Finite Group Databasebeta

← #13140190

Subgroups — #13140190

Order 1728.

696 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#88740C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic, complemented
3C2×#88439C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic, complemented
5#63657⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic, complemented
8#10591173—31nomaximal
9#10591170—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12#88740—22no—
13#88740—22no—
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272—21no—
16C2×#6229C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18#3757⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20#3750⋊C3—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88439—22no—
24#88437—22no—
25C2×C3×#235—31no—
26C2×C3×#235—31no—
27C2×C3×#235—61no—
28#64133D311yescharacteristic, complemented
29C3×#924—31no—
30C2×#3719—31no—
31C2×#3718—31no—
32C2×#3716—31no—
33C2×#3715—31no—
34#63657D311yescharacteristic, complemented
35C3×#830—31no—
36#63566—31no—
37#63565—31no—
38#63563—31no—
39#63562—31no—
40C23×H(3,𝔽3)D411yescharacteristic
41C22×#272—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)D411yescharacteristic
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46#6235—24no—
47C2×#1084—21no—
48C4⋊#272—21no—
49C4⋊#272D412yes—
50C4×#272—24no—
51#6229—21no—
52#6229D412yes—
53C2×(C3⋊Dic3)⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3—21no—
55#634⋊C3—21no—
56#634⋊C3—21no—
57(C3⋊(C3⋊C8))⋊C3—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59#5239—91no—
60#5047—91no—
61#4733—91no—
62C22×C32×C4—21no—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C2×#634C2×D311yescharacteristic
66C2×C3×#67—62no—
67C2×C3×#67—61no—
68C2×C3×#67—61no—
69C3×#235—62no—
70C3×#235—62no—
71C3×#235—62no—
72C3×#235—62no—
73C3×#235—64no—
74C3×#235—64no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×C4×D3—62no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—61no—
85#3757—21no—
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic
87#3750—21no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×#217—61no—
91C3×C4⋊Dic3—31no—
92C3×Dic3⋊C4—61no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—31no—
95#3719—62no—
96#3718—62no—
97#3716—62no—
98#3715—62no—
99C22×H(3,𝔽3)—21no—
100C22×H(3,𝔽3)—21no—
101C22×H(3,𝔽3)—21no—
102C22×H(3,𝔽3)C2×D411yescharacteristic
103C2×#272—22no—
104C2×#272—41no—
105C2×#272—41no—
106C4×H(3,𝔽3)—21no—
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2×D411yescharacteristic
109#1084—22no—
110(C3⋊Dic3)⋊C3—41no—
111(C3⋊Dic3)⋊C3—41no—
112C2×C3×#41—91no—
113C2×#235—31no—
114C2×#235—31no—
115C2×#235—61no—
116C3×C22⋊Q8—91no—
117C2×#216—91no—
118C2×#216—91no—
119C2×#215—91no—
120C2×#214—91no—
121#924—31no—
122#924—91no—
123C2×Dic12—91no—
124C2×#204—91no—
125C3×C22⋊C8—91no—
126#835—91no—
127#832—91no—
128#830—31no—
129#830—91no—
130#821—91no—
131#816—91no—
132#810—91no—
133#809—91no—
134C23×C32—21no—
135C23×C32D1211yescharacteristic
136C23×C32#6711yescharacteristic
137C22×C3×D3—62no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4D1211yescharacteristic
149C2×C32×C4#6711yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—21no—
154#634—21no—
155#634—21no—
156C3×#67—68no—
157C3×#67—64no—
158C3×#67—64no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C3×D12—32no—
165C3×D12—32no—
166C3×D12—62no—
167C3×D12—62no—
168C3×D12—61no—
169C3×D12—61no—
170C3×C4×D3—68no—
171C3×C4×D3—64no—
172C3×C4×D3—64no—
173C3×Dic6—32no—
174C3×Dic6—32no—
175C3×Dic6—62no—
176C3×Dic6—62no—
177C3×Dic6—61no—
178C3×Dic6—61no—
179C3×Dic6—61no—
180C3×Dic6—61no—
181C3⋊(C3⋊C8)—21no—
182C3⋊(C3⋊C8)—21no—
183C3×C3⋊C8—61no—
184C3×C3⋊C8—61no—
185#452—271noSylow
186C2×H(3,𝔽3)—41no—
187C2×H(3,𝔽3)#12011yescharacteristic
188C2×H(3,𝔽3)#13511yescharacteristic
189C2×H(3,𝔽3)#13711yescharacteristic
190#272—42no—
191C3×#41—182no—
192C3×#41—182no—
193C2×C3×Q8—91no—
194C2×C3×Q8—91no—
195C2×C3×D4—91no—
196C2×C3×D4—181no—
197C22×C3×C4—31no—
198C22×C3×C4—31no—
199C22×C3×C4—61no—
200C22×C3×C4—181no—
201C22×C3×C4D3211yescharacteristic
202C2×#67—61no—
203C2×#67—61no—
204C2×#67—62no—
205#235—62no—
206#235—62no—
207#235—62no—
208#235—62no—
209#235—64no—
210#235—64no—
211C2×D12—31no—
212C2×D12—31no—
213C2×D12—61no—
214C2×C4×D3—61no—
215C2×C4×D3—61no—
216C2×C4×D3—62no—
217C2×Dic6—31no—
218C2×Dic6—31no—
219C2×Dic6—31no—
220C2×Dic6—61no—
221C2×Dic6—91no—
222C2×C3×C8—91no—
223C2×C3×C8—91no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—181no—
226C3×C22⋊C4—181no—
227#217—61no—
228#217—181no—
229#216—182no—
230#216—182no—
231#215—182no—
232#214—182no—
233C4⋊Dic3—31no—
234C4⋊Dic3—91no—
235Dic3⋊C4—61no—
236Dic3⋊C4—181no—
237C2×C3⋊C8—31no—
238C2×C3⋊C8—31no—
239C2×C3⋊C8—91no—
240C2×C3⋊C8—91no—
241Dic12—182no—
242#204—182no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—41no—
252C22×C32—41no—
