Finite Group Databasebeta

← #13151451

Subgroups — #13151451

Order 1728.

998 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721847C211yesmaximal, characteristic
2#64145⋊C3C211yesmaximal, characteristic, complemented
3#64144⋊C3C211yesmaximal, characteristic, complemented
4#11721320C212yesmaximal, complemented
5#11721319C212yesmaximal, complemented
6#11721297C212yesmaximal, complemented
7#11721293C212yesmaximal, complemented
8#11721148C212yesmaximal, complemented
9#11721076C212yesmaximal, complemented
10#10595178—31nomaximal
11#10595176—31nomaximal
12C23×#272C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×#6235C2212yes—
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
16C2×#634⋊C3C2211yescharacteristic, complemented
17C2×#6199C2212yes—
18C2×#6199C2212yes—
19#88552C2211yescharacteristic
20#88542C2212yes—
21#3757⋊C3C2212yes—
22#3757⋊C3C2212yes—
23#3757⋊C3C2212yescomplemented
24#3757⋊C3C2211yescharacteristic, complemented
25#3750⋊C3C2212yescomplemented
26C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
27#88451C2212yes—
28#88451C2212yes—
29#88449C2212yescomplemented
30#88448C2212yescomplemented
31#88448C2212yescomplemented
32#88447C2212yescomplemented
33#64145D311yescharacteristic, complemented
34#64144D311yescharacteristic, complemented
35C3×#937—31no—
36C3×#937—31no—
37C3×#937—61no—
38C3×#936—31no—
39C3×#935—31no—
40#67⋊Dic3—32no—
41Dic3×#67—32no—
42#63969—32no—
43#63969—32no—
44D3×#217—32no—
45D3×#217—32no—
46#63956—32no—
47#63956—32no—
48#63896—32no—
49#63887—32no—
50#63878—32no—
51#63877—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C23×H(3,𝔽3)C2311yescharacteristic
54C22×#272—24no—
55C22×#272—41no—
56C22×#272C2312yes—
57D4×H(3,𝔽3)—41no—
58C2×C4×H(3,𝔽3)C2311yescharacteristic
59#6235—24no—
60C2×#1084C2312yes—
61C2×(C3⋊Dic3)⋊C3—41no—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66#634⋊C3—41no—
67#6199—24no—
68#6199—42no—
69#5679—91no—
70#5659—91no—
71#5587—91no—
72C23×C3×D3—31no—
73C23×C3×D3—31no—
74C23×C3×D3—61no—
75C2×C32×D4—21no—
76C2×C32×D4C2×D311yescharacteristic
77C2×C32×D4C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic, complemented
79C2×#634C2×D311yescharacteristic, complemented
80C2×C3×#67—32no—
81C2×C3×#67—32no—
82C2×C3×#67—62no—
83C22×C3×Dic3—31no—
84C2×C3×Dic6—31no—
85C2×D3×Dic3—32no—
86C2×D3×Dic3—32no—
87C2×D3×Dic3—32no—
88C2×D3×Dic3—32no—
89#3757C2×D312yes—
90#3757C2×D312yes—
91#3757C2×D311yescharacteristic, complemented
92#3757C2×D312yescomplemented
93#3750C2×D312yescomplemented
94C4×C3⋊Dic3C2×D311yescharacteristic, complemented
95C3×#217—32no—
96C3×#217—31no—
97C3×#217—32no—
98C3×#217—32no—
99C3×#217—31no—
100C3×#217—31no—
101C3×#217—61no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—62no—
105C3×Dic3⋊C4—32no—
106C3×C4×Dic3—31no—
107#3724—32no—
108#3724—32no—
109Dic3⋊Dic3—32no—
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112Dic3⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115(C2×D3)⋊Dic3—32no—
116(C2×D3)⋊Dic3—32no—
117Dic32—32no—
118Dic32—32no—
119C22×H(3,𝔽3)—22no—
120C22×H(3,𝔽3)—22no—
121C22×H(3,𝔽3)—21no—
122C22×H(3,𝔽3)—21no—
123C22×H(3,𝔽3)C2411yescharacteristic
124C2×#272—22no—
125C2×#272—22no—
126C2×#272—44no—
127C2×#272#4114yes—
128C4×H(3,𝔽3)—21no—
129#1084—22no—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132(C3⋊Dic3)⋊C3—22no—
133(C3⋊Dic3)⋊C3—22no—
134C3×#124—91no—
135C3×#123—91no—
136C3×#120—91no—
137C2×#217—92no—
138#946—92no—
139#937—31no—
140#937—31no—
141#937—61no—
142#936—31no—
143#936—91no—
144#935—31no—
145#935—91no—
146#935—92no—
147D4×Dic3—92no—
148C4×#67—92no—
149#896—92no—
150#892—92no—
151#883—92no—
152#883—92no—
153#67⋊C4—92no—
154D3×C22⋊C4—92no—
155D3×C22⋊C4—92no—
156#879—92no—
157#879—92no—
158#879—92no—
159#878—92no—
160#877—92no—
161#872—92no—
162C23×C32—21no—
163C23×C32—21no—
164C23×C32C22×D311yescharacteristic
165C23×C32C22×D311yescharacteristic
