Finite Group Databasebeta

← #13140528

Subgroups — #13140528

Order 1728.

528 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 77 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
1D4×#1084C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic
3C2×#3754⋊C3C211yesmaximal, characteristic, complemented
4#11721113C211yesmaximal, characteristic, complemented
5#11720309C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10591471—31nomaximal
9#10591468—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16C4×#1084C2211yescharacteristic
17#3753⋊C3—22no—
18#3754⋊C3—22no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88450—21no—
23C2×#3754D311yescharacteristic, complemented
24C2×#3753D311yescharacteristic
25C3×D4×Dic3—31no—
26C3×D4×Dic3—31no—
27C3×D4×Dic3—61no—
28C2×C3×#214—31no—
29C2×C3×#213—31no—
30#63864—31no—
31#63860—31no—
32Dic3⋊(C3⋊C8)—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63567—31no—
35#63565—31no—
36#63564—31no—
37#63562—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)D412yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×#1084—42no—
45C2×#1084D411yescharacteristic
46C2×#1084D411yescharacteristic
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5165—91no—
55#5153—91no—
56#5016—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C22×C3×Dic3—61no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3—62no—
65C2×C3×D12—31no—
66C2×C3×Dic6—31no—
67#3754—22no—
68#3753—22no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#217—61no—
72C3×#217—61no—
73C3×#217—62no—
74C3×#214—62no—
75C3×#213—62no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—61no—
79C3×C4×Dic3—31no—
80C3×C4×Dic3—31no—
81C3×C4×Dic3—61no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84#3722—61no—
85#3722—61no—
86(C2×D3)⋊Dic3—61no—
87(C2×D3)⋊Dic3—61no—
88C22×H(3,𝔽3)—22no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97#1084—21no—
98#1084—21no—
99#1084—41no—
100(C3⋊Dic3)⋊C3—41no—
101C2×C3×SD16—91no—
102C2×C3×D8—91no—
103C3×C4×D4—91no—
104#947—91no—
105#936—91no—
106D4×Dic3—31no—
107D4×Dic3—31no—
108D4×Dic3—61no—
109C2×#214—31no—
110C2×#214—91no—
111C2×#213—31no—
112C2×#213—91no—
113#875—91no—
114#835—91no—
115#832—91no—
116#821—91no—
117#816—91no—
118Dic3⋊C8—91no—
119Dic3⋊C8—91no—
120#810—91no—
121#809—91no—
122C4⋊(C3⋊C8)—91no—
123C23×C32—22no—
124C23×C32—21no—
125C23×C32—21no—
126C22×C3⋊D3—21no—
127C22×C3×D3—61no—
128C32×D4—22no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—22no—
132C32×D4#6712yes—
133C32×D4#6712yes—
134C2×C32×C4—21no—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140#634—21no—
141#634—21no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—31no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—124no—
153C2×C3×Dic3—122no—
154C2×C3×Dic3—122no—
155C3×D12—61no—
156C3×D12—61no—
157C3×Dic6—61no—
158C3×Dic6—61no—
159C3⋊(C3⋊C8)—21no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163#463—271noSylow
164C2×H(3,𝔽3)—42no—
165C2×H(3,𝔽3)#12111yescharacteristic
166C2×H(3,𝔽3)#13511yescharacteristic
167C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
168(C3⋊D3)⋊C3—81no—
169C2×C3×Q8—91no—
170C2×C3×D4—31no—
171C2×C3×D4—31no—
172C2×C3×D4—61no—
173C2×C3×D4—91no—
174C2×C3×D4D3211yescharacteristic
175C22×C3×C4—181no—
176C22×Dic3—61no—
177C22×Dic3—61no—
178C22×Dic3—62no—
179C2×D12—31no—
180C2×D12—91no—
181C2×Dic6—31no—
182C2×Dic6—91no—
183C3×SD16—182no—
184C3×D8—182no—
185C2×C3×C8—91no—
186C2×C3×C8—91no—
187C3×C4⋊C4—91no—
188C3×C22⋊C4—181no—
189C3×C42—91no—
190#217—61no—
191#217—61no—
192#217—62no—
193#217—181no—
194#217—181no—
195#214—62no—
196#214—182no—
197#213—62no—
198#213—182no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—181no—
201(C2×D3)⋊C4—181no—
202(C2×D3)⋊C4—181no—
203C4⋊Dic3—31no—
204C4⋊Dic3—31no—
205C4⋊Dic3—61no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3—61no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—91no—
213C22×C32—24no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—42no—
218C22×C32—42no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32D3×D411yescharacteristic
224C22×C32D3×D411yescharacteristic
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4#23711yescharacteristic
234C32×C4#23711yescharacteristic
235C32×C4C2×#6711yescharacteristic
236C32×C4C2×#6711yescharacteristic