253C22×C32C2×D1211yescharacteristic
254C22×C32C2×#6711yescharacteristic
255C2×C3×D3—64no—
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C32×C4—22no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4D3×D411yescharacteristic
270C32×C4D3×D411yescharacteristic
271C32×C4D3×D411yescharacteristic
272C32×C4D3×D411yescharacteristic
273C3⋊Dic3—41no—
274C3⋊Dic3—41no—
275C3×Dic3—64no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—121no—
279C3×Dic3—121no—
280C2×#41—91no—
281C2×Q16—271no—
282C2×SD16—271no—
283C22⋊Q8—91no—
284#103—271no—
285#102—271no—
286C22⋊C8—91no—
287H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
288C23×C3—31no—
289C23×C3—31no—
290C23×C3—61no—
291C23×C3—181no—
292C23×C3#62611yescharacteristic
293C22×D3—61no—
294C22×D3—61no—
295C22×D3—62no—
296C3×Q8—92no—
297C3×Q8—181no—
298C3×Q8—181no—
299C3×Q8—181no—
300C3×D4—92no—
301C3×D4—184no—
302C3×D4—181no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—184no—
318C2×C3×C4—181no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4#62611yescharacteristic
322C2×C3×C4C2×D3211yescharacteristic
323#67—64no—
324#67—64no—
325#67—68no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328C2×Dic3—62no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—181no—
332C2×Dic3—181no—
333D12—32no—
334D12—32no—
335D12—62no—
336D12—61no—
337D12—61no—
338D12—62no—
339C4×D3—64no—
340C4×D3—64no—
341C4×D3—68no—
342Dic6—32no—
343Dic6—32no—
344Dic6—62no—
345Dic6—61no—
346Dic6—61no—
347Dic6—62no—
348Dic6—61no—
349Dic6—61no—
350Dic6—181no—
351Dic6—181no—
352C3×C8—181no—
353C3×C8—181no—
354C3⋊C8—61no—
355C3⋊C8—61no—
356C3⋊C8—181no—
357C3⋊C8—181no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—42no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32#88211yescharacteristic
365C2×C32#90411yescharacteristic
366C2×C32#90911yescharacteristic
367C2×C32#93711yescharacteristic
368C2×C32#94211yescharacteristic
369C2×C32#95011yescharacteristic
370C3×D3—124no—
371C3×D3—122no—
372C3×D3—122no—
373#41—182no—
374#41—182no—
375C2×Q8—91no—
376C2×Q8—91no—
377C2×D4—91no—
378C2×D4—181no—
379C22×C4—181no—
380C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
381Q16—542no—
382SD16—542no—
383C2×C8—91no—
384C2×C8—91no—
385C4⋊C4—91no—
386C4⋊C4—181no—
387C22⋊C4—181no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—61no—
398C22×C3—61no—
399C22×C3—62no—
400C22×C3—61no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—182no—
404C22×C3—361no—
405C22×C3—361no—
406C22×C3C2×#62611yescharacteristic
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—64no—
410C2×D3—121no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—122no—
415C2×D3—122no—
416C3×C4—21no—
417C3×C4—31no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—61no—
422C3×C4—61no—
423C3×C4—62no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—182no—
427C3×C4—361no—
428C3×C4—361no—
429C3×C4#381111yescharacteristic
430C3×C4#381111yescharacteristic
431Dic3—62no—
432Dic3—62no—
433Dic3—64no—
434Dic3—121no—
435Dic3—121no—
436Dic3—361no—
437Dic3—361no—
438C32—21no—
439C32#473311yescharacteristic, complemented
440C32#504711yescharacteristic, complemented
441C23—181no—
442C23#620011yescharacteristic
443Q8—92no—
444Q8—181no—
445Q8—181no—
446Q8—181no—
447D4—92no—
448D4—184no—
449D4—181no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—184no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4#620011yescharacteristic
457C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
458C8—181no—
459C8—181no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—31no—
466C2×C3—41no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—61no—
470C2×C3—121no—
471C2×C3—121no—
472C2×C3—122no—
473C2×C3—362no—
474C2×C3#6382911yescharacteristic
475C2×C3#6383311yescharacteristic
476C2×C3#6398111yescharacteristic
477D3—122no—
478D3—122no—
479D3—124no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—182no—
484C22—361no—
485C22—361no—
486C22C2×#620011yescharacteristic
487C4—21no—
488C4—182no—
489C4—361no—
490C4—361no—
491C4#8867711yescharacteristic
492C4#8867711yescharacteristic
493C3—31no—
494C3—31no—
495C3—61no—
496C3#1059118411yescharacteristic
497C2—41no—
498C2—362no—
499C2#1172106911yescharacteristic
500C2#1172107311yescharacteristic
501C2#1172132311yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×#88740C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#452
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #64133⋊C3C2semidirect
3. C2×#88439C2semidirect
4. C2×#88437C2semidirect
5. #63657⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720324C2semidirect
28. #64133D3semidirect
34. #63657D3semidirect
287. H(3,𝔽3)#452semidirect
439. C32#4733semidirect
440. C32#5047semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
#64133⋊C32
C2×#884392
C2×#884372
#63657⋊C32
#117203262
#117203242
#105911733
#105911703