Finite Group Databasebeta

← #13140962

Subgroups — #13140962

Order 1728.

621 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 87 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×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic, complemented
3#11721156C211yesmaximal, characteristic
4C2×#88437C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10591592—31nomaximal
9#10591583—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12Q8×(C3⋊D3)⋊C3—22no—
13Q8×(C3⋊D3)⋊C3—22no—
14C2×C4×(C3⋊D3)⋊C3—21no—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3—21no—
17C2×#634⋊C3C2211yescharacteristic
18C2×#6200—21no—
19#88549—22no—
20C2×#6180C2211yescharacteristic
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88451—21no—
24#88437—22no—
25C2×Q8×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×Q8—31no—
27C2×C3×#215—31no—
28C2×C3×#215—31no—
29C2×C3×#215—61no—
30#63916—31no—
31#63908—31no—
32C2×#3716—31no—
33C2×#3715—31no—
34#63662D311yescharacteristic, complemented
35C3×#835—31no—
36#63567—31no—
37#63566—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×#272—21no—
41C22×(C3⋊D3)⋊C3D411yescharacteristic
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)D412yescomplemented
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C4⋊#272—21no—
47C4⋊#272—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4×(C3⋊D3)⋊C3—24no—
52C4×(C3⋊D3)⋊C3—21no—
53C4×(C3⋊D3)⋊C3—21no—
54#634⋊C3—24no—
55#634⋊C3—21no—
56#634⋊C3D412yescomplemented
57#6200—42no—
58#6180—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
66Q8×C3⋊D3—22no—
67Q8×C3⋊D3—22no—
68C2×C4×C3⋊D3—21no—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×#634—21no—
71C2×#634C2×D311yescharacteristic
72C3×D3×Q8—62no—
73C3×D3×Q8—62no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×C4×D3—31no—
78C2×C3×C4×D3—61no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81C2×#626—61no—
82C2×#625—61no—
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C3×#215—64no—
86C3×#215—62no—
87C3×#215—62no—
88C3×C4⋊Dic3—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—61no—
93(C2×D3)⋊Dic3—61no—
94(C2×D3)⋊Dic3—61no—
95#3716—62no—
96#3715—62no—
97C22×H(3,𝔽3)C2×D411yescharacteristic
98C2×#272—41no—
99C2×#272—41no—
100C2×#272—41no—
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3SD1612yescomplemented
104C4×H(3,𝔽3)—22no—
105C4×H(3,𝔽3)C2×D411yescharacteristic
106C4×H(3,𝔽3)C2×D411yescharacteristic
107(C3⋊Dic3)⋊C3—21no—
108(C3⋊Dic3)⋊C3—22no—
109(C3⋊Dic3)⋊C3—41no—
110C22×C3×Q8—91no—
111C2×D3×Q8—31no—
112C2×D3×Q8—91no—
113C2×C3×SD16—91no—
114C2×#215—31no—
115C2×#215—31no—
116C2×#215—61no—
117C2×#215—91no—
118#938—91no—
119C2×#214—91no—
120#930—91no—
121C2×#204—91no—
122#895—91no—
123C3×#103—91no—
124#835—31no—
125#835—91no—
126#835—91no—
127#830—91no—
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8—91no—
130#816—91no—
131#810—91no—
132C22×C3⋊D3#6711yescharacteristic
133C22×C3×D3—31no—
134C22×C3×D3—61no—
135C22×C3×D3—61no—
136C22×C3×D3—121no—
137C32×Q8—21no—
138C32×Q8—21no—
139C32×Q8—22no—
140C32×Q8—41no—
141C32×Q8D1212yescomplemented
142C32×Q8#6712yescomplemented
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—41no—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3#6711yescharacteristic
152C4×C3⋊D3—24no—
153C4×C3⋊D3—21no—
154C4×C3⋊D3—21no—
155#634—24no—
156#634—21no—
