Finite Group Databasebeta

← #13140915

Subgroups — #13140915

Order 1728.

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

normal subgroups only

This group has 595 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×#3755⋊C3C211yesmaximal, characteristic, complemented
3#11721141C211yesmaximal, characteristic
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10591592—31nomaximal
9#10591584—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×(C3⋊D3)⋊C3—22no—
12Q8×(C3⋊D3)⋊C3—22no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3—21no—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3—21no—
17C2×#634⋊C3C2211yescharacteristic
18C2×#6198—21no—
19C4⋊#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#3755⋊C3—22no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88450—21no—
24#88431—22no—
25C2×Q8×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×Q8—31no—
27C2×#3755D311yescharacteristic, complemented
28C2×C3×#215—31no—
29#63916—31no—
30#63909—31no—
31C2×#3717—31no—
32C2×#3716—31no—
33C3×#835—31no—
34C3×#835—31no—
35C3×#835—61no—
36#63567—31no—
37#63565—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×(C3⋊D3)⋊C3—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
46C2×#1084—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
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#6198—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)C2×D311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×#634—21no—
72C2×#634C2×D311yescharacteristic
73C3×D3×Q8—62no—
74C3×D3×Q8—62no—
75C2×C3×D12—31no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—61no—
80C2×#626—61no—
81C2×#626—61no—
82#3755—22no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×#215—62no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—61no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—61no—
92#3722—61no—
93(C2×D3)⋊Dic3—61no—
94#3717—62no—
95#3716—62no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—21no—
99C2×(C3⋊D3)⋊C3—41no—
100C2×(C3⋊D3)⋊C3—41no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3SD1612yescomplemented
103C4×H(3,𝔽3)—22no—
104C4×H(3,𝔽3)C2×D411yescharacteristic
105C4×H(3,𝔽3)C2×D411yescharacteristic
106#1084—41no—
107(C3⋊Dic3)⋊C3—21no—
108(C3⋊Dic3)⋊C3—22no—
109C22×C3×Q8—91no—
110C2×D3×Q8—31no—
111C2×D3×Q8—91no—
112C2×C3×SD16—91no—
113C2×#215—31no—
114C2×#215—91no—
115C2×#215—91no—
116#938—91no—
117C2×#214—91no—
118#930—91no—
119C2×#204—91no—
120#895—91no—
121C3×#103—91no—
122#835—31no—
123#835—31no—
124#835—61no—
125#835—91no—
126#830—91no—
127(C2×D3)⋊C8—91no—
128(C2×D3)⋊C8—91no—
129#816—91no—
130#810—91no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3#6711yescharacteristic
133C22×C3×D3—31no—
134C22×C3×D3—61no—
135C32×Q8—21no—
136C32×Q8—21no—
137C32×Q8—22no—
138C32×Q8—41no—
139C32×Q8#6712yescomplemented
140C32×Q8#6712yescomplemented
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—41no—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3#6711yescharacteristic
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—21no—
151C4×C3⋊D3—24no—
152C4×C3⋊D3—21no—
153C4×C3⋊D3—21no—
154#634—24no—
155#634—21no—
156#634#6712yescomplemented
