Finite Group Databasebeta

← #13140946

Subgroups — #13140946

Order 1728.

709 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 81 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×Q8×#272C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3#11721159C211yesmaximal, characteristic
4C2×#88438C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720323C211yesmaximal, characteristic, complemented
8#10591584—31nomaximal
9#10591583—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×#272—22no—
12Q8×#272—22no—
13C2×C4×#272—21no—
14C2×C4×#272C2211yescharacteristic
15C2×#6229—21no—
16C2×#6229C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×#6200—21no—
19#3755⋊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×D3×Q8—31no—
26C2×C3×D3×Q8—31no—
27C2×C3×D3×Q8—61no—
28C2×#3755D311yescharacteristic, complemented
29C2×C3×#215—31no—
30#63909—31no—
31#63908—31no—
32C2×#3717—31no—
33C2×#3715—31no—
34#63662D311yescharacteristic, complemented
35C3×#835—31no—
36#63566—31no—
37#63565—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×#272D411yescharacteristic
41C22×(C3⋊D3)⋊C3—21no—
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)D412yescomplemented
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×#1084—21no—
47C2×#1084D411yescharacteristic
48C4×#272—24no—
49C4×#272—21no—
50C4×#272—21no—
51#6229—24no—
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#5227—91no—
61#5170—91no—
62#4806—91no—
63C2×C32×Q8—21no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C2×D311yescharacteristic
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C3×D3×Q8—62no—
68C3×D3×Q8—62no—
69C3×D3×Q8—64no—
70C3×D3×Q8—64no—
71C3×D3×Q8—62no—
72C3×D3×Q8—62no—
73C2×C3×D12—31no—
74C2×C3×C4×D3—31no—
75C2×C3×C4×D3—31no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—61no—
78C2×C3×C4×D3—61no—
79C2×C3×C4×D3—62no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—61no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6—61no—
85C2×C3×Dic6—62no—
86C2×#626—61no—
87C2×#625—61no—
88#3755—22no—
89C4⋊(C3⋊Dic3)C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#215—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)C2×D411yescharacteristic
101C2×#272—21no—
102C2×#272—21no—
103C2×#272SD1612yescomplemented
104C2×(C3⋊D3)⋊C3—41no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—41no—
107C4×H(3,𝔽3)—22no—
108C4×H(3,𝔽3)C2×D411yescharacteristic
109C4×H(3,𝔽3)C2×D411yescharacteristic
110#1084—21no—
111#1084—22no—
112(C3⋊Dic3)⋊C3—41no—
113C22×C3×Q8—91no—
114C2×D3×Q8—31no—
115C2×D3×Q8—31no—
116C2×D3×Q8—61no—
117C2×C3×SD16—91no—
118C2×#215—31no—
119C2×#215—91no—
120C2×#215—91no—
121#938—91no—
122C2×#214—91no—
123#930—91no—
124C2×#204—91no—
125#895—91no—
126C3×#103—91no—
127#835—31no—
128#835—91no—
129#835—91no—
130#830—91no—
131(C2×D3)⋊C8—91no—
132(C2×D3)⋊C8—91no—
133#816—91no—
134#810—91no—
135C22×C3⋊D3—21no—
136C22×C3×D3—31no—
137C22×C3×D3—31no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C32×Q8—22no—
141C32×Q8—21no—
142C32×Q8—21no—
143C32×Q8—22no—
144C32×Q8D1212yescomplemented
145C32×Q8#6712yescomplemented
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C3⋊Dic3—21no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)—21no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C3×D12—61no—
163C3×D12—61no—
164C3×C4×D3—68no—
165C3×C4×D3—64no—
166C3×C4×D3—64no—
167C3×C4×D3—62no—
168C3×C4×D3—62no—
