Finite Group Databasebeta

← #13147887

Subgroups — #13147887

Order 1728.

846 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 129 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#11721721C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic, complemented
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721157C212yesmaximal, complemented
5#11721148C212yesmaximal, complemented
6#11721135C212yesmaximal, complemented
7#11721130C212yesmaximal, complemented
8#11721127C212yesmaximal, complemented
9#11721102C212yesmaximal, complemented
10#10593985—31nomaximal
11#10593953—31nomaximal
12C2×C4⋊#272C2211yescharacteristic
13C2×#6229C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×#634⋊C3C2211yescharacteristic
16C2×#6200C2212yes—
17C2×#6200C2212yes—
18C2×#6199C2212yes—
19C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
20C42×H(3,𝔽3)C2211yescharacteristic
21#88542C2212yes—
22#88542C2212yes—
23#3752⋊C3C2212yes—
24#3752⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29#88451C2212yes—
30#88450C2212yes—
31#88449C2212yes—
32#88448C2212yes—
33#64084D311yescharacteristic, complemented
34#64079D311yescharacteristic, complemented
35C3×#875—31no—
36C3×#875—31no—
37C3×#875—31no—
38C3×#875—61no—
39C3×#870—31no—
40#63910—32no—
41#63896—32no—
42#63891—32no—
43#63891—32no—
44#63890—32no—
45#63888—32no—
46#63887—32no—
47#63887—32no—
48#63886—32no—
49#63885—32no—
50#63885—32no—
51#63854—32no—
52C22×#272C2312yes—
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×#1084C2312yes—
58C4⋊#272—41no—
59#6229—41no—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C4⋊((C3⋊D3)⋊C3)—41no—
64#634⋊C3—41no—
65#6200—42no—
66#6200—42no—
67#6199—42no—
68((C3⋊D3)⋊C3)⋊C4—42no—
69#5700—91no—
70#5684—91no—
71#5684—91no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×#634C2×D311yescharacteristic
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C2×#626—32no—
83C2×#626—32no—
84C2×#625—32no—
85C2×#625—32no—
86C2×#624—32no—
87C2×D3×Dic3—32no—
88C2×D3×Dic3—32no—
89C2×D3×Dic3—32no—
90C32×C42—21no—
91C32×C42C2×D311yescharacteristic
92C32×C42C2×D311yescharacteristic
93#3752C2×D312yes—
94#3752C2×D312yes—
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96C4⋊(C3⋊Dic3)C2×D311yescharacteristic
97#3750C2×D312yes—
98#3750C2×D312yes—
99C3×(C2×D3)⋊C4—32no—
100C3×(C2×D3)⋊C4—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—62no—
106C3×(C2×D3)⋊C4—62no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111#3724—32no—
112Dic3⋊Dic3—32no—
113Dic3⋊Dic3—32no—
114Dic3⋊Dic3—32no—
115#3722—32no—
116(C2×D3)⋊Dic3—32no—
117(C2×D3)⋊Dic3—32no—
118(C2×D3)⋊Dic3—32no—
119C22×H(3,𝔽3)C2411yescharacteristic
120C2×#272—22no—
121C2×#272—22no—
122C2×#272—22no—
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3—22no—
125C2×(C3⋊D3)⋊C3—22no—
126C4×H(3,𝔽3)—21no—
127C4×H(3,𝔽3)—21no—
128C4×H(3,𝔽3)—21no—
129#1084—22no—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132(C3⋊Dic3)⋊C3—22no—
133C3×#125—91no—
134C3×#124—91no—
135C3×#124—91no—
136#946—92no—
137#946—92no—
138#939—92no—
139#939—92no—
140#896—92no—
141#896—92no—
142#895—92no—
143#895—92no—
144#894—92no—
145#894—92no—
146#886—92no—
147#886—92no—
148#884—92no—
149#884—92no—
150#883—92no—
151#883—92no—
152#878—92no—
153#878—92no—
154#875—31no—
155#875—31no—
156#875—31no—
157#875—61no—
