Finite Group Databasebeta

← #13140589

Subgroups — #13140589

Order 1728.

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

normal subgroups only

This group has 731 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×#88744C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3#11721159C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720323C211yesmaximal, characteristic, complemented
8#10591452—31nomaximal
9#10591450—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C22×#1084—21no—
13#88744—22no—
14#88744—22no—
15C2×C4×#272C2211yescharacteristic
16C2×#6229C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×#6200—21no—
19#3753⋊C3—22no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88450—21no—
24#88438—22no—
25C2×C3×#237—31no—
26C2×C3×#237—31no—
27C2×C3×#237—61no—
28C2×#3753D311yescharacteristic, complemented
29C2×C3×#213—31no—
30#63909—31no—
31#63908—31no—
32C2×#3717—31no—
33C2×#3715—31no—
34#63659D311yescharacteristic, complemented
35C3×#832—31no—
36#63566—31no—
37#63565—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C23×H(3,𝔽3)—21no—
41C22×#272D411yescharacteristic
42C22×(C3⋊D3)⋊C3—21no—
43D4×H(3,𝔽3)—21no—
44D4×H(3,𝔽3)D412yescomplemented
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46#6235—24no—
47C2×#1084—24no—
48C2×#1084—21no—
49C2×#1084D411yescharacteristic
50C4×#272—21no—
51C4×#272—21no—
52#6229—21no—
53#6229D412yescomplemented
54C2×(C3⋊Dic3)⋊C3—21no—
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)—21no—
57#6200—42no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59(C3⋊(C3⋊C8))⋊C3—21no—
60#5238—91no—
61#5158—91no—
62#4781—91no—
63C2×C32×D4—21no—
64C2×C32×D4C2×D311yescharacteristic
65C2×C32×D4C2×D311yescharacteristic
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C3×#67—61no—
68C2×C3×#67—61no—
69C2×C3×#67—62no—
70C22×C3×Dic3—61no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3—62no—
73C3×#237—62no—
74C3×#237—62no—
75C3×#237—64no—
76C3×#237—64no—
77C3×#237—62no—
78C3×#237—62no—
79C2×C3×D12—31no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—61no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C2×#626—61no—
87C2×#625—61no—
88#3753—22no—
89C4⋊(C3⋊Dic3)C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#213—62no—
93C3×C4⋊Dic3—31no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96#3722—61no—
97(C2×D3)⋊Dic3—61no—
98#3717—62no—
99#3715—62no—
100C22×H(3,𝔽3)—22no—
101C22×H(3,𝔽3)—41no—
102C22×H(3,𝔽3)—41no—
103C22×H(3,𝔽3)C2×D411yescharacteristic
104C2×#272—21no—
105C2×#272—21no—
106C2×#272—21no—
107C2×(C3⋊D3)⋊C3—41no—
108C2×(C3⋊D3)⋊C3—41no—
109C2×(C3⋊D3)⋊C3—41no—
110C4×H(3,𝔽3)C2×D411yescharacteristic
111C4×H(3,𝔽3)C2×D411yescharacteristic
112#1084—22no—
113#1084—21no—
114(C3⋊Dic3)⋊C3—41no—
115C2×C3×#41—91no—
116C2×#237—31no—
117C2×#237—31no—
118C2×#237—61no—
119C2×C3×D8—91no—
120C2×#215—91no—
121#938—91no—
122C2×#214—91no—
123C2×#213—31no—
124C2×#213—91no—
125#930—91no—
126C2×#204—91no—
127#895—91no—
128C3×#102—91no—
129#835—91no—
130#832—31no—
131#832—91no—
132#830—91no—
133(C2×D3)⋊C8—91no—
134(C2×D3)⋊C8—91no—
135#816—91no—
136#810—91no—
137C23×C32—22no—
138C23×C32—21no—
139C23×C32—21no—
140C22×C3⋊D3—21no—
141C22×C3×D3—31no—
142C22×C3×D3—31no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C32×D4—22no—
146C32×D4—21no—
147C32×D4—21no—
148C32×D4—22no—
149C32×D4D1212yescomplemented
150C32×D4#6712yescomplemented
151C2×C32×C4—21no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)—21no—
157C3×#67—68no—
158C3×#67—64no—