166C23×C32C22×D311yescharacteristic
167C23×C32C22×D311yescharacteristic
168C22×C3×D3—32no—
169C22×C3×D3—32no—
170C22×C3×D3—62no—
171C22×C3×D3—64no—
172C22×C3×D3—64no—
173C22×C3×D3—124no—
174C22×C3×D3—121no—
175C22×C3×D3—121no—
176C22×C3×D3—122no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—42no—
180C2×C32×C4—21no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3—41no—
184C2×C3⋊Dic3C22×D312yes—
185C2×C3⋊Dic3C22×D312yes—
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D312yes—
188#634—41no—
189C3×#67—64no—
190C3×#67—64no—
191C3×#67—124no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—121no—
200C3×Dic6—121no—
201D3×Dic3—64no—
202D3×Dic3—64no—
203D3×Dic3—122no—
204D3×Dic3—122no—
205#551—271noSylow
206C2×H(3,𝔽3)—41no—
207C2×H(3,𝔽3)—41no—
208C2×H(3,𝔽3)C2×#4112yes—
209C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
210#272—24no—
211C24×C3—91no—
212C23×D3—31no—
213C23×D3—31no—
214C23×D3—61no—
215C2×C3×Q8—91no—
216C2×C3×D4—31no—
217C2×C3×D4—31no—
218C2×C3×D4—61no—
219C2×C3×D4—92no—
220C2×C3×D4D3211yescharacteristic
221C22×C3×C4—91no—
222C2×#67—32no—
223C2×#67—32no—
224C2×#67—62no—
225C22×Dic3—31no—
226C22×Dic3—91no—
227C22×Dic3—92no—
228C22×Dic3—92no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×C4×D3—92no—
232C2×C4×D3—92no—
233C2×Dic6—31no—
234C2×Dic6—91no—
235C3×C4⋊C4—92no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4—92no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—91no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C42—91no—
243#217—31no—
244#217—31no—
245#217—32no—
246#217—32no—
247#217—32no—
248#217—31no—
249#217—61no—
250#217—92no—
251#217—92no—
252#217—91no—
253#217—92no—
254#217—92no—
255#217—92no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—62no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263C4⋊Dic3—92no—
264C4⋊Dic3—92no—
265C4⋊Dic3—92no—
266Dic3⋊C4—32no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274C4×Dic3—31no—
275C4×Dic3—91no—
276C4×Dic3—92no—
277C4×Dic3—92no—
278C4×Dic3—92no—
279C22×C32—22no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—22no—
285C22×C32—22no—
286C22×C32—22no—
287C22×C32—22no—
288C22×C32—21no—
289C22×C32—42no—
290C22×C32—42no—
291C22×C32—41no—
292C22×C32C23×D311yescharacteristic
293C22×C32C23×D311yescharacteristic
294C2×C3×D3—34no—
295C2×C3×D3—34no—
296C2×C3×D3—64no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—122no—
302C2×C3×D3—122no—
303C2×C3×D3—124no—
304C2×C3×D3—124no—
305C2×C3×D3—128no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—41no—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3—22no—
312C3⋊Dic3—22no—
313C3×Dic3—62no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—62no—
318C3×Dic3—62no—
319C3×Dic3—122no—
320#126—272no—
321#124—91no—
322#123—91no—
323#123—272no—
324C22⋊Q8—272no—
325#120—91no—
326C4×D4—272no—
327#117—272no—
328C2×C22⋊C4—272no—
329H(3,𝔽3)#55111yescharacteristic, complemented, Sylow
330C23×C3—31no—
331C23×C3—31no—
332C23×C3—31no—
333C23×C3—31no—
334C23×C3—61no—
335C23×C3—61no—
336C23×C3—92no—
337C23×C3—184no—
338C23×C3—361no—
339C23×C3C2×D3211yescharacteristic
340C23×C3C2×D3211yescharacteristic
341C22×D3—32no—
342C22×D3—32no—
343C22×D3—62no—
344C22×D3—64no—
345C22×D3—64no—
346C22×D3—124no—
347C22×D3—121no—
348C22×D3—121no—
349C22×D3—122no—
350C3×Q8—361no—
351C3×D4—41no—
352C3×D4—121no—
353C3×D4—121no—
354C3×D4—122no—
355C3×D4—184no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—61no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4—92no—
362C2×C3×C4—92no—
363C2×C3×C4—92no—
364C2×C3×C4—361no—
365C2×C3×C4C2×D3211yescharacteristic
366#67—64no—
367#67—64no—
368#67—124no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—62no—
376C2×Dic3—92no—
377C2×Dic3—92no—
378C2×Dic3—92no—
379C2×Dic3—92no—
380C2×Dic3—121no—
381C2×Dic3—184no—
382C2×Dic3—361no—
383C2×Dic3—362no—
384C4×D3—184no—
385C4×D3—184no—
386C4×D3—362no—