157C2×C3×Dic3—31no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—121no—
162C3×D12—61no—
163C3×D12—61no—
164C3×C4×D3—64no—
165C3×C4×D3—61no—
166C3×C4×D3—61no—
167C3×Dic6—32no—
168C3×Dic6—64no—
169C3×Dic6—61no—
170#626—122no—
171#626—122no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—61no—
174C3×C3⋊C8—61no—
175C3×C3⋊C8—61no—
176C3×C3⋊C8—121no—
177#448—271noSylow
178C2×H(3,𝔽3)#12011yescharacteristic
179C2×H(3,𝔽3)C2×SD1611yescharacteristic
180C2×H(3,𝔽3)#13711yescharacteristic
181(C3⋊D3)⋊C3—22no—
182(C3⋊D3)⋊C3—81no—
183C2×C3×Q8—31no—
184C2×C3×Q8—31no—
185C2×C3×Q8—61no—
186C2×C3×Q8—91no—
187C2×C3×Q8—181no—
188C2×C3×Q8—182no—
189C2×C3×Q8—182no—
190C2×C3×Q8D3211yescharacteristic
191C2×C3×D4—91no—
192C22×C3×C4—91no—
193C22×C3×C4—181no—
194C2×#67—181no—
195C2×#67—181no—
196D3×Q8—62no—
197D3×Q8—62no—
198D3×Q8—182no—
199D3×Q8—182no—
200C2×D12—31no—
201C2×D12—91no—
202C2×D12—181no—
203C2×C4×D3—31no—
204C2×C4×D3—61no—
205C2×C4×D3—91no—
206C2×C4×D3—181no—
207C2×Dic6—31no—
208C2×Dic6—61no—
209C2×Dic6—91no—
210C2×Dic6—181no—
211C3×SD16—182no—
212C2×C3×C8—91no—
213C2×C3×C8—91no—
214C3×C4⋊C4—91no—
215#217—181no—
216#215—62no—
217#215—182no—
218#215—182no—
219#214—182no—
220(C2×D3)⋊C4—181no—
221(C2×D3)⋊C4—181no—
222C4⋊Dic3—31no—
223C4⋊Dic3—31no—
224C4⋊Dic3—61no—
225C2×C3⋊C8—31no—
226C2×C3⋊C8—31no—
227C2×C3⋊C8—31no—
228C2×C3⋊C8—61no—
229C2×C3⋊C8—91no—
230#204—182no—
231C22×C32—21no—
232C22×C32D3×D411yescharacteristic
233C22×C32C2×#6711yescharacteristic
234C2×C3⋊D3—21no—
235C2×C3⋊D3—21no—
236C2×C3⋊D3—41no—
237C2×C3⋊D3—41no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3#21512yescomplemented
240C2×C3×D3—32no—
241C2×C3×D3—61no—
242C2×C3×D3—61no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C32×C4—22no—
247C32×C4—22no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—42no—
251C32×C4D3×D411yescharacteristic
252C32×C4C2×#6711yescharacteristic
253C32×C4C2×#6711yescharacteristic
254C32×C4C2×#6711yescharacteristic
255C3⋊Dic3—21no—
256C3⋊Dic3—22no—
257C3×Dic3—62no—
258C3×Dic3—61no—
259C3×Dic3—121no—
260C3×Dic3—121no—
261C3×Dic3—241no—
262C22×Q8—91no—
263C2×SD16—91no—
264C2×SD16—271no—
265#121—271no—
266#103—91no—
267#103—271no—
268C22⋊C8—271no—
269H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
270C23×C3—91no—
271C23×C3—181no—
272C22×D3—31no—
273C22×D3—61no—
274C22×D3—91no—
275C22×D3—181no—
276C3×Q8—21no—
277C3×Q8—32no—
278C3×Q8—32no—
279C3×Q8—61no—
280C3×Q8—61no—
281C3×Q8—62no—
282C3×Q8—92no—
283C3×Q8—121no—
284C3×Q8—181no—
285C3×Q8—184no—
286C3×Q8#62512yes—
287C3×D4—181no—
288C3×D4—181no—
289C2×C3×C4—21no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—91no—
296C2×C3×C4—121no—
297C2×C3×C4—181no—
298C2×C3×C4—181no—
299C2×C3×C4—184no—
300C2×C3×C4—181no—
301C2×C3×C4—181no—
302C2×C3×C4C2×D3211yescharacteristic
303#67—362no—
304#67—362no—
305C2×Dic3—31no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—91no—
310C2×Dic3—121no—
311C2×Dic3—181no—
312D12—61no—
313D12—61no—
314D12—181no—
315D12—181no—
316D12—362no—
317C4×D3—64no—
318C4×D3—61no—
319C4×D3—61no—
320C4×D3—181no—
321C4×D3—181no—
322C4×D3—184no—
323Dic6—32no—
324Dic6—64no—
325Dic6—61no—
326Dic6—92no—
327Dic6—181no—
328Dic6—184no—
329C3×C8—181no—
330C3×C8—181no—
331C3⋊C8—61no—
