×C3C22×C3⋊D311yescharacteristic
311C22×D3—34no—
312C3×Q8—361no—
313C3×Q8—361no—
314C3×Q8—361no—
315C3×D4—361no—
316C2×C3×C4—34no—
317C2×C3×C4—34no—
318C2×C3×C4—34no—
319C2×C3×C4—34no—
320C2×C3×C4—34no—
321C2×C3×C4—34no—
322C2×C3×C4—91no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—361no—
330C2×C3×C4—361no—
331C2×C3×C4C22×C3⋊D311yescharacteristic
332C2×C3×C4C22×C3⋊D311yescharacteristic
333C2×C3×C4C22×C3⋊D311yescharacteristic
334C2×C3×C4C22×C3⋊D311yescharacteristic
335C2×C3×C4C22×C3⋊D311yescharacteristic
336C2×C3×C4C22×C3⋊D311yescharacteristic
337#67—124no—
338C2×Dic3—34no—
339C2×Dic3—34no—
340C2×Dic3—34no—
341C2×Dic3—34no—
342C2×Dic3—34no—
343C2×Dic3—34no—
344C2×Dic3—34no—
345C2×Dic3—124no—
346C4×D3—124no—
347Dic6—124no—
348Dic6—124no—
349Dic6—124no—
350C2×C32—44no—
351C2×C32#100914yes—
352C2×C32#101014yes—
353C2×C32#101014yes—
354C3×D3—124no—
355C2×Q8—91no—
356C2×Q8—91no—
357C2×Q8—91no—
358C2×D4—91no—
359C22×C4—91no—
360C22×C4—91no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C4⋊C4—91no—
364C4⋊C4—91no—
365C4⋊C4—91no—
366C4⋊C4—91no—
367C4⋊C4—91no—
368C4⋊C4—91no—
369C4⋊C4—91no—
370C4⋊C4—91no—
371C4⋊C4—91no—
372C4⋊C4—91no—
373C4⋊C4—91no—
374C4⋊C4C2×#27211yescharacteristic
375C4⋊C4C2×#27211yescharacteristic
376C4⋊C4C2×#27211yescharacteristic
377C22⋊C4—91no—
378C22⋊C4—91no—
379C22⋊C4—91no—
380C22⋊C4—91no—
381C22⋊C4—91no—
382C22⋊C4—91no—
383C22⋊C4—91no—
384C22⋊C4C2×#27211yescharacteristic
385C22⋊C4C2×#27211yescharacteristic
386C22⋊C4C2×#27211yescharacteristic
387C42—91no—
388C42—91no—
389C42C2×#27211yescharacteristic
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—34no—
394C22×C3—64no—
395C22×C3—64no—
396C22×C3—64no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3—181no—
400C22×C3C23×C3⋊D311yescharacteristic
401C2×D3—64no—
402C2×D3—64no—
403C2×D3—64no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—64no—
411C3×C4—64no—
412C3×C4—64no—
413C3×C4—64no—
414C3×C4—64no—
415C3×C4—64no—
416C3×C4—181no—
417C3×C4—181no—
418C3×C4—181no—
419C3×C4—181no—
420C3×C4—181no—
421C3×C4—181no—
422C3×C4—181no—
423Dic3—64no—
424Dic3—64no—
425Dic3—64no—
426Dic3—64no—
427Dic3—64no—
428Dic3—64no—
429Dic3—64no—
430C32#571214yescomplemented
431C23—91no—
432C23C22×#27211yescharacteristic
433Q8—361no—
434Q8—361no—
435Q8—361no—
436D4—361no—
437C2×C4—91no—
438C2×C4—91no—
439C2×C4—91no—
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—361no—
445C2×C4—361no—
446C2×C4C22×#27211yescharacteristic
447C2×C4C22×#27211yescharacteristic
448C2×C4C22×#27211yescharacteristic
449C2×C4C22×#27211yescharacteristic
450C2×C4C22×#27211yescharacteristic
451C2×C4C22×#27211yescharacteristic
452C2×C3—34no—
453C2×C3—34no—
454C2×C3—34no—
455C2×C3—41no—
456C2×C3—124no—
457C2×C3—361no—
458C2×C3#6436611yescharacteristic
459C2×C3#6436711yescharacteristic
460C2×C3#6436711yescharacteristic
461D3—124no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—181no—
466C22—181no—
467C22—181no—
468C22C23×#27211yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—181no—
476C4—181no—
477C4—181no—
478C4—181no—
479C4—181no—
480C4—181no—
481C4—181no—
482C3—34no—
483C3#1059618311yescharacteristic
484C2—41no—
485C2—361no—
486C2#1172267511yescharacteristic
487C2#1172267611yescharacteristic
488C2#1172267611yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#126C2
prime radicalC2×#6235C22
hypercenterC3×#126C3⋊D3
hypocenterH(3,𝔽3)#563
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#126

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#126C2semidirect
2. #11721766C2semidirect
3. #11721765C2semidirect
4. #11721764C2semidirect
5. #11721758C2semidirect
6. #11721757C2semidirect
7. #11721756C2semidirect
11. #11721735C2semidirect
12. #11721734C2semidirect
13. #11721734C2semidirect
14. #11721722C2semidirect
15. #11721715C2semidirect
52. C32×#126D3semidirect
297. #126#272semidirect
307. H(3,𝔽3)#563semidirect
430. C32#5712semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1262
#117217662
#117217652
#117217642
#117217582
#117217572
#117217562
#117217422
#117217412
#117217392
#117217352
#117217342
#117217222
#117217152
C3×#57123