159C3×#67—64no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—68no—
165C2×C3×Dic3—64no—
166C2×C3×Dic3—64no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C3×D12—61no—
171C3×D12—61no—
172C3×C4×D3—62no—
173C3×C4×D3—62no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—61no—
178C3×Dic6—32no—
179C3×Dic6—32no—
180C3×Dic6—62no—
181C3×Dic6—62no—
182C3×Dic6—61no—
183C3×Dic6—61no—
184#626—122no—
185#625—122no—
186C3⋊(C3⋊C8)—21no—
187C3⋊(C3⋊C8)—21no—
188C3×C3⋊C8—61no—
189C3×C3⋊C8—61no—
190#449—271noSylow
191C2×H(3,𝔽3)—42no—
192C2×H(3,𝔽3)#12011yescharacteristic
193C2×H(3,𝔽3)#13511yescharacteristic
194C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
195#272—41no—
196(C3⋊D3)⋊C3—81no—
197C3×#41—182no—
198C3×#41—182no—
199C2×C3×Q8—91no—
200C2×C3×D4—31no—
201C2×C3×D4—31no—
202C2×C3×D4—61no—
203C2×C3×D4—91no—
204C2×C3×D4—181no—
205C2×C3×D4D3211yescharacteristic
206C22×C3×C4—91no—
207C22×C3×C4—181no—
208C2×#67—61no—
209C2×#67—61no—
210C2×#67—62no—
211C2×#67—181no—
212C2×#67—181no—
213C22×Dic3—61no—
214C22×Dic3—61no—
215C22×Dic3—62no—
216#237—62no—
217#237—62no—
218#237—62no—
219#237—62no—
220#237—64no—
221#237—64no—
222C2×D12—31no—
223C2×D12—91no—
224C2×D12—181no—
225C2×C4×D3—31no—
226C2×C4×D3—31no—
227C2×C4×D3—61no—
228C2×Dic6—31no—
229C2×Dic6—31no—
230C2×Dic6—61no—
231C3×D8—182no—
232C2×C3×C8—91no—
233C2×C3×C8—91no—
234C3×C4⋊C4—91no—
235#217—181no—
236#215—182no—
237#214—182no—
238#213—62no—
239#213—182no—
240(C2×D3)⋊C4—181no—
241(C2×D3)⋊C4—181no—
242C4⋊Dic3—31no—
243C4⋊Dic3—91no—
244C2×C3⋊C8—31no—
245C2×C3⋊C8—31no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—91no—
248#204—182no—
249C22×C32—24no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32—41no—
258C22×C32—41no—
259C22×C32D3×D411yescharacteristic
260C22×C32D3×D411yescharacteristic
261C2×C3⋊D3—41no—
262C2×C3⋊D3—41no—
263C2×C3⋊D3—41no—
264C2×C3×D3—62no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4C2×D1211yescharacteristic
279C32×C4D3×D411yescharacteristic
280C32×C4D3×D411yescharacteristic
281C32×C4C2×#6711yescharacteristic
282C3⋊Dic3—41no—
283C3×Dic3—64no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—61no—
287C3×Dic3—62no—
288C3×Dic3—61no—
289C3×Dic3—121no—
290C2×#41—91no—
291C2×SD16—271no—
292C2×D8—91no—
293#121—271no—
294#103—271no—
295#102—91no—
296C22⋊C8—271no—
297H(3,𝔽3)#44911yescharacteristic, complemented, Sylow
298C23×C3—21no—
299C23×C3—61no—
300C23×C3—61no—
301C23×C3—62no—
302C23×C3—91no—
303C23×C3—181no—
304C22×D3—31no—
305C22×D3—31no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—181no—
309C3×Q8—92no—
310C3×Q8—181no—
311C3×D4—21no—
312C3×D4—32no—
313C3×D4—32no—
314C3×D4—61no—
315C3×D4—61no—
316C3×D4—62no—
317C3×D4—62no—
318C3×D4—181no—
319C3×D4—181no—
320C3×D4—184no—
321C3×D4#62612yes—
322C2×C3×C4—31no—
323C2×C3×C4—31no—
324C2×C3×C4—61no—
325C2×C3×C4—91no—
326C2×C3×C4—184no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4C2×D3211yescharacteristic
332#67—64no—
333#67—64no—
334#67—68no—
335#67—362no—
336#67—362no—
337C2×Dic3—31no—
338C2×Dic3—31no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—62no—
343C2×Dic3—64no—
344C2×Dic3—64no—
345C2×Dic3—68no—
346C2×Dic3—61no—
347C2×Dic3—181no—
348D12—61no—
349D12—61no—
350D12—181no—
351D12—181no—
352D12—362no—
353C4×D3—61no—
354C4×D3—61no—
355C4×D3—61no—
356C4×D3—61no—
357C4×D3—62no—
358C4×D3—62no—
359Dic6—32no—
360Dic6—32no—
361Dic6—62no—
362Dic6—61no—
363Dic6—61no—
364Dic6—62no—
365C3×C8—181no—
366C3×C8—181no—
367C3⋊C8—61no—
368C3⋊C8—61no—
