Finite Group Databasebeta

← #13140923

Subgroups — #13140923

Order 1728.

595 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 107 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×#88722C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic, complemented
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720583C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591560—31nomaximal
9#10591556—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88722—22no—
12#88722C414yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C4⋊#1084C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88447—21no—
23((C3⋊D3)⋊C3)⋊C8—22no—
24C2×#3832D311yescharacteristic
25C2×C3×#239—31no—
26#63849—31no—
27#63841—31no—
28C2×#3710—31no—
29C2×D3×C3⋊C8—31no—
30#63662D311yescharacteristic, complemented
31C3×#835—31no—
32C3×#835—31no—
33C3×#835—31no—
34C3×#835—61no—
35#63564—31no—
36#63564—31no—
37#63563—31no—
38#63562—31no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)C2×C412yes—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4⋊((C3⋊D3)⋊C3)—24no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)C2×C412yes—
51C4×(C3⋊D3)⋊C3—24no—
52C4×(C3⋊D3)⋊C3C2×C412yes—
53C4×(C3⋊D3)⋊C3D412yes—
54#6180—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5233—91no—
57#4804—91no—
58#4804—91no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C2×D311yescharacteristic
62#3832—22no—
63#3832Dic314yescomplemented
64C2×C4⋊(C3⋊D3)—21no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C4×C3⋊D3—21no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C3×#239—34no—
69C3×#239—62no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—61no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—61no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—61no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86Dic32—61no—
87Dic32—61no—
88#3710—62no—
89D3×C3⋊C8—62no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—22no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3C22⋊C412yes—
97C4×H(3,𝔽3)—22no—
98C4×H(3,𝔽3)C22×C411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100#1084—41no—
101(C3⋊Dic3)⋊C3—41no—
102(C3⋊Dic3)⋊C3C22⋊C412yes—
103C2×C3×#41—91no—
104C2×#239—31no—
105C2×#239—91no—
106#926—91no—
107C22×C3⋊C8—91no—
108C2×D3×C8—91no—
109C2×D3×C8—91no—
110#892—91no—
111#892—91no—
112C3×#103—91no—
113C3×#103—91no—
114#835—31no—
115#835—31no—
116#835—31no—
117#835—61no—
118#835—91no—
119#832—91no—
120#832—91no—
121#821—91no—
122#821—91no—
123#809—91no—
124#809—91no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3#6711yescharacteristic
127C22×C3×D3—31no—
128C22×C3×D3—61no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—22no—
132C32×Q8—41no—
133C32×Q8C4×D312yes—
134C32×Q8C2×Dic312yes—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—41no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3#6711yescharacteristic
143C4⋊(C3⋊D3)—24no—
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)C2×Dic312yes—
146C4×C3⋊D3—24no—
147C4×C3⋊D3C2×Dic312yes—
148C4×C3⋊D3#6712yes—
149C2×C3×Dic3—31no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—121no—
154C3×D12—32no—
155C3×D12—64no—
156C3×D12—61no—
157C3×C4×D3—32no—
158C3×C4×D3—32no—
159C3×C4×D3—64no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—121no—
165#416—271noSylow
166C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
167C2×H(3,𝔽3)#13511yescharacteristic
168C2×H(3,𝔽3)#13511yescharacteristic
169(C3⋊D3)⋊C3—22no—
170(C3⋊D3)⋊C3—42no—
171C3×#41—94no—
172C3×#41—182no—
173C2×C3×Q8—31no—
174C2×C3×Q8—31no—
175C2×C3×Q8—61no—
176C2×C3×Q8D3211yescharacteristic
177C2×C3×D4—91no—
178C2×C3×D4—181no—
179C22×C3×C4—91no—
180C22×C3×C4—181no—
181#239—34no—
182#239—62no—
183#239—94no—
184#239—182no—
185C2×D12—31no—
186C2×D12—61no—
187C2×D12—91no—
188C2×D12—181no—
189C2×C4×D3—31no—
190C2×C4×D3—61no—
191C2×C4×D3—91no—
192C2×C4×D3—181no—
193C2×C3×C8—91no—
194C2×C3×C8—91no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4—91no—
197#217—181no—
198(C2×D3)⋊C4—181no—
199(C2×D3)⋊C4—181no—
200C4⋊Dic3—31no—
201C4⋊Dic3—31no—
202C4⋊Dic3—31no—
203C4⋊Dic3—61no—
204C4⋊Dic3—91no—
205C4×Dic3—181no—
206C4×Dic3—181no—
207C4×Dic3—181no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—61no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—182no—
214D3×C8—182no—
215D3×C8—182no—
216C22×C32—21no—
217C22×C32D3×D411yescharacteristic
218C22×C32C2×#6711yescharacteristic
219C2×C3⋊D3—21no—
220C2×C3⋊D3—21no—
221C2×C3⋊D3—22no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3#21712yes—
225C2×C3×D3—32no—