157#634#6712yescomplemented
158C2×C3×Dic3—31no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C3×D12—62no—
162C3×D12—61no—
163C3×D12—61no—
164C3×D12—61no—
165C3×D12—61no—
166C3×D12—121no—
167C3×C4×D3—64no—
168C3×C4×D3—61no—
169C3×C4×D3—61no—
170C3×Dic6—32no—
171C3×Dic6—64no—
172C3×Dic6—61no—
173#626—122no—
174#625—122no—
175C3⋊(C3⋊C8)—21no—
176C3×C3⋊C8—62no—
177C3×C3⋊C8—61no—
178C3×C3⋊C8—61no—
179C3×C3⋊C8—61no—
180#448—271noSylow
181C2×H(3,𝔽3)#12011yescharacteristic
182C2×H(3,𝔽3)C2×SD1611yescharacteristic
183C2×H(3,𝔽3)#13711yescharacteristic
184#272—81no—
185(C3⋊D3)⋊C3—22no—
186C2×C3×Q8—31no—
187C2×C3×Q8—31no—
188C2×C3×Q8—61no—
189C2×C3×Q8—91no—
190C2×C3×Q8—181no—
191C2×C3×Q8—182no—
192C2×C3×Q8—182no—
193C2×C3×Q8D3211yescharacteristic
194C2×C3×D4—91no—
195C22×C3×C4—91no—
196C22×C3×C4—181no—
197C2×#67—181no—
198C2×#67—181no—
199D3×Q8—62no—
200D3×Q8—62no—
201D3×Q8—182no—
202D3×Q8—182no—
203C2×D12—31no—
204C2×D12—31no—
205C2×D12—61no—
206C2×D12—181no—
207C2×C4×D3—31no—
208C2×C4×D3—61no—
209C2×C4×D3—91no—
210C2×C4×D3—181no—
211C2×Dic6—31no—
212C2×Dic6—61no—
213C2×Dic6—91no—
214C2×Dic6—181no—
215C3×SD16—182no—
216C2×C3×C8—91no—
217C2×C3×C8—91no—
218C3×C4⋊C4—91no—
219#217—181no—
220#215—62no—
221#215—62no—
222#215—64no—
223#215—182no—
224#214—182no—
225(C2×D3)⋊C4—181no—
226(C2×D3)⋊C4—181no—
227C4⋊Dic3—31no—
228C4⋊Dic3—91no—
229C2×C3⋊C8—31no—
230C2×C3⋊C8—31no—
231C2×C3⋊C8—31no—
232C2×C3⋊C8—61no—
233C2×C3⋊C8—91no—
234#204—182no—
235C22×C32—21no—
236C22×C32D3×D411yescharacteristic
237C22×C32C2×#6711yescharacteristic
238C2×C3⋊D3—21no—
239C2×C3⋊D3—21no—
240C2×C3⋊D3#21512yescomplemented
241C2×C3×D3—32no—
242C2×C3×D3—61no—
243C2×C3×D3—61no—
244C2×C3×D3—122no—
245C2×C3×D3—121no—
246C2×C3×D3—121no—
247C2×C3×D3—121no—
248C2×C3×D3—121no—
249C2×C3×D3—121no—
250C2×C3×D3—121no—
251C2×C3×D3—241no—
252C2×C3×D3—241no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—42no—
258C32×C4C2×D1211yescharacteristic
259C32×C4D3×D411yescharacteristic
260C32×C4C2×#6711yescharacteristic
261C32×C4C2×#6711yescharacteristic
262C3⋊Dic3—21no—
263C3⋊Dic3—22no—
264C3⋊Dic3—41no—
265C3×Dic3—62no—
266C3×Dic3—61no—
267C3×Dic3—121no—
268C22×Q8—91no—
269C2×SD16—91no—
270C2×SD16—271no—
271#121—271no—
272#103—91no—
273#103—271no—
274C22⋊C8—271no—
275H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
276C23×C3—91no—
277C23×C3—181no—
278C22×D3—31no—
279C22×D3—61no—
280C22×D3—61no—
281C22×D3—91no—
282C22×D3—121no—
283C3×Q8—21no—
284C3×Q8—32no—
285C3×Q8—32no—
286C3×Q8—61no—
287C3×Q8—61no—
288C3×Q8—62no—
289C3×Q8—92no—
290C3×Q8—121no—
291C3×Q8—181no—
292C3×Q8—184no—
293C3×Q8#62612yes—
294C3×D4—181no—
295C3×D4—181no—
296C2×C3×C4—21no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—91no—
303C2×C3×C4—121no—
304C2×C3×C4—181no—
305C2×C3×C4—181no—
306C2×C3×C4—184no—
307C2×C3×C4—181no—
308C2×C3×C4—181no—
309C2×C3×C4C2×D3211yescharacteristic
310#67—362no—
311#67—362no—
312C2×Dic3—31no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—91no—
316C2×Dic3—181no—
317C2×Dic3—181no—
318D12—61no—
319D12—61no—
320D12—61no—
321D12—61no—
322D12—62no—
323D12—121no—
324D12—362no—
325C4×D3—64no—
326C4×D3—61no—
327C4×D3—61no—
328C4×D3—181no—
329C4×D3—181no—
330C4×D3—184no—
331Dic6—32no—
332Dic6—64no—
333Dic6—61no—
334Dic6—92no—
335Dic6—181no—
336Dic6—184no—
337C3×C8—181no—
338C3×C8—181no—
