Finite Group Databasebeta

← #13177493

Subgroups — #13177493

Order 1728.

2420 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89078C211yesmaximal, characteristic
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723622C212yesmaximal, complemented
5#11723618C212yesmaximal, complemented
6C2×#88806C212yesmaximal, complemented
7#11723598C212yesmaximal, complemented
8#11723503C211yesmaximal, characteristic, complemented
9#11723452C211yesmaximal, characteristic, complemented
10#11723406C211yesmaximal, characteristic, complemented
11#11723399C211yesmaximal, characteristic, complemented
12#10596017—31nomaximal
13#10595219—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C22×C3⋊(C3⋊Dic3)C2212yes—
17#89078C418yescomplemented
18C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C22×C3×C3⋊Dic3C2212yes—
21C22×C32×Dic3C2212yes—
22C2×#6289C2212yes—
23C3×#3757C2212yes—
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88806C2211yescharacteristic
30#88806C2212yes—
31#88806C2211yescharacteristic
32#88806C2212yes—
33#88806C2212yes—
34#88804C2211yescharacteristic
35#88804C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37Dic3×C3⋊Dic3C2212yes—
38C2×#3830—31no—
39C2×#3830—34no—
40C2×#3830—64no—
41C32×C4×D4—31no—
42D4×C3⋊Dic3D311yescharacteristic, complemented
43C4×#638—32no—
44C2×#3750—32no—
45C4⋊C4×C3⋊D3—31no—
46#64096—31no—
47#64090—32no—
48#64086—32no—
49#64081—31no—
50C4×#634—31no—
51C3×D4×Dic3—34no—
52C3×D4×Dic3D314yescomplemented
53C2×#3724—38no—
54#63957—38no—
55#63952—38no—
56#63952—38no—
57#63900—34no—
58#63874—34no—
59#63844—34no—
60#63844—34no—
61C23×C33C2312yes—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C33×D4C2×C414yescomplemented
64C2×C33×C4C2311yescharacteristic
65#6312C2×C418yescomplemented
66C2×C3⋊(C3⋊Dic3)C2×C418yescomplemented
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C4×C3⋊(C3⋊D3)C2×C414yescomplemented
70#6308C2×C414yescomplemented
71C2×C3×C3⋊Dic3—42no—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2312yes—
75C2×C32×Dic3—42no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2312yes—
78C2×C32×Dic3C2311yescharacteristic
79#6289—42no—
80#5546—94no—
81#5546—91no—
82C2×C32×D4—24no—
83C2×C32×D4C2×D314yes—
84C2×C32×D4C2×D311yescharacteristic
85C22×C32×C4—32no—
86C2×#638—32no—
87C2×#638—38no—
88C2×#638—68no—
89C22×C3⋊Dic3—32no—
90C22×C3⋊Dic3—38no—
91C22×C3⋊Dic3—68no—
92C22×C3⋊Dic3C2×D312yes—
93#3830—38no—
94#3830—332no—
95#3830—632no—
96C2×C4×C3⋊D3—31no—
97C2×C4×C3⋊D3—34no—
98C2×C4×C3⋊D3—32no—
99C2×C4×C3⋊D3—64no—
100C2×#634—31no—
101C2×#634—34no—
102C2×#634—64no—
103C22×C3×Dic3—38no—
104C22×C3×Dic3C2×D318yes—
105C2×#624—38no—
106C32×C4⋊C4—31no—
107C32×C22⋊C4—32no—
108C32×C42—31no—
109#3757C2×D312yes—
110#3752—31no—
111#3752—31no—
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113#3750—32no—
114#3750—31no—
115#3750—31no—
116#3750—32no—
117#3750—32no—
118C4×C3⋊Dic3—32no—
119C4×C3⋊Dic3—32no—
120C4×C3⋊Dic3C2×D311yescharacteristic
121C3×#217—38no—
122C3×#217C2×D318yes—