158#875—91no—
159#870—31no—
160#870—91no—
161C22×C3⋊D3C22×D312yes—
162C22×C3×D3—32no—
163C22×C3×D3—32no—
164C22×C3×D3—32no—
165C22×C3×D3—62no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3C22×D312yes—
176C2×C3⋊Dic3C22×D312yes—
177C2×C3⋊Dic3C22×D312yes—
178C4⋊(C3⋊D3)—41no—
179#634—41no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—62no—
186C3×D12—121no—
187C3×D12—121no—
188C3×D12—121no—
189C3×D12—122no—
190C3×Dic6—121no—
191C3×Dic6—121no—
192C3×Dic6—121no—
193C3×Dic6—122no—
194#626—122no—
195#626—122no—
196#625—122no—
197#625—122no—
198#624—122no—
199D3×Dic3—122no—
200D3×Dic3—122no—
201D3×Dic3—122no—
202#562—271noSylow
203C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
204C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
205C2×H(3,𝔽3)2−1+411yescharacteristic
206#272—42no—
207(C3⋊D3)⋊C3—42no—
208C2×C3×Q8—91no—
209C2×C3×Q8—91no—
210C2×C3×D4—91no—
211C2×C3×D4—91no—
212C2×#67—92no—
213C2×#67—92no—
214C2×#67—92no—
215C2×#67—92no—
216C22×Dic3—92no—
217C22×Dic3—92no—
218C2×D12—31no—
219C2×D12—31no—
220C2×D12—31no—
221C2×D12—61no—
222C2×D12—91no—
223C2×D12—92no—
224C2×D12—92no—
225C2×C4×D3—92no—
226C2×C4×D3—92no—
227C2×C4×D3—92no—
228C2×C4×D3—92no—
229C2×Dic6—31no—
230C2×Dic6—31no—
231C2×Dic6—31no—
232C2×Dic6—61no—
233C2×Dic6—91no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—92no—
238C3×C22⋊C4—92no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—92no—
241C3×C22⋊C4—92no—
242C3×C42—31no—
243C3×C42—31no—
244C3×C42—61no—
245C3×C42D3211yescharacteristic
246#217—92no—
247#217—92no—
248(C2×D3)⋊C4—32no—
249(C2×D3)⋊C4—32no—
250(C2×D3)⋊C4—32no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—62no—
255(C2×D3)⋊C4—62no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262C4⋊Dic3—31no—
263C4⋊Dic3—31no—
264C4⋊Dic3—91no—
265C4⋊Dic3—91no—
266C4⋊Dic3—92no—
267C4⋊Dic3—92no—
268Dic3⋊C4—32no—
269Dic3⋊C4—32no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276C22×C32—21no—
277C22×C32C23×D311yescharacteristic
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—22no—
280C2×C3⋊D3—22no—
281C2×C3⋊D3—22no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—122no—
292C2×C3×D3—122no—
293C2×C3×D3—122no—
294C32×C4—22no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—41no—
302C32×C4—41no—
303C3⋊Dic3—22no—
304C3⋊Dic3—22no—
305C3⋊Dic3—22no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C3×Dic3—122no—
312#125—91no—
313#124—91no—
314#124—91no—
315#123—272no—
316#123—272no—
317C22⋊Q8—272no—
318C22⋊Q8—272no—
319#121—272no—
320#121—272no—
321H(3,𝔽3)#56211yescharacteristic, complemented, Sylow
322C23×C3—92no—
323C23×C3—92no—
324C22×D3—32no—
325C22×D3—32no—
326C22×D3—32no—
327C22×D3—62no—
328C22×D3—92no—
329C3×Q8—361no—
330C3×Q8—361no—
331C3×D4—361no—
332C3×D4—361no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—92no—
343C2×C3×C4—92no—
344C2×C3×C4—92no—
345C2×C3×C4—92no—
346C2×C3×C4C2×D3211yescharacteristic
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349#67—362no—
350#67—362no—
351#67—362no—
352#67—362no—
353C2×Dic3—32no—
354C2×Dic3—32no—
355C2×Dic3—32no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—62no—
359C2×Dic3—92no—
360C2×Dic3—92no—
361C2×Dic3—92no—
362C2×Dic3—362no—
363C2×Dic3—362no—
364D12—121no—
365D12—121no—
366D12—121no—
367D12—122no—
368D12—361no—
369D12—362no—
370D12—362no—