237C3⋊Dic3—41no—
238C3×Dic3—62no—
239C3×Dic3—61no—
240C3×Dic3—61no—
241C3×Dic3—61no—
242C3×Dic3—61no—
243C3×Dic3—121no—
244C3×Dic3—122no—
245C3×Dic3—121no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C2×SD16—91no—
249C2×D8—91no—
250#124—271no—
251C4×D4—91no—
252C4⋊C8—271no—
253#103—271no—
254#102—271no—
255H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
256C23×C3—21no—
257C23×C3—61no—
258C23×C3—61no—
259C23×C3—62no—
260C23×C3—181no—
261C22×D3—61no—
262C22×D3—181no—
263C3×Q8—181no—
264C3×Q8—181no—
265C3×D4—21no—
266C3×D4—32no—
267C3×D4—32no—
268C3×D4—61no—
269C3×D4—61no—
270C3×D4—62no—
271C3×D4—62no—
272C3×D4—181no—
273C3×D4—181no—
274C3×D4#62512yes—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—61no—
278C2×C3×C4—91no—
279C2×C3×C4—91no—
280C2×C3×C4—181no—
281C2×C3×C4—181no—
282C2×C3×C4—362no—
283C2×C3×C4C2×D3211yescharacteristic
284C2×Dic3—31no—
285C2×Dic3—31no—
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—62no—
293C2×Dic3—61no—
294C2×Dic3—122no—
295C2×Dic3—122no—
296C2×Dic3—124no—
297C2×Dic3—181no—
298D12—61no—
299D12—61no—
300D12—181no—
301D12—181no—
302Dic6—61no—
303Dic6—61no—
304Dic6—181no—
305Dic6—181no—
306C3×C8—181no—
307C3×C8—181no—
308C3⋊C8—61no—
309C3⋊C8—61no—
310C3⋊C8—181no—
311C3⋊C8—181no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—44no—
316C2×C32—42no—
317C2×C32—42no—
318C2×C32#91111yescharacteristic
319C2×C32#91211yescharacteristic
320C2×C32#91411yescharacteristic
321C2×C32#91611yescharacteristic
322C2×C32#93911yescharacteristic
323C2×C32#93911yescharacteristic
324C3⋊D3—81no—
325C3×D3—241no—
326C2×Q8—91no—
327C2×D4—91no—
328C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C22×C4—181no—
330SD16—182no—
331D8—182no—
332C2×C8—91no—
333C2×C8—91no—
334C4⋊C4—91no—
335C22⋊C4—181no—
336C22⋊C4—541no—
337C22⋊C4—541no—
338C42—91no—
339C22×C3—22no—
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—62no—
345C22×C3—62no—
346C22×C3—64no—
347C22×C3—61no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—122no—
353C22×C3—122no—
354C22×C3—361no—
355C22×C3—361no—
356C22×C3—361no—
357C22×C3#381111yescharacteristic
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—361no—
362C2×D3—361no—
363C2×D3—361no—
364C3×C4—31no—
365C3×C4—31no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—61no—
369C3×C4—61no—
370C3×C4—181no—
371C3×C4—181no—
372C3×C4—361no—
373C3×C4—361no—
374C3×C4#380511yescharacteristic
375C3×C4C2×#62511yescharacteristic
376Dic3—61no—
377Dic3—61no—
378Dic3—61no—
379Dic3—61no—
380Dic3—62no—
381Dic3—121no—
382Dic3—121no—
383Dic3—121no—
384Dic3—122no—
385Dic3—121no—
386Dic3—361no—
387C32—21no—
388C32#515311yescharacteristic, complemented
389C32#516511yescharacteristic, complemented
390C23—21no—
391C23—181no—
392Q8—181no—
393Q8—181no—
394D4—21no—
395D4—181no—
396D4—181no—
397D4#619812yes—
398C2×C4—91no—
399C2×C4—91no—
400C2×C4—181no—
401C2×C4—181no—
402C2×C4—362no—
403C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
404C8—181no—
405C8—181no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—42no—
413C2×C3—61no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—122no—
417C2×C3—122no—
418C2×C3—124no—
419C2×C3—721no—
420C2×C3#6393611yescharacteristic
421C2×C3#6393711yescharacteristic
422C2×C3#6398011yescharacteristic
423D3—241no—
424D3—721no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—361no—
429C22—361no—
430C22—361no—
431C22#8867711yescharacteristic
432C4—181no—
433C4—181no—
434C4—361no—
435C4—361no—
436C4#8867211yescharacteristic
437C4C2×#619811yescharacteristic
438C3—31no—
439C3—31no—
440C3—61no—
441C3#1059147611yescharacteristic
442C2—42no—
443C2—721no—
444C2#1172119211yescharacteristic
445C2#1172119311yescharacteristic
446C2#1172131811yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×#3753⋊C3C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#463
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. D4×#1084C2semidirect
3. C2×#3754⋊C3C2semidirect
4. #11721113C2semidirect
5. #11720309C2semidirect
6. #11720308C2semidirect
7. #11720304C2semidirect
23. C2×#3754D3semidirect
255. H(3,𝔽3)#463semidirect
388. C32#5153semidirect
389. C32#5165semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
C2×#3753⋊C32
C2×#3754⋊C32
#117211132
#117203092
#117203082
#117203042
#105914713
#105914683