Finite Group Databasebeta

← #13140182

Subgroups — #13140182

Order 1728.

661 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 83 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×#3828⋊C3C211yesmaximal, characteristic, complemented
2#11721830C211yesmaximal, characteristic, complemented
3C2×#88437C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63657⋊C3C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591182—31nomaximal
9#10591161—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×C4⋊#272C2211yescharacteristic
13C2×#638⋊C3—21no—
14#3828⋊C3—22no—
15#3828⋊C3—22no—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3—21no—
18C2×#634⋊C3C2211yescharacteristic
19#88542—21no—
20C4⋊#1084C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88437—22no—
24#88436—22no—
25C2×#3828D311yescharacteristic, complemented
26C2×C3×#235—31no—
27C3×#930—31no—
28C3×#930—31no—
29C3×#930—61no—
30C2×#3716—31no—
31C2×#3715—31no—
32C2×#3714—31no—
33C2×#3713—31no—
34#63657D311yescharacteristic, complemented
35C3×#830—31no—
36#63567—31no—
37#63565—31no—
38#63564—31no—
39#63562—31no—
40C23×H(3,𝔽3)D411yescharacteristic
41C22×#272—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)D411yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47#6235—42no—
48C2×#1084—21no—
49C4⋊#272—21no—
50C4⋊#272—21no—
51#638⋊C3—24no—
52C2×(C3⋊Dic3)⋊C3—21no—
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)D412yescomplemented
55C4×(C3⋊D3)⋊C3—24no—
56#634⋊C3—21no—
57#634⋊C3D412yescomplemented
58(C3⋊(C3⋊C8))⋊C3—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60#5238—91no—
61#5037—91no—
62#4734—91no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C2×#638—21no—
67#3828—22no—
68#3828—22no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4×C3⋊D3—21no—
71C2×#634C2×D311yescharacteristic
72C2×C3×#67—61no—
73C2×C3×#67—62no—
74C2×C3×#67—61no—
75C2×C3×#67—61no—
76C3×#235—62no—
77C3×#235—62no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×C4×D3—61no—
83C2×C3×Dic6—31no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×(C2×D3)⋊C4—62no—
87C3×(C2×D3)⋊C4—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94#3716—62no—
95#3715—62no—
96#3714—62no—
97#3713—62no—
98C22×H(3,𝔽3)—21no—
99C22×H(3,𝔽3)—21no—
100C22×H(3,𝔽3)—21no—
101C22×H(3,𝔽3)C2×D411yescharacteristic
102C2×#272—41no—
103C2×#272—41no—
104C2×#272—41no—
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3—41no—
108C4×H(3,𝔽3)—21no—
109C4×H(3,𝔽3)C2×D411yescharacteristic
110C4×H(3,𝔽3)C2×D411yescharacteristic
111#1084—41no—
112(C3⋊Dic3)⋊C3—22no—
113C2×C3×#41—91no—
114C2×#235—31no—
115C2×#235—91no—
116C3×#121—91no—
117C2×#215—91no—
118C2×#214—91no—
119C2×#213—91no—
120C2×#213—91no—
121#930—31no—
122#930—31no—
123#930—61no—
124C2×D24—91no—
125C2×#204—91no—
126C3×C22⋊C8—91no—
127#835—91no—
128#832—91no—
129#830—31no—
130#830—91no—
131#821—91no—
132#816—91no—
133#810—91no—
134#809—91no—
135C23×C32—21no—
136C23×C32D1211yescharacteristic
137C23×C32#6711yescharacteristic
138C22×C3⋊D3—21no—
139C22×C3×D3—61no—
140C22×C3×D3—62no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4D1211yescharacteristic
152C2×C32×C4#6711yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155#638—24no—
156C2×C3⋊Dic3—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)#6712yescomplemented
159C4×C3⋊D3—24no—
160#634—21no—
161#634#6712yescomplemented
162C3×#67—64no—
163C3×#67—124no—
164C3×#67—122no—
