Finite Group Databasebeta

← #13137890

Subgroups — #13137890

Order 1728.

392 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#11721758C211yesmaximal, characteristic, complemented
2#11721113C211yesmaximal, characteristic
3#63636⋊C3C211yesmaximal, characteristic
4#63637⋊C3C211yesmaximal, characteristic, complemented
5#11720309C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic
7#11720302C211yesmaximal, characteristic, complemented
8#10590537—31nomaximal
9#10590533—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084—21no—
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16C4×#1084C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88450—21no—
21C3×#890—31no—
22C3×#890—31no—
23C3×#890—61no—
24#63864—31no—
25#63860—31no—
26#63637D311yescharacteristic, complemented
27#63636D311yescharacteristic
28C3×#810—31no—
29C3×#809—31no—
30#63567—31no—
31#63565—31no—
32#63564—31no—
33#63562—31no—
34Dic3×C3⋊C8—31no—
35Dic3×C3⋊C8—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084D411yescharacteristic
41C2×#1084D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5068—91no—
50#4800—91no—
51#4768—91no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×#634C2×D311yescharacteristic
54C2×C3×D12—31no—
55C2×C3×Dic6—31no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—61no—
64C3×C4⋊Dic3—61no—
65C3×C4⋊Dic3—61no—
66C3×C4⋊Dic3—62no—
67C3×Dic3⋊C4—61no—
68C3×Dic3⋊C4—61no—
69C3×Dic3⋊C4—62no—
70C3×C4×Dic3—31no—
71C3×C4×Dic3—31no—
72C3×C4×Dic3—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75#3722—61no—
76#3722—61no—
77(C2×D3)⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×(C3⋊D3)⋊C3—41no—
81C2×(C3⋊D3)⋊C3—41no—
82C2×(C3⋊D3)⋊C3—41no—
83C4×H(3,𝔽3)—41no—
84C4×H(3,𝔽3)#4111yescharacteristic
85C4×H(3,𝔽3)#4111yescharacteristic
86#1084—21no—
87#1084—21no—
88#1084—41no—
89(C3⋊Dic3)⋊C3—41no—
90C3×#125—91no—
91#947—91no—
92#936—91no—
93#890—31no—
94#890—31no—
95#890—61no—
96#875—91no—
97C3×#103—91no—
98C3×#102—91no—
99#835—91no—
100#832—91no—
101#821—91no—
102#816—91no—
103C8×Dic3—91no—
104C8×Dic3—91no—
105#810—31no—
106#810—91no—
107#810—91no—
108#809—31no—
109#809—91no—
110#809—91no—
111C4×C3⋊C8—91no—
112C22×C3⋊D3—21no—
113C22×C3×D3—61no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C4⋊(C3⋊D3)—21no—
122C4⋊(C3⋊D3)—21no—
123#634—21no—
124#634—21no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C3×D12—61no—
136C3×D12—61no—
137C3×Dic6—61no—
138C3×Dic6—61no—
139C3⋊(C3⋊C8)—21no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143#488—271noSylow
144C2×H(3,𝔽3)#12411yescharacteristic
145C2×H(3,𝔽3)#13511yescharacteristic
146C2×H(3,𝔽3)#13511yescharacteristic
147(C3⋊D3)⋊C3—81no—
148C2×C3×Q8—91no—
149C2×C3×D4—91no—
150C2×D12—31no—
151C2×D12—91no—
152C2×Dic6—31no—
153C2×Dic6—91no—
154C2×C3×C8—91no—
155C2×C3×C8—91no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4D3211yescharacteristic
163C3×C42—91no—
164#217—181no—
165#217—181no—
166(C2×D3)⋊C4—181no—
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—61no—
173C4⋊Dic3—61no—
174C4⋊Dic3—61no—
175C4⋊Dic3—62no—
176Dic3⋊C4—61no—
177Dic3⋊C4—61no—
178Dic3⋊C4—62no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—61no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—91no—
186C22×C32—21no—
187C22×C32D3×D411yescharacteristic
188C22×C32D3×D411yescharacteristic
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C2×C3×D3—121no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—42no—
198C32×C4—41no—
199C32×C4—41no—