123C3×C4⋊Dic3—34no—
124C3×C4⋊Dic3C2×D314yes—
125C3×C4×Dic3—34no—
126C3×C4×Dic3C2×D314yes—
127#3724—38no—
128#3724—38no—
129#3724—34no—
130#3724—34no—
131#3724—38no—
132#3722—34no—
133#3722—34no—
134Dic32—38no—
135Dic32—38no—
136C22×C33—22no—
137C22×C33—22no—
138C22×C33C22×C414yes—
139C22×C33C2411yescharacteristic
140C2×C3⋊(C3⋊D3)—21no—
141C2×C3⋊(C3⋊D3)—21no—
142C2×C3⋊(C3⋊D3)C22×C412yes—
143C33×C4C22×C412yes—
144C3⋊(C3⋊Dic3)C22×C414yes—
145C3⋊(C3⋊Dic3)C22×C412yes—
146C3×C3⋊Dic3—22no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—21no—
149C32×Dic3—22no—
150C32×Dic3—21no—
151C32×Dic3—21no—
152C2×#237—94no—
153C2×#237—91no—
154C2×#237—184no—
155C3×C4×D4—34no—
156C3×C4×D4—91no—
157D4×Dic3—34no—
158D4×Dic3C3⋊D311yescharacteristic, complemented
159C4×#67—98no—
160C4×#67—92no—
161C2×Dic3⋊C4—98no—
162C2×Dic3⋊C4—92no—
163D3×C4⋊C4—94no—
164D3×C4⋊C4—91no—
165#887—94no—
166#887—91no—
167#67⋊C4—98no—
168#67⋊C4—92no—
169#877—98no—
170#877—92no—
171#872—94no—
172#872—91no—
173C4×Dic6—94no—
174C4×Dic6—91no—
175C23×C32—28no—
176C23×C32C22×D318yes—
177C23×C32C22×D312yes—
178C22×C3⋊D3—31no—
179C22×C3⋊D3—34no—
180C22×C3⋊D3—64no—
181C32×D4—216no—
182C32×D4C4×D3116yescomplemented
183C32×D4C4×D314yescomplemented
184C2×C32×C4—24no—
185C2×C32×C4—31no—
186C2×C32×C4—32no—
187C2×C32×C4—31no—
188C2×C32×C4—122no—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D311yescharacteristic
191#638—38no—
192#638—332no—
193#638—632no—
194C2×C3⋊Dic3—38no—
195C2×C3⋊Dic3—32no—
196C2×C3⋊Dic3—31no—
197C2×C3⋊Dic3—332no—
198C2×C3⋊Dic3—38no—
199C2×C3⋊Dic3—34no—
200C2×C3⋊Dic3—42no—
201C2×C3⋊Dic3—632no—
202C2×C3⋊Dic3—68no—
203C2×C3⋊Dic3—64no—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C2×C3⋊Dic3C22×D312yes—
206C2×C3⋊Dic3C22×D311yescharacteristic
207C4×C3⋊D3—34no—
208C4×C3⋊D3—316no—
209C4×C3⋊D3—616no—
210C4×C3⋊D3—122no—
211#634—34no—
212#634—316no—
213#634—616no—
214C2×C3×Dic3—34no—
215C2×C3×Dic3—38no—
216C2×C3×Dic3—34no—
217C2×C3×Dic3—48no—
218C2×C3×Dic3—128no—
219C2×C3×Dic3C22×D314yes—
220C2×C3×Dic3C22×D318yes—
221C2×C3×Dic3C22×D314yes—
222#624—128no—
223#520—271noSylow
224C2×C33—24no—
225C2×C33C23×C411yescharacteristic
226C2×C33H(5,𝔽2)11yescharacteristic
227C2×C332−1+411yescharacteristic
228C3⋊(C3⋊D3)—22no—
229C2×C3×D4—24no—
230C2×C3×D4D3214yes—
231C2×C3×D4C2×C3⋊D311yescharacteristic
232C22×C3×C4—38no—
233C22×C3×C4—92no—
234C2×#67—98no—
235C2×#67—92no—
236C2×#67—188no—
237C22×Dic3—38no—
238C22×Dic3—98no—
239C22×Dic3—92no—
240C22×Dic3—188no—
241C22×Dic3C2×C3⋊D312yes—
242#237—932no—
243#237—98no—
244#237—1832no—
245C2×C4×D3—94no—
246C2×C4×D3—91no—
247C2×C4×D3—98no—
248C2×C4×D3—92no—
249C2×C4×D3—184no—
250C2×Dic6—94no—
251C2×Dic6—91no—
252C2×Dic6—184no—
253C3×C4⋊C4—34no—
254C3×C4⋊C4—91no—
255C3×C22⋊C4—38no—
256C3×C22⋊C4—92no—
257C3×C42—34no—
258C3×C42—91no—
259#217—38no—
260#217C2×C3⋊D312yes—
261(C2×D3)⋊C4—94no—
262(C2×D3)⋊C4—94no—
263(C2×D3)⋊C4—91no—
264(C2×D3)⋊C4—91no—
265C4⋊Dic3—34no—
266C4⋊Dic3C2×C3⋊D311yescharacteristic
267Dic3⋊C4—98no—
268Dic3⋊C4—94no—