371C4×D3—362no—
372C4×D3—362no—
373C4×D3—362no—
374C4×D3—362no—
375Dic6—121no—
376Dic6—121no—
377Dic6—121no—
378Dic6—122no—
379Dic6—361no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32#100411yescharacteristic
384C2×C32#100711yescharacteristic
385C2×C32#100911yescharacteristic
386C2×C32#100911yescharacteristic
387C2×C32#100911yescharacteristic
388C2×C32#101011yescharacteristic
389C3⋊D3—42no—
390C3×D3—122no—
391C3×D3—122no—
392C3×D3—122no—
393C3×D3—242no—
394C2×Q8—91no—
395C2×Q8—91no—
396C2×D4—91no—
397C2×D4—91no—
398C2×D4—272no—
399C2×D4—272no—
400C22×C4—272no—
401C22×C4—272no—
402C4⋊C4—92no—
403C4⋊C4—92no—
404C4⋊C4—91no—
405C4⋊C4—91no—
406C4⋊C4—272no—
407C4⋊C4—272no—
408C22⋊C4—92no—
409C22⋊C4—92no—
410C22⋊C4—92no—
411C22⋊C4—92no—
412C22⋊C4—272no—
413C22⋊C4—272no—
414C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C22×C3—31no—
416C22×C3—31no—
417C22×C3—61no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3—182no—
421C22×C3—182no—
422C22×C3—182no—
423C22×C3—182no—
424C22×C3C22×D3211yescharacteristic
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—122no—
435C2×D3—122no—
436C2×D3—122no—
437C2×D3—182no—
438C2×D3—182no—
439C2×D3—182no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—61no—
444C3×C4—61no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—62no—
450C3×C4—121no—
451C3×C4—121no—
452C3×C4—182no—
453C3×C4—182no—
454C3×C4—182no—
455C3×C4—182no—
456Dic3—62no—
457Dic3—62no—
458Dic3—62no—
459Dic3—62no—
460Dic3—62no—
461Dic3—122no—
462Dic3—182no—
463Dic3—182no—
464Dic3—182no—
465C32—21no—
466C32#568411yescharacteristic, complemented
467C32#570011yescharacteristic, complemented
468C23—92no—
469C23—92no—
470Q8—361no—
471Q8—361no—
472D4—361no—
473D4—361no—
474D4—1082no—
475D4—1082no—
476C2×C4—92no—
477C2×C4—92no—
478C2×C4—92no—
479C2×C4—92no—
480C2×C4—1082no—
481C2×C4—1082no—
482C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
483C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
484C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—61no—
492C2×C3—61no—
493C2×C3—61no—
494C2×C3—362no—
495C2×C3—362no—
496C2×C3#6429711yescharacteristic
497C2×C3#6430711yescharacteristic
498C2×C3#6430811yescharacteristic
499D3—122no—
500D3—122no—
501D3—122no—
502D3—242no—
503D3—362no—
504C22—182no—
505C22—182no—
506C22—182no—
507C22—182no—
508C22—182no—
509C22—182no—
510C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—182no—
515C4—182no—
516C4—182no—
517C4—182no—
518C3—31no—
519C3—31no—
520C3—61no—
521C3#1059395311yescharacteristic
522C2—362no—
523C2—362no—
524C2#1172248211yescharacteristic
525C2#1172249211yescharacteristic
526C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#562
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
1. #11721721C2semidirect
2. #64084⋊C3C2semidirect
3. #64079⋊C3C2semidirect
4. #11721157C2semidirect
5. #11721148C2semidirect
6. #11721135C2semidirect
7. #11721130C2semidirect
8. #11721127C2semidirect
9. #11721102C2semidirect
33. #64084D3semidirect
34. #64079D3semidirect
321. H(3,𝔽3)#562semidirect
466. C32#5684semidirect
467. C32#5700semidirect

Maximal subgroups

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

subgroupindex
#117217212
#64084⋊C32
#64079⋊C32
#117211572
#117211482
#117211352
#117211302
#117211272
#117211022
#105939853
#105939533