332C3⋊C8—61no—
333C3⋊C8—61no—
334C3⋊C8—121no—
335C3⋊C8—181no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32D3×SD1611yescharacteristic
340C2×C32#91511yescharacteristic
341C2×C32#93711yescharacteristic
342C2×C32C2×#21511yescharacteristic
343C2×C32#94211yescharacteristic
344C2×C32#95311yescharacteristic
345C3⋊D3—22no—
346C3⋊D3—81no—
347C3×D3—62no—
348C3×D3—241no—
349C2×Q8—91no—
350C2×Q8—181no—
351C2×Q8—182no—
352C2×Q8—182no—
353C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
354C2×D4—91no—
355C2×D4—541no—
356C22×C4—91no—
357C22×C4—181no—
358SD16—182no—
359SD16—542no—
360C2×C8—91no—
361C2×C8—91no—
362C4⋊C4—91no—
363C22⋊C4—541no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—61no—
367C22×C3—92no—
368C22×C3—181no—
369C22×C3—181no—
370C22×C3—361no—
371C22×C3—361no—
372C22×C3—361no—
373C22×C3D3×#6711yescharacteristic
374C2×D3—32no—
375C2×D3—61no—
376C2×D3—61no—
377C2×D3—92no—
378C2×D3—121no—
379C2×D3—121no—
380C2×D3—121no—
381C2×D3—181no—
382C2×D3—181no—
383C2×D3—361no—
384C2×D3—361no—
385C2×D3—361no—
386C3×C4—22no—
387C3×C4—31no—
388C3×C4—31no—
389C3×C4—31no—
390C3×C4—31no—
391C3×C4—62no—
392C3×C4—62no—
393C3×C4—61no—
394C3×C4—61no—
395C3×C4—122no—
396C3×C4—181no—
397C3×C4—182no—
398C3×C4—361no—
399C3×C4C2×#62511yescharacteristic
400C3×C4D3×#6711yescharacteristic
401Dic3—62no—
402Dic3—61no—
403Dic3—121no—
404Dic3—121no—
405Dic3—182no—
406Dic3—181no—
407Dic3—241no—
408C32—21no—
409C32#517011yescharacteristic, complemented
410C32#522711yescharacteristic, complemented
411C23—91no—
412C23—181no—
413Q8—21no—
414Q8—92no—
415Q8—184no—
416Q8—181no—
417Q8#619812yescomplemented
418D4—181no—
419D4—181no—
420D4—1082no—
421C2×C4—21no—
422C2×C4—91no—
423C2×C4—184no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
429C8—181no—
430C8—181no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—61no—
440C2×C3—182no—
441C2×C3—721no—
442C2×C3D3×#21511yescharacteristic
443C2×C3#6394611yescharacteristic
444C2×C3#6397611yescharacteristic
445D3—62no—
446D3—182no—
447D3—241no—
448D3—721no—
449C22—92no—
450C22—181no—
451C22—181no—
452C22—361no—
453C22—361no—
454C22—361no—
455C22#8867611yescharacteristic
456C4—22no—
457C4—181no—
458C4—182no—
459C4—361no—
460C4C2×#619811yescharacteristic
461C4#8867611yescharacteristic
462C3—31no—
463C3—31no—
464C3—61no—
465C3#1059158311yescharacteristic
466C2—182no—
467C2—721no—
468C2#1172123211yescharacteristic
469C2#1172123811yescharacteristic
470C2#1172131611yescharacteristic
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#11721141C2
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×#3755⋊C3C2semidirect
4. C2×#88431C2semidirect
5. #11720583C2semidirect
6. #11720317C2semidirect
7. #11720309C2semidirect
25. C2×Q8×C3⋊D3D3semidirect
27. C2×#3755D3semidirect
43. Q8×H(3,𝔽3)D4semidirect
56. #634⋊C3D4semidirect
102. C2×(C3⋊D3)⋊C3SD16semidirect
139. C32×Q8#67semidirect
140. C32×Q8#67semidirect
156. #634#67semidirect
239. C2×C3⋊D3#215semidirect
269. H(3,𝔽3)#448semidirect
409. C32#5170semidirect
410. C32#5227semidirect
417. Q8#6198semidirect

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×#3755⋊C32
#117211412
C2×#884312
#117205832
#117203172
#117203092
#105915923
#105915843