339C3⋊C8—61no—
340C3⋊C8—61no—
341C3⋊C8—62no—
342C3⋊C8—61no—
343C3⋊C8—181no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32#88211yescharacteristic
348C2×C32D3×SD1611yescharacteristic
349C2×C32#91811yescharacteristic
350C2×C32C2×#21511yescharacteristic
351C2×C32#94211yescharacteristic
352C2×C32#95311yescharacteristic
353C3⋊D3—22no—
354C3×D3—62no—
355C3×D3—242no—
356C3×D3—241no—
357C3×D3—241no—
358C2×Q8—91no—
359C2×Q8—181no—
360C2×Q8—182no—
361C2×Q8—182no—
362C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
363C2×D4—91no—
364C2×D4—541no—
365C22×C4—91no—
366C22×C4—181no—
367SD16—182no—
368SD16—542no—
369C2×C8—91no—
370C2×C8—91no—
371C4⋊C4—91no—
372C22⋊C4—541no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—61no—
376C22×C3—92no—
377C22×C3—181no—
378C22×C3—181no—
379C22×C3—361no—
380C22×C3—361no—
381C22×C3—361no—
382C22×C3D3×#6711yescharacteristic
383C2×D3—32no—
384C2×D3—61no—
385C2×D3—61no—
386C2×D3—92no—
387C2×D3—121no—
388C2×D3—121no—
389C2×D3—121no—
390C2×D3—121no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—122no—
394C2×D3—181no—
395C2×D3—181no—
396C2×D3—241no—
397C2×D3—241no—
398C3×C4—22no—
399C3×C4—31no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—31no—
403C3×C4—62no—
404C3×C4—62no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—122no—
408C3×C4—181no—
409C3×C4—182no—
410C3×C4—361no—
411C3×C4C2×#62611yescharacteristic
412C3×C4D3×#6711yescharacteristic
413Dic3—62no—
414Dic3—61no—
415Dic3—121no—
416Dic3—182no—
417Dic3—181no—
418Dic3—361no—
419C32—21no—
420C32#480611yescharacteristic, complemented
421C32#522711yescharacteristic, complemented
422C23—91no—
423C23—181no—
424Q8—21no—
425Q8—92no—
426Q8—184no—
427Q8—181no—
428Q8#620012yescomplemented
429D4—181no—
430D4—181no—
431D4—1082no—
432C2×C4—21no—
433C2×C4—91no—
434C2×C4—184no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
440C8—181no—
441C8—181no—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—61no—
449C2×C3—61no—
450C2×C3—61no—
451C2×C3—182no—
452C2×C3—721no—
453C2×C3D3×#21511yescharacteristic
454C2×C3#6394811yescharacteristic
455C2×C3#6397711yescharacteristic
456D3—62no—
457D3—182no—
458D3—241no—
459D3—241no—
460D3—242no—
461C22—92no—
462C22—181no—
463C22—181no—
464C22—361no—
465C22—361no—
466C22—361no—
467C22#8867611yescharacteristic
468C4—22no—
469C4—181no—
470C4—182no—
471C4—361no—
472C4C2×#620011yescharacteristic
473C4#8867611yescharacteristic
474C3—31no—
475C3—31no—
476C3—61no—
477C3#1059158411yescharacteristic
478C2—182no—
479C2—721no—
480C2#1172123211yescharacteristic
481C2#1172123411yescharacteristic
482C2#1172132111yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#11721156C2
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×(C3⋊D3)⋊C3C2semidirect
2. C2×#88549C2semidirect
4. C2×#88437C2semidirect
5. #63662⋊C3C2semidirect
6. #11720317C2semidirect
7. #11720313C2semidirect
25. C2×Q8×C3⋊D3D3semidirect
34. #63662D3semidirect
43. Q8×H(3,𝔽3)D4semidirect
56. #634⋊C3D4semidirect
103. C2×(C3⋊D3)⋊C3SD16semidirect
141. C32×Q8D12semidirect
142. C32×Q8#67semidirect
157. #634#67semidirect
240. C2×C3⋊D3#215semidirect
275. H(3,𝔽3)#448semidirect
420. C32#4806semidirect
421. C32#5227semidirect
428. Q8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
C2×#885492
#117211562
C2×#884372
#63662⋊C32
#117203172
#117203132
#105915923
#105915833