369C3⋊C8—181no—
370C3⋊C8—181no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—44no—
375C2×C32—42no—
376C2×C32—42no—
377C2×C32#88211yescharacteristic
378C2×C32#91111yescharacteristic
379C2×C32#91111yescharacteristic
380C2×C32#91211yescharacteristic
381C2×C32#91611yescharacteristic
382C2×C32#93711yescharacteristic
383C3⋊D3—81no—
384C3×D3—122no—
385C3×D3—121no—
386C3×D3—121no—
387C3×D3—241no—
388#41—182no—
389#41—182no—
390C2×Q8—91no—
391C2×D4—91no—
392C2×D4—181no—
393C2×D4—541no—
394C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
395C22×C4—91no—
396C22×C4—181no—
397SD16—542no—
398D8—182no—
399C2×C8—91no—
400C2×C8—91no—
401C4⋊C4—91no—
402C22⋊C4—541no—
403C22×C3—22no—
404C22×C3—31no—
405C22×C3—31no—
406C22×C3—41no—
407C22×C3—41no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—64no—
411C22×C3—61no—
412C22×C3—121no—
413C22×C3—121no—
414C22×C3—121no—
415C22×C3—121no—
416C22×C3—122no—
417C22×C3—122no—
418C22×C3—181no—
419C22×C3—181no—
420C22×C3—181no—
421C22×C3—361no—
422C22×C3—361no—
423C22×C3—361no—
424C22×C3#381111yescharacteristic
425C2×D3—61no—
426C2×D3—61no—
427C2×D3—61no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—62no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—121no—
436C2×D3—121no—
437C2×D3—361no—
438C2×D3—361no—
439C2×D3—361no—
440C3×C4—31no—
441C3×C4—31no—
442C3×C4—31no—
443C3×C4—31no—
444C3×C4—61no—
445C3×C4—61no—
446C3×C4—181no—
447C3×C4—182no—
448C3×C4—361no—
449C3×C4C2×#62611yescharacteristic
450C3×C4#381111yescharacteristic
451Dic3—62no—
452Dic3—62no—
453Dic3—61no—
454Dic3—61no—
455Dic3—64no—
456Dic3—62no—
457Dic3—121no—
458Dic3—361no—
459C32—21no—
460C32#478111yescharacteristic, complemented
461C32#515811yescharacteristic, complemented
462C23—21no—
463C23—91no—
464C23—181no—
465Q8—92no—
466Q8—181no—
467D4—21no—
468D4—184no—
469D4—181no—
470D4—181no—
471D4—1082no—
472D4#620012yescomplemented
473C2×C4—91no—
474C2×C4—184no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C8—181no—
481C8—181no—
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—42no—
489C2×C3—61no—
490C2×C3—61no—
491C2×C3—61no—
492C2×C3—122no—
493C2×C3—122no—
494C2×C3—124no—
495C2×C3—361no—
496C2×C3—721no—
497C2×C3#6392711yescharacteristic
498C2×C3#6393611yescharacteristic
499C2×C3#6398111yescharacteristic
500D3—121no—
501D3—121no—
502D3—122no—
503D3—241no—
504D3—721no—
505C22—22no—
506C22—41no—
507C22—41no—
508C22—181no—
509C22—181no—
510C22—181no—
511C22—361no—
512C22—361no—
513C22—361no—
514C22#8867711yescharacteristic
515C4—181no—
516C4—182no—
517C4—361no—
518C4C2×#620011yescharacteristic
519C4#8867711yescharacteristic
520C3—31no—
521C3—31no—
522C3—61no—
523C3#1059145711yescharacteristic
524C2—42no—
525C2—361no—
526C2—721no—
527C2#1172119111yescharacteristic
528C2#1172119211yescharacteristic
529C2#1172132311yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#449
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88744C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721159C2semidirect
4. C2×#88438C2semidirect
5. #63659⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720323C2semidirect
28. C2×#3753D3semidirect
34. #63659D3semidirect
44. D4×H(3,𝔽3)D4semidirect
53. #6229D4semidirect
149. C32×D4D12semidirect
150. C32×D4#67semidirect
297. H(3,𝔽3)#449semidirect
460. C32#4781semidirect
461. C32#5158semidirect
472. D4#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
C2×#3753⋊C32
#117211592
C2×#884382
#63659⋊C32
#117203262
#117203232
#105914523
#105914503