226C2×C3×D3—61no—
227C2×C3×D3—61no—
228C2×C3×D3—62no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—42no—
236C32×C4C2×C4×D311yescharacteristic
237C32×C4D3×D411yescharacteristic
238C32×C4C22×Dic311yescharacteristic
239C32×C4C2×#6711yescharacteristic
240C3⋊Dic3—41no—
241C3⋊Dic3#21712yes—
242C3×Dic3—32no—
243C3×Dic3—121no—
244C3×Dic3—121no—
245C3×Dic3—121no—
246C3×Dic3—241no—
247C2×#41—91no—
248C22×C8—271no—
249#117—271no—
250#103—91no—
251#103—91no—
252#102—271no—
253#102—271no—
254H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
255C23×C3—91no—
256C23×C3—181no—
257C22×D3—31no—
258C22×D3—61no—
259C22×D3—91no—
260C22×D3—181no—
261C3×Q8—21no—
262C3×Q8—32no—
263C3×Q8—32no—
264C3×Q8—61no—
265C3×Q8—61no—
266C3×Q8—62no—
267C3×Q8—121no—
268C3×Q8D3×Dic312yes—
269C3×D4—92no—
270C3×D4—181no—
271C3×D4—184no—
272C2×C3×C4—21no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—92no—
279C2×C3×C4—92no—
280C2×C3×C4—91no—
281C2×C3×C4—121no—
282C2×C3×C4—184no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C2×D3211yescharacteristic
286C2×Dic3—31no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—91no—
291C2×Dic3—121no—
292C2×Dic3—181no—
293D12—32no—
294D12—64no—
295D12—61no—
296D12—92no—
297D12—181no—
298D12—184no—
299C4×D3—32no—
300C4×D3—32no—
301C4×D3—64no—
302C4×D3—92no—
303C4×D3—92no—
304C4×D3—184no—
305C3×C8—181no—
306C3×C8—181no—
307C3⋊C8—61no—
308C3⋊C8—61no—
309C3⋊C8—61no—
310C3⋊C8—121no—
311C3⋊C8—181no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32D3×C22⋊C411yescharacteristic
316C2×C32#91611yescharacteristic
317C2×C32#91911yescharacteristic
318C2×C32#95011yescharacteristic
319C2×C32#95011yescharacteristic
320C2×C32C2×#21711yescharacteristic
321C3⋊D3—22no—
322C3⋊D3—42no—
323C3×D3—62no—
324C3×D3—122no—
325#41—94no—
326#41—182no—
327C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
328C2×D4—91no—
329C2×D4—181no—
330C22×C4—91no—
331C22×C4—181no—
332C2×C8—91no—
333C2×C8—91no—
334C2×C8—542no—
335C4⋊C4—91no—
336C4⋊C4—91no—
337C22⋊C4—541no—
338C42—541no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—61no—
342C22×C3—92no—
343C22×C3—181no—
344C22×C3—181no—
345C22×C3—182no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3D3×#6711yescharacteristic
349C2×D3—32no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—62no—
353C2×D3—92no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—181no—
357C2×D3—181no—
358C2×D3—182no—
359C2×D3—361no—
360C2×D3—361no—
361C3×C4—22no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—31no—
365C3×C4—31no—
366C3×C4—62no—
367C3×C4—62no—
368C3×C4—61no—
369C3×C4—61no—
370C3×C4—92no—
371C3×C4—122no—
372C3×C4—361no—
373C3×C4—361no—
374C3×C4C2×D3×Dic311yescharacteristic
375C3×C4D3×#6711yescharacteristic
376Dic3—32no—
377Dic3—92no—
378Dic3—121no—
379Dic3—121no—
380Dic3—121no—
381Dic3—241no—
382Dic3—361no—
383C32—21no—
384C32#480411yescharacteristic, complemented
385C32#523311yescharacteristic, complemented
386C23—91no—
387C23—181no—
388Q8—21no—
389Q8((C3⋊D3)⋊C3)⋊C412yes—
390D4—92no—
391D4—184no—
392D4—181no—
393C2×C4—21no—
394C2×C4—91no—
395C2×C4—92no—
396C2×C4—92no—
397C2×C4—184no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
401C8—181no—
402C8—181no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—61no—
410C2×C3—61no—
411C2×C3—61no—
412C2×C3—182no—
413C2×C3—362no—
414C2×C3#6394711yescharacteristic
415C2×C3#6394911yescharacteristic
416C2×C3D3×#21711yescharacteristic
417D3—62no—
418D3—122no—
419D3—182no—
420D3—362no—
421C22—92no—
422C22—181no—
423C22—181no—
424C22—182no—
425C22—361no—
426C22—361no—
427C22#8867611yescharacteristic
428C4—22no—
429C4—92no—
430C4—361no—
431C4—361no—
432C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
433C4#8867611yescharacteristic
434C3—31no—
435C3—31no—
436C3—61no—
437C3#1059155611yescharacteristic
438C2—182no—
439C2—362no—
440C2#1172123511yescharacteristic
441C2#1172123911yescharacteristic
442C2#1172131011yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88722C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#416
LCSG ▷ 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
2. #11721081C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720583C2semidirect
5. #63662⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720308C2semidirect
12. #88722C4semidirect
30. #63662D3semidirect
63. #3832Dic3semidirect
254. H(3,𝔽3)#416semidirect
384. C32#4804semidirect
385. C32#5233semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
#117210812
C2×((C3⋊D3)⋊C3)⋊C82
#117205832
#63662⋊C32
#117203182
#117203082
#105915603
#105915563