269Dic3⋊C4—94no—
270Dic3⋊C4—98no—
271Dic3⋊C4—98no—
272Dic3⋊C4—92no—
273Dic3⋊C4—91no—
274Dic3⋊C4—91no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277C4×Dic3—34no—
278C4×Dic3—98no—
279C4×Dic3—98no—
280C4×Dic3—92no—
281C4×Dic3—92no—
282C4×Dic3C2×C3⋊D311yescharacteristic
283C22×C32—22no—
284C22×C32—22no—
285C22×C32—24no—
286C22×C32—216no—
287C22×C32—28no—
288C22×C32—28no—
289C22×C32—48no—
290C22×C32—48no—
291C22×C32C2×C4×D3116yes—
292C22×C32C2×C4×D314yes—
293C22×C32C23×D314yes—
294C22×C32C23×D311yescharacteristic
295C2×C3⋊D3—32no—
296C2×C3⋊D3—38no—
297C2×C3⋊D3—68no—
298C2×C3⋊D3—61no—
299C2×C3⋊D3—61no—
300C2×C3⋊D3—64no—
301C2×C3⋊D3—64no—
302C2×C3⋊D3—124no—
303C2×C3⋊D3—124no—
304C32×C4—28no—
305C32×C4—62no—
306C32×C4—61no—
307C32×C4—61no—
308C32×C4C2×C4×D318yes—
309C32×C4C2×C4×D312yes—
310C3⋊Dic3—22no—
311C3⋊Dic3—21no—
312C3⋊Dic3—21no—
313C3⋊Dic3—34no—
314C3⋊Dic3—32no—
315C3⋊Dic3—316no—
316C3⋊Dic3—38no—
317C3⋊Dic3—616no—
318C3⋊Dic3—68no—
319C3×Dic3—28no—
320C3×Dic3—24no—
321C3×Dic3—24no—
322C3×Dic3—68no—
323C3×Dic3—64no—
324C3×Dic3—64no—
325C2×#41—271no—
326C4×Q8—271no—
327C4×Q8—271no—
328C4×D4—31no—
329C4×D4—91no—
330C4×D4—272no—
331C4×D4—272no—
332#117—271no—
333#117—272no—
334C2×C4⋊C4—271no—
335C2×C4⋊C4—272no—
336C33#52011yescharacteristic, complemented, Sylow
337C23×C3—28no—
338C23×C3C2×D3218yes—
339C23×C3C22×C3⋊D312yes—
340C22×D3—94no—
341C22×D3—91no—
342C22×D3—184no—
343C3×D4—216no—
344C3×D4#624116yescomplemented
345C3×D4C4×C3⋊D314yescomplemented
346C2×C3×C4—24no—
347C2×C3×C4—34no—
348C2×C3×C4—38no—
349C2×C3×C4—34no—
350C2×C3×C4—91no—
351C2×C3×C4—92no—
352C2×C3×C4—91no—
353C2×C3×C4—128no—
354C2×C3×C4—362no—
355C2×C3×C4C2×D3214yes—
356C2×C3×C4C22×C3⋊D311yescharacteristic
357#67—932no—
358#67—98no—
359#67—1832no—
360C2×Dic3—34no—
361C2×Dic3—34no—
362C2×Dic3—38no—
363C2×Dic3—42no—
364C2×Dic3—98no—
365C2×Dic3—94no—
366C2×Dic3—92no—
367C2×Dic3—91no—
368C2×Dic3—932no—
369C2×Dic3—98no—
370C2×Dic3—128no—
371C2×Dic3—188no—
372C2×Dic3—184no—
373C2×Dic3—1832no—
374C2×Dic3C22×C3⋊D311yescharacteristic
375C2×Dic3C22×C3⋊D312yes—
376C2×Dic3C22×C3⋊D311yescharacteristic
377C4×D3—916no—
378C4×D3—94no—
379C4×D3—1816no—
380C4×D3—362no—
381C4×D3—368no—
382Dic6—916no—
383Dic6—94no—
384Dic6—1816no—
385C2×C32—24no—
386C2×C32—24no—
387C2×C32—24no—
388C2×C32—24no—
389C2×C32—216no—
390C2×C32—416no—
391C2×C32C22×C4×D314yes—
392C2×C32C22×C4×D311yescharacteristic
393C2×C32#100414yes—
394C2×C32#100411yescharacteristic
395C2×C32#101014yes—
396C2×C32#101011yescharacteristic
397C3⋊D3—62no—
398C3⋊D3—68no—
399C3⋊D3—128no—
400#41—278no—
401C2×Q8—271no—
402C2×D4—272no—
403C2×D4D3×C3⋊D311yescharacteristic
404C22×C4—32no—
405C22×C4—92no—
406C22×C4—272no—
407C22×C4—272no—
408C22×C4—271no—
409C4⋊C4—31no—
410C4⋊C4—91no—
411C4⋊C4—271no—
412C4⋊C4—272no—
413C4⋊C4—272no—
414C4⋊C4—272no—
415C4⋊C4—271no—
416C22⋊C4—32no—
417C22⋊C4—92no—
418C22⋊C4—271no—
419C22⋊C4—271no—
420C42—31no—
421C42—91no—
422C42—272no—
423C42—272no—
424C22×C3—28no—