387C4×D3—362no—
388Dic6—121no—
389Dic6—361no—
390C2×C32—22no—
391C2×C32—21no—
392C2×C32—42no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32—41no—
396C2×C32—41no—
397C2×C32—81no—
398C2×C32#100411yescharacteristic
399C2×C32#100411yescharacteristic
400C2×C32D3×#4112yes—
401C2×C32D3×#4112yes—
402C3×D3—64no—
403C3×D3—64no—
404C3×D3—124no—
405C24—91no—
406C2×Q8—91no—
407C2×D4—92no—
408C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
409C22×C4—91no—
410C22×C4—272no—
411C22×C4—272no—
412C4⋊C4—92no—
413C4⋊C4—272no—
414C4⋊C4—272no—
415C4⋊C4—272no—
416C22⋊C4—92no—
417C22⋊C4—92no—
418C22⋊C4—92no—
419C22⋊C4—91no—
420C22⋊C4—91no—
421C22⋊C4—92no—
422C22⋊C4—272no—
423C22⋊C4—272no—
424C42—91no—
425C42—272no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—31no—
431C22×C3—31no—
432C22×C3—61no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—61no—
438C22×C3—62no—
439C22×C3—62no—
440C22×C3—62no—
441C22×C3—61no—
442C22×C3—94no—
443C22×C3—122no—
444C22×C3—122no—
445C22×C3—121no—
446C22×C3—182no—
447C22×C3—182no—
448C22×C3—364no—
449C22×C3C22×D3211yescharacteristic
450C2×D3—34no—
451C2×D3—34no—
452C2×D3—64no—
453C2×D3—62no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—122no—
458C2×D3—122no—
459C2×D3—124no—
460C2×D3—124no—
461C2×D3—128no—
462C3×C4—21no—
463C3×C4—61no—
464C3×C4—61no—
465C3×C4—121no—
466C3×C4—182no—
467C3×C4—182no—
468C3×C4—182no—
469C3×C4—182no—
470C3×C4—182no—
471Dic3—62no—
472Dic3—62no—
473Dic3—62no—
474Dic3—62no—
475Dic3—62no—
476Dic3—62no—
477Dic3—122no—
478Dic3—182no—
479Dic3—182no—
480Dic3—182no—
481Dic3—182no—
482C32—21no—
483C32#565911yescharacteristic, complemented
484C32#567911yescharacteristic, complemented
485C23—92no—
486C23—184no—
487C23—361no—
488C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
489C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
490Q8—361no—
491D4—41no—
492D4—184no—
493C2×C4—92no—
494C2×C4—92no—
495C2×C4—92no—
496C2×C4—92no—
497C2×C4—92no—
498C2×C4—361no—
499C2×C4—544no—
500C2×C4—1082no—
501C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
502C2×C3—32no—
503C2×C3—32no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—62no—
509C2×C3—61no—
510C2×C3—121no—
511C2×C3—121no—
512C2×C3—121no—
513C2×C3—121no—
514C2×C3—122no—
515C2×C3—184no—
516C2×C3—241no—
517C2×C3#6431212yes—
518C2×C3#6433011yescharacteristic
519D3—64no—
520D3—64no—
521D3—124no—
522C22—22no—
523C22—22no—
524C22—21no—
525C22—21no—
526C22—94no—
527C22—182no—
528C22—182no—
529C22—364no—
530C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
531C4—21no—
532C4—182no—
533C4—182no—
534C4—182no—
535C4—182no—
536C4—182no—
537C3—31no—
538C3—31no—
539C3—61no—
540C3#1059520811yescharacteristic
541C2—41no—
542C2—41no—
543C2—184no—
544C2#1172250512yes—
545C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
prime radical#11721847C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#551
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #64145⋊C3C2semidirect
3. #64144⋊C3C2semidirect
4. #11721320C2semidirect
5. #11721319C2semidirect
6. #11721297C2semidirect
7. #11721293C2semidirect
8. #11721148C2semidirect
9. #11721076C2semidirect
15. C22×(C3⋊Dic3)⋊C3C22semidirect
16. C2×#634⋊C3C22semidirect
23. #3757⋊C3C22semidirect
24. #3757⋊C3C22semidirect
25. #3750⋊C3C22semidirect
26. C4×(C3⋊Dic3)⋊C3C22semidirect
29. #88449C22semidirect
30. #88448C22semidirect
31. #88448C22semidirect
32. #88447C22semidirect
33. #64145D3semidirect
34. #64144D3semidirect
78. C22×C3⋊Dic3C2×D3semidirect
79. C2×#634C2×D3semidirect
91. #3757C2×D3semidirect
92. #3757C2×D3semidirect
93. #3750C2×D3semidirect
94. C4×C3⋊Dic3C2×D3semidirect
329. H(3,𝔽3)#551semidirect
483. C32#5659semidirect
484. C32#5679semidirect

Maximal subgroups

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

subgroupindex
#117218472
#64145⋊C32
#64144⋊C32
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