165C3×#67—122no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C3×D12—32no—
171C3×D12—61no—
172C3×D12—61no—
173C3×D12—61no—
174C3×D12—61no—
175C3×D12—61no—
176C3×D12—121no—
177C3×D12—121no—
178C3×C4×D3—64no—
179C3×Dic6—32no—
180C3×Dic6—61no—
181C3⋊(C3⋊C8)—21no—
182C3⋊(C3⋊C8)—21no—
183C3×C3⋊C8—61no—
184C3×C3⋊C8—61no—
185#449—271noSylow
186C2×H(3,𝔽3)—41no—
187C2×H(3,𝔽3)#12011yescharacteristic
188C2×H(3,𝔽3)#13511yescharacteristic
189C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
190#272—81no—
191(C3⋊D3)⋊C3—42no—
192C3×#41—182no—
193C3×#41—182no—
194C2×C3×Q8—91no—
195C2×C3×D4—91no—
196C2×C3×D4—91no—
197C2×C3×D4—181no—
198C2×C3×D4—181no—
199C22×C3×C4—31no—
200C22×C3×C4—31no—
201C22×C3×C4—61no—
202C22×C3×C4—181no—
203C22×C3×C4D3211yescharacteristic
204C2×#67—61no—
205C2×#67—61no—
206C2×#67—61no—
207C2×#67—62no—
208C2×#67—181no—
209#235—62no—
210#235—62no—
211#235—182no—
212#235—182no—
213C2×D12—31no—
214C2×D12—31no—
215C2×D12—31no—
216C2×D12—61no—
217C2×D12—91no—
218C2×C4×D3—61no—
219C2×C4×D3—181no—
220C2×Dic6—31no—
221C2×Dic6—91no—
222C2×C3×C8—91no—
223C2×C3×C8—91no—
224C3×C4⋊C4—91no—
225C3×C22⋊C4—181no—
226#215—182no—
227#214—182no—
228#213—182no—
229#213—182no—
230(C2×D3)⋊C4—61no—
231(C2×D3)⋊C4—61no—
232(C2×D3)⋊C4—62no—
233C4⋊Dic3—31no—
234C4⋊Dic3—31no—
235C4⋊Dic3—61no—
236C2×C3⋊C8—31no—
237C2×C3⋊C8—31no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—91no—
240D24—182no—
241#204—182no—
242C22×C32—22no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—41no—
251C22×C32—41no—
252C22×C32C2×D1211yescharacteristic
253C22×C32C2×#6711yescharacteristic
254C2×C3⋊D3—22no—
255C2×C3⋊D3—41no—
256C2×C3⋊D3—41no—
257C2×C3×D3—62no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—122no—
261C2×C3×D3—122no—
262C2×C3×D3—122no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—41no—
274C32×C4D3×D411yescharacteristic
275C32×C4D3×D411yescharacteristic
276C32×C4C2×#6711yescharacteristic
277C32×C4C2×#6711yescharacteristic
278C3⋊Dic3—22no—
279C3×Dic3—62no—
280C3×Dic3—122no—
281C3×Dic3—121no—
282C3×Dic3—121no—
283C2×#41—91no—
284C2×SD16—271no—
285C2×D8—271no—
286#121—91no—
287#103—271no—
288#102—271no—
289C22⋊C8—91no—
290H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
291C23×C3—31no—
292C23×C3—31no—
293C23×C3—61no—
294C23×C3—181no—
295C23×C3—181no—
296C23×C3#62611yescharacteristic
297C22×D3—61no—
298C22×D3—61no—
299C22×D3—61no—
300C22×D3—62no—
301C22×D3—181no—
302C3×Q8—92no—
303C3×Q8—181no—
304C3×D4—92no—
305C3×D4—181no—
306C3×D4—184no—
307C3×D4—181no—
308C3×D4—181no—
309C3×D4—362no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—121no—
323C2×C3×C4—121no—
324C2×C3×C4—184no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4#62611yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329#67—64no—
330#67—122no—
331#67—122no—
332#67—124no—
333#67—184no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—62no—
338C2×Dic3—181no—
339D12—32no—
340D12—61no—
341D12—61no—
342D12—61no—
343D12—61no—
344D12—61no—
345D12—92no—
346D12—121no—
347D12—121no—
348D12—181no—
349C4×D3—64no—
350C4×D3—184no—
351Dic6—32no—
352Dic6—61no—
353Dic6—92no—
354Dic6—181no—
355C3×C8—181no—
356C3×C8—181no—
357C3⋊C8—61no—
358C3⋊C8—61no—
359C3⋊C8—181no—
360C3⋊C8—181no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—42no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32#88211yescharacteristic
368C2×C32#90411yescharacteristic