425C22×C3—28no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—24no—
429C22×C3—216no—
430C22×C3—48no—
431C22×C3—48no—
432C22×C3C2×#624116yes—
433C22×C3C2×C4×C3⋊D314yes—
434C22×C3C22×D3214yes—
435C22×C3C23×C3⋊D311yescharacteristic
436C2×D3—98no—
437C2×D3—92no—
438C2×D3—188no—
439C2×D3—184no—
440C2×D3—184no—
441C2×D3—181no—
442C2×D3—181no—
443C2×D3—364no—
444C2×D3—364no—
445C3×C4—28no—
446C3×C4—64no—
447C3×C4—68no—
448C3×C4—64no—
449C3×C4—182no—
450C3×C4—181no—
451C3×C4—181no—
452C3×C4C2×#62418yes—
453C3×C4C2×C4×C3⋊D312yes—
454Dic3—22no—
455Dic3—21no—
456Dic3—21no—
457Dic3—64no—
458Dic3—68no—
459Dic3—64no—
460Dic3—916no—
461Dic3—98no—
462Dic3—94no—
463Dic3—92no—
464Dic3—1816no—
465Dic3—188no—
466C32—24no—
467C32#554614yescomplemented
468C32#554611yescharacteristic, complemented
469C23—271no—
470C23C2×D3×C3⋊D312yes—
471Q8—274no—
472D4—278no—
473D4#628914yescomplemented
474C2×C4—31no—
475C2×C4—31no—
476C2×C4—32no—
477C2×C4—91no—
478C2×C4—91no—
479C2×C4—92no—
480C2×C4—122no—
481C2×C4—272no—
482C2×C4—271no—
483C2×C4—278no—
484C2×C4—274no—
485C2×C4—362no—
486C2×C4—1082no—
487C2×C4C2×D3×C3⋊D311yescharacteristic
488C2×C3—216no—
489C2×C3—24no—
490C2×C3—24no—
491C2×C3—24no—
492C2×C3—24no—
493C2×C3—416no—
494C2×C3#6430914yes—
495C2×C3#6431314yes—
496C2×C3C22×#62414yes—
497C2×C3C22×C4×C3⋊D311yescharacteristic
498C2×C3#6436111yescharacteristic
499C2×C3#6436711yescharacteristic
500D3—188no—
501D3—182no—
502D3—368no—
503C22—22no—
504C22—22no—
505C22—272no—
506C22—541no—
507C22—541no—
508C22C2×#628914yes—
509C22C22×D3×C3⋊D311yescharacteristic
510C4—61no—
511C4—61no—
512C4—62no—
513C4—181no—
514C4—181no—
515C4—182no—
516C4—272no—
517C4—274no—
518C4C2×#628912yes—
519C3—24no—
520C3#1059521914yescomplemented
521C3#1059601711yescharacteristic, complemented
522C2—24no—
523C2—542no—
524C2#1172458211yescharacteristic
525C2#1172458711yescharacteristic
526C2C22×#628911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC2×C33×D4C22
socle, cosocleC22×C33C24
prime radicalC2×#89078C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×D4×C3⋊Dic3C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723622C2semidirect
5. #11723618C2semidirect
6. C2×#88806C2semidirect
7. #11723598C2semidirect
8. #11723503C2semidirect
9. #11723452C2semidirect
10. #11723406C2semidirect
11. #11723399C2semidirect
17. #89078C4semidirect
42. D4×C3⋊Dic3D3semidirect
52. C3×D4×Dic3D3semidirect
63. C33×D4C2×C4semidirect
65. #6312C2×C4semidirect
66. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
69. C4×C3⋊(C3⋊D3)C2×C4semidirect
70. #6308C2×C4semidirect
158. D4×Dic3C3⋊D3semidirect
182. C32×D4C4×D3semidirect
183. C32×D4C4×D3semidirect
336. C33#520semidirect
344. C3×D4#624semidirect
345. C3×D4C4×C3⋊D3semidirect
467. C32#5546semidirect
468. C32#5546semidirect
473. D4#6289semidirect
520. C3#10595219semidirect
521. C3#10596017semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C3×D4×C3⋊Dic32
C32×D4×Dic32
#117236222
#117236182
C2×#888062
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