169C3×C4×D3—61no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×C4×D3—61no—
173C3×Dic6—32no—
174C3×Dic6—32no—
175C3×Dic6—62no—
176C3×Dic6—68no—
177C3×Dic6—64no—
178C3×Dic6—64no—
179C3×Dic6—62no—
180C3×Dic6—61no—
181C3×Dic6—61no—
182#626—122no—
183#625—122no—
184C3⋊(C3⋊C8)—21no—
185C3⋊(C3⋊C8)—21no—
186C3×C3⋊C8—61no—
187C3×C3⋊C8—61no—
188#448—271noSylow
189C2×H(3,𝔽3)#12011yescharacteristic
190C2×H(3,𝔽3)C2×SD1611yescharacteristic
191C2×H(3,𝔽3)#13711yescharacteristic
192#272—22no—
193(C3⋊D3)⋊C3—81no—
194C2×C3×Q8—31no—
195C2×C3×Q8—31no—
196C2×C3×Q8—61no—
197C2×C3×Q8—91no—
198C2×C3×Q8—181no—
199C2×C3×Q8—182no—
200C2×C3×Q8—182no—
201C2×C3×Q8D3211yescharacteristic
202C2×C3×D4—91no—
203C22×C3×C4—91no—
204C22×C3×C4—181no—
205C2×#67—181no—
206C2×#67—181no—
207D3×Q8—62no—
208D3×Q8—62no—
209D3×Q8—62no—
210D3×Q8—62no—
211D3×Q8—64no—
212D3×Q8—64no—
213C2×D12—31no—
214C2×D12—91no—
215C2×D12—181no—
216C2×C4×D3—31no—
217C2×C4×D3—31no—
218C2×C4×D3—61no—
219C2×C4×D3—61no—
220C2×C4×D3—61no—
221C2×C4×D3—62no—
222C2×Dic6—31no—
223C2×Dic6—31no—
224C2×Dic6—61no—
225C2×Dic6—61no—
226C2×Dic6—61no—
227C2×Dic6—62no—
228C3×SD16—182no—
229C2×C3×C8—91no—
230C2×C3×C8—91no—
231C3×C4⋊C4—91no—
232#217—181no—
233#215—62no—
234#215—182no—
235#215—182no—
236#214—182no—
237(C2×D3)⋊C4—181no—
238(C2×D3)⋊C4—181no—
239C4⋊Dic3—31no—
240C4⋊Dic3—91no—
241C2×C3⋊C8—31no—
242C2×C3⋊C8—31no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—91no—
245#204—182no—
246C22×C32—21no—
247C22×C32D3×D411yescharacteristic
248C22×C32D3×D411yescharacteristic
249C2×C3⋊D3—41no—
250C2×C3⋊D3—41no—
251C2×C3⋊D3—41no—
252C2×C3×D3—32no—
253C2×C3×D3—32no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—62no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C32×C4—24no—
265C32×C4—22no—
266C32×C4—22no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4C2×D1211yescharacteristic
270C32×C4D3×D411yescharacteristic
271C32×C4D3×D411yescharacteristic
272C32×C4C2×#6711yescharacteristic
273C3⋊Dic3—41no—
274C3×Dic3—62no—
275C3×Dic3—64no—
276C3×Dic3—62no—
277C3×Dic3—61no—
278C3×Dic3—62no—
279C3×Dic3—61no—
280C3×Dic3—121no—
281C22×Q8—91no—
282C2×SD16—91no—
283C2×SD16—271no—
284#121—271no—
285#103—91no—
286#103—271no—
287C22⋊C8—271no—
288H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
289C23×C3—91no—
290C23×C3—181no—
291C22×D3—31no—
292C22×D3—31no—
293C22×D3—61no—
294C22×D3—61no—
295C22×D3—181no—
296C3×Q8—21no—
297C3×Q8—32no—
298C3×Q8—32no—
299C3×Q8—61no—
300C3×Q8—61no—
301C3×Q8—62no—
302C3×Q8—62no—
303C3×Q8—92no—
304C3×Q8—184no—
305C3×Q8—181no—
306C3×Q8#62612yes—
307C3×D4—181no—
308C3×D4—181no—
309C2×C3×C4—21no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—62no—
315C2×C3×C4—61no—
316C2×C3×C4—91no—
317C2×C3×C4—184no—
318C2×C3×C4—181no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4C2×D3211yescharacteristic
323#67—362no—
324#67—362no—
325C2×Dic3—31no—
326C2×Dic3—31no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—62no—
331C2×Dic3—61no—
332C2×Dic3—181no—
333D12—61no—
334D12—61no—
335D12—181no—
336D12—181no—
337D12—362no—
338C4×D3—61no—
339C4×D3—61no—
340C4×D3—61no—
341C4×D3—61no—
342C4×D3—64no—
343C4×D3—64no—
344C4×D3—68no—
345C4×D3—62no—
346C4×D3—62no—
347Dic6—32no—
348Dic6—32no—
349Dic6—62no—
350Dic6—61no—
351Dic6—61no—
352Dic6—64no—
353Dic6—64no—
354Dic6—68no—
355Dic6—62no—
356C3×C8—181no—
357C3×C8—181no—
358C3⋊C8—61no—
359C3⋊C8—61no—