200C32×C4#23511yescharacteristic
201C32×C4#23511yescharacteristic
202C32×C4#23711yescharacteristic
203C32×C4#23711yescharacteristic
204C3⋊Dic3—41no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—122no—
212C3×Dic3—121no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215#125—91no—
216#124—271no—
217#103—91no—
218#103—271no—
219#102—91no—
220#102—271no—
221C4×C8—271no—
222H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
223C23×C3—181no—
224C22×D3—61no—
225C22×D3—181no—
226C3×Q8—181no—
227C3×Q8—181no—
228C3×D4—181no—
229C3×D4—181no—
230C2×C3×C4—21no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—62no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—62no—
251C2×Dic3—61no—
252C2×Dic3—181no—
253D12—61no—
254D12—61no—
255D12—181no—
256D12—181no—
257Dic6—61no—
258Dic6—61no—
259Dic6—181no—
260Dic6—181no—
261C3×C8—181no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8—61no—
265C3⋊C8—181no—
266C3⋊C8—181no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32#88511yescharacteristic
271C2×C32#88511yescharacteristic
272C2×C32#91211yescharacteristic
273C2×C32#91611yescharacteristic
274C2×C32#91611yescharacteristic
275C2×C32#91911yescharacteristic
276C3⋊D3—81no—
277C3×D3—241no—
278C2×Q8—91no—
279C2×D4—91no—
280C2×C8—91no—
281C2×C8—91no—
282C4⋊C4—91no—
283C4⋊C4—181no—
284C4⋊C4—181no—
285C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C22⋊C4—541no—
287C22⋊C4—541no—
288C42—91no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—61no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3#381111yescharacteristic
296C2×D3—121no—
297C2×D3—121no—
298C2×D3—121no—
299C2×D3—361no—
300C2×D3—361no—
301C2×D3—361no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—41no—
307C3×C4—61no—
308C3×C4—61no—
309C3×C4—121no—
310C3×C4—121no—
311C3×C4—122no—
312C3×C4—181no—
313C3×C4—181no—
314C3×C4—361no—
315C3×C4—361no—
316C3×C4#379511yescharacteristic
317C3×C4#379511yescharacteristic
318Dic3—61no—
319Dic3—61no—
320Dic3—61no—
321Dic3—61no—
322Dic3—121no—
323Dic3—121no—
324Dic3—121no—
325Dic3—121no—
326Dic3—122no—
327Dic3—121no—
328Dic3—361no—
329C32—21no—
330C32#476811yescharacteristic, complemented
331C32#480011yescharacteristic, complemented
332C23—181no—
333Q8—181no—
334Q8—181no—
335D4—181no—
336D4—181no—
337C2×C4—21no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
343C8—181no—
344C8—181no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—61no—
352C2×C3—61no—
353C2×C3—61no—
354C2×C3—721no—
355C2×C3#6391311yescharacteristic
356C2×C3#6393611yescharacteristic
357C2×C3#6395011yescharacteristic
358D3—241no—
359D3—721no—
360C22—361no—
361C22—361no—
362C22—361no—
363C22#8867711yescharacteristic
364C4—41no—
365C4—181no—
366C4—181no—
367C4—361no—
368C4—361no—
369C4#8864811yescharacteristic
370C4#8864811yescharacteristic
371C3—31no—
372C3—31no—
373C3—61no—
374C3#1059053911yescharacteristic
375C2—721no—
376C2#1172114311yescharacteristic
377C2#1172119211yescharacteristic
378C2#1172123011yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#488
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721758C2semidirect
4. #63637⋊C3C2semidirect
5. #11720309C2semidirect
7. #11720302C2semidirect
26. #63637D3semidirect
222. H(3,𝔽3)#488semidirect
330. C32#4768semidirect
331. C32#4800semidirect

Maximal subgroups

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

subgroupindex
#117217582
#117211132
#63636⋊C32
#63637⋊C32
#117203092
#117203082
#117203022
#105905373
#105905333