369C2×C32#90811yescharacteristic
370C2×C32#94911yescharacteristic
371C2×C32#95011yescharacteristic
372C2×C32#95311yescharacteristic
373C3⋊D3—42no—
374C3×D3—122no—
375C3×D3—242no—
376C3×D3—241no—
377C3×D3—241no—
378#41—182no—
379#41—182no—
380C2×Q8—91no—
381C2×D4—91no—
382C2×D4—91no—
383C2×D4—181no—
384C2×D4—181no—
385C22×C4—181no—
386C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
387SD16—542no—
388D8—542no—
389C2×C8—91no—
390C2×C8—91no—
391C4⋊C4—91no—
392C22⋊C4—181no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—31no—
397C22×C3—31no—
398C22×C3—61no—
399C22×C3—61no—
400C22×C3—61no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—61no—
404C22×C3—62no—
405C22×C3—61no—
406C22×C3—121no—
407C22×C3—121no—
408C22×C3—182no—
409C22×C3—361no—
410C22×C3—361no—
411C22×C3—361no—
412C22×C3—361no—
413C22×C3—361no—
414C22×C3C2×#62611yescharacteristic
415C2×D3—62no—
416C2×D3—121no—
417C2×D3—121no—
418C2×D3—121no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—122no—
425C2×D3—122no—
426C2×D3—122no—
427C2×D3—182no—
428C2×D3—361no—
429C2×D3—361no—
430C3×C4—21no—
431C3×C4—31no—
432C3×C4—31no—
433C3×C4—31no—
434C3×C4—31no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—61no—
438C3×C4—61no—
439C3×C4—121no—
440C3×C4—182no—
441C3×C4—361no—
442C3×C4D3×#6711yescharacteristic
443C3×C4D3×#6711yescharacteristic
444Dic3—62no—
445Dic3—121no—
446Dic3—121no—
447Dic3—122no—
448Dic3—182no—
449C32—21no—
450C32#473411yescharacteristic, complemented
451C32#523811yescharacteristic, complemented
452C23—181no—
453C23—181no—
454C23#620011yescharacteristic
455Q8—92no—
456Q8—181no—
457D4—92no—
458D4—184no—
459D4—181no—
460D4—181no—
461D4—181no—
462D4—362no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—184no—
466C2×C4—181no—
467C2×C4—181no—
468C2×C4#620011yescharacteristic
469C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
470C8—181no—
471C8—181no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—31no—
477C2×C3—31no—
478C2×C3—41no—
479C2×C3—61no—
480C2×C3—61no—
481C2×C3—61no—
482C2×C3—121no—
483C2×C3—121no—
484C2×C3—122no—
485C2×C3—362no—
486C2×C3—721no—
487C2×C3#6382511yescharacteristic
488C2×C3#6382911yescharacteristic
489C2×C3#6397711yescharacteristic
490D3—122no—
491D3—241no—
492D3—241no—
493D3—242no—
494D3—362no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—182no—
499C22—361no—
500C22—361no—
501C22—361no—
502C22—361no—
503C22—361no—
504C22C2×#620011yescharacteristic
505C4—21no—
506C4—182no—
507C4—361no—
508C4#8867611yescharacteristic
509C4#8867611yescharacteristic
510C3—31no—
511C3—31no—
512C3—61no—
513C3#1059116311yescharacteristic
514C2—41no—
515C2—362no—
516C2—721no—
517C2#1172106811yescharacteristic
518C2#1172107311yescharacteristic
519C2#1172132111yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#449
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
1. C2×#3828⋊C3C2semidirect
2. #11721830C2semidirect
3. C2×#88437C2semidirect
4. C2×#88436C2semidirect
5. #63657⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720308C2semidirect
25. C2×#3828D3semidirect
34. #63657D3semidirect
54. C4⋊((C3⋊D3)⋊C3)D4semidirect
57. #634⋊C3D4semidirect
158. C4⋊(C3⋊D3)#67semidirect
161. #634#67semidirect
290. H(3,𝔽3)#449semidirect
450. C32#4734semidirect
451. C32#5238semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
#117218302
C2×#884372
C2×#884362
#63657⋊C32
#117203092
#117203082
#105911823
#105911613