360C3⋊C8—181no—
361C3⋊C8—181no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32#88211yescharacteristic
366C2×C32D3×SD1611yescharacteristic
367C2×C32D3×SD1611yescharacteristic
368C2×C32#91511yescharacteristic
369C2×C32#91811yescharacteristic
370C2×C32#93711yescharacteristic
371C3⋊D3—81no—
372C3×D3—64no—
373C3×D3—62no—
374C3×D3—62no—
375C3×D3—241no—
376C2×Q8—91no—
377C2×Q8—181no—
378C2×Q8—182no—
379C2×Q8—182no—
380C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C2×D4—91no—
382C2×D4—541no—
383C22×C4—91no—
384C22×C4—181no—
385SD16—182no—
386SD16—542no—
387C2×C8—91no—
388C2×C8—91no—
389C4⋊C4—91no—
390C22⋊C4—541no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—61no—
394C22×C3—92no—
395C22×C3—181no—
396C22×C3—181no—
397C22×C3—361no—
398C22×C3—361no—
399C22×C3—361no—
400C22×C3#381111yescharacteristic
401C2×D3—32no—
402C2×D3—32no—
403C2×D3—62no—
404C2×D3—61no—
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—61no—
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—121no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—361no—
414C2×D3—361no—
415C2×D3—361no—
416C3×C4—22no—
417C3×C4—31no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—31no—
421C3×C4—62no—
422C3×C4—62no—
423C3×C4—64no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—181no—
427C3×C4—182no—
428C3×C4—361no—
429C3×C4C2×#62611yescharacteristic
430C3×C4#381111yescharacteristic
431Dic3—62no—
432Dic3—62no—
433Dic3—61no—
434Dic3—61no—
435Dic3—62no—
436Dic3—64no—
437Dic3—121no—
438Dic3—361no—
439C32—21no—
440C32#480611yescharacteristic, complemented
441C32#517011yescharacteristic, complemented
442C23—91no—
443C23—181no—
444Q8—21no—
445Q8—92no—
446Q8—184no—
447Q8—181no—
448Q8#620012yescomplemented
449D4—181no—
450D4—181no—
451D4—1082no—
452C2×C4—21no—
453C2×C4—91no—
454C2×C4—184no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4—181no—
459C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
460C8—181no—
461C8—181no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—61no—
469C2×C3—61no—
470C2×C3—61no—
471C2×C3—182no—
472C2×C3—721no—
473C2×C3#6394011yescharacteristic
474C2×C3#6395111yescharacteristic
475C2×C3#6398111yescharacteristic
476D3—62no—
477D3—62no—
478D3—64no—
479D3—241no—
480D3—721no—
481C22—92no—
482C22—181no—
483C22—181no—
484C22—361no—
485C22—361no—
486C22—361no—
487C22#8867711yescharacteristic
488C4—22no—
489C4—181no—
490C4—182no—
491C4—361no—
492C4C2×#620011yescharacteristic
493C4#8867711yescharacteristic
494C3—31no—
495C3—31no—
496C3—61no—
497C3#1059159211yescharacteristic
498C2—182no—
499C2—721no—
500C2#1172122811yescharacteristic
501C2#1172123111yescharacteristic
502C2#1172132311yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721159C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#448
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#272C2semidirect
2. C2×#3755⋊C3C2semidirect
4. C2×#88438C2semidirect
5. #63662⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720323C2semidirect
28. C2×#3755D3semidirect
34. #63662D3semidirect
43. Q8×H(3,𝔽3)D4semidirect
53. #6229D4semidirect
103. C2×#272SD16semidirect
144. C32×Q8D12semidirect
145. C32×Q8#67semidirect
288. H(3,𝔽3)#448semidirect
440. C32#4806semidirect
441. C32#5170semidirect
448. Q8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×#3755⋊C32
#117211592
C2×#884382
#63662⋊C32
#117203262
#117203232
#105915843
#105915833