Finite Group Databasebeta

← #13137993

Subgroups — #13137993

Order 1728.

428 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 75 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#64097⋊C3C211yesmaximal, characteristic, complemented
2#11721149C211yesmaximal, characteristic
3#11720531C211yesmaximal, characteristic
4#63634⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic
6#11720315C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10590471—31nomaximal
9#10590463—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#634⋊C3—21no—
12C2×#6199—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21#64097D311yescharacteristic, complemented
22C3×#888—31no—
23#63898—31no—
24Dic3×D12—31no—
25#63634D311yescharacteristic, complemented
26C3×#809—31no—
27C3×#809—31no—
28C3×#809—61no—
29C3×#807—31no—
30#63575—31no—
31#63575—31no—
32#63572—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63563—31no—
35#63562—31no—
36C22×#272—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C4⋊#272—21no—
40C4⋊#272Q812yescomplemented
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#634⋊C3—42no—
46#6199—42no—
47#6180—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5087—91no—
50#4890—91no—
51#4759—91no—
52C2×#634—21no—
53C2×C3×D12—31no—
54C2×C3×D12—31no—
55C2×C3×D12—61no—
56C2×C3×Dic6—61no—
57C2×D3×Dic3—61no—
58C2×D3×Dic3—61no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—21no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×Dic3⋊C4—61no—
70C3×C4×Dic3—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—61no—
75(C2×D3)⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×#272—22no—
79C2×#272—41no—
80C2×#272—41no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)C2×Q811yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3D812yes—
88C3×C4⋊Q8—91no—
89D4×Dic3—91no—
90#893—91no—
91#888—31no—
92#888—91no—
93C4×D12—91no—
94C3×#107—91no—
95C3×#102—91no—
96#832—91no—
97#821—91no—
98#818—91no—
99Dic3⋊C8—91no—
100Dic3⋊C8—91no—
101#809—31no—
102#809—31no—
103#809—61no—
104#809—91no—
105#807—31no—
106#807—91no—
107#807—91no—
108#807—91no—
109C4⋊(C3⋊C8)—91no—
110C22×C3×D3—61no—
111C22×C3×D3—61no—
112C22×C3×D3—121no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3⋊Dic3#6711yescharacteristic
123#634—42no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—32no—
129C3×D12—32no—
130C3×D12—62no—
131C3×D12—61no—
132C3×D12—61no—
133C3×D12—121no—
134C3×Dic6—122no—
135D3×Dic3—122no—
136D3×Dic3—122no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—121no—
142#474—271noSylow
143C2×H(3,𝔽3)C22⋊Q811yescharacteristic
144C2×H(3,𝔽3)C2×D811yescharacteristic
145C2×H(3,𝔽3)#13711yescharacteristic
146#272—42no—
147C2×C3×Q8—181no—
148C2×C3×D4—91no—
149C22×Dic3—181no—
150C2×D12—31no—
151C2×D12—31no—
152C2×D12—61no—
153C2×C4×D3—181no—
154C2×C4×D3—181no—
155C2×Dic6—61no—
156C2×Dic6—181no—
157C2×C3×C8—91no—
158C2×C3×C8—91no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—181no—
165C3×C4⋊C4D3211yescharacteristic
166C3×C42—91no—
167#217—181no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—91no—
173C4⋊Dic3—91no—
174Dic3⋊C4—61no—
175Dic3⋊C4—181no—
176C4×Dic3—31no—
177C4×Dic3—91no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32D3×D411yescharacteristic
185C22×C32C2×#6711yescharacteristic
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C2×C3×D3—122no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C2×C3×D3—241no—
194C2×C3×D3—241no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—81no—
200C32×C4#23511yescharacteristic
201C32×C4D3×Q811yescharacteristic
202C32×C4D3×Q811yescharacteristic
203C32×C4#23911yescharacteristic
204C3⋊Dic3—21no—
205C3⋊Dic3—41no—
206C3⋊Dic3—41no—
207C3⋊Dic3#21312yes—
208C3×Dic3—32no—
209C3×Dic3—61no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C4⋊Q8—91no—
213C4×D4—271no—
214#107—91no—
215#107—271no—
216C4⋊C8—271no—
217#102—91no—
218#102—271no—
219H(3,𝔽3)#47411yescharacteristic, complemented, Sylow
220C23×C3—181no—
221C22×D3—61no—
222C22×D3—61no—
223C22×D3—121no—
224C3×Q8—362no—
225C3×D4—92no—
226C3×D4—181no—
227C2×C3×C4—21no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4—121no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4C2×D3211yescharacteristic
239C2×Dic3—31no—
240C2×Dic3—31no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—91no—
244C2×Dic3—91no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—362no—
248D12—32no—
249D12—32no—
250D12—62no—
251D12—61no—
252D12—61no—
253D12—121no—
254C4×D3—362no—
255C4×D3—362no—
256Dic6—122no—
257Dic6—362no—
258C3×C8—181no—
259C3×C8—181no—
260C3⋊C8—61no—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C3⋊C8—121no—
264C3⋊C8—181no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32#89611yescharacteristic
269C2×C32D3×D811yescharacteristic
270C2×C32#91811yescharacteristic
271C2×C32C2×#21311yescharacteristic
272C2×C32#94211yescharacteristic
273C2×C32#94611yescharacteristic
274C3×D3—122no—
275C3×D3—122no—
276C3×D3—242no—
277C2×Q8—181no—
278C2×D4—91no—
279C22×C4—541no—
280C2×C8—91no—
281C2×C8—91no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4—181no—
285C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C22⋊C4—541no—
287C42—91no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—61no—
291C22×C3—182no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3D3×#6711yescharacteristic
295C2×D3—62no—
296C2×D3—62no—
297C2×D3—122no—
298C2×D3—121no—
299C2×D3—121no—
300C2×D3—121no—
301C2×D3—121no—
302C2×D3—241no—
303C2×D3—241no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—41no—
309C3×C4—61no—
310C3×C4—61no—
311C3×C4—92no—
312C3×C4—121no—
313C3×C4—121no—
314C3×C4—181no—
315C3×C4—241no—
316C3×C4—361no—
317C3×C4—361no—
318C3×C4#379711yescharacteristic
319C3×C4#379911yescharacteristic
320Dic3—32no—
321Dic3—61no—
322Dic3—92no—
323Dic3—121no—
324Dic3—121no—
325Dic3—181no—
326Dic3—361no—
327Dic3—361no—
328C32—21no—
329C32#489011yescharacteristic, complemented
330C32#508711yescharacteristic, complemented
331C23—181no—
332Q8—362no—
333D4—92no—
334D4—181no—
335C2×C4—21no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—1082no—
341C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C8—181no—
343C8—181no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—61no—
351C2×C3—61no—
352C2×C3—61no—
353C2×C3—362no—
354C2×C3#6388811yescharacteristic
355C2×C3D3×#21311yescharacteristic
356C2×C3#6394811yescharacteristic
357D3—122no—
358D3—122no—
359D3—242no—
360C22—182no—
361C22—361no—
362C22—361no—
363C22#8867611yescharacteristic
364C4—41no—
365C4—92no—
366C4—181no—
367C4—361no—
368C4—361no—
369C4#8865111yescharacteristic
370C4#8865211yescharacteristic
371C3—31no—
372C3—31no—
373C3—61no—
374C3#1059046411yescharacteristic
375C2—362no—
376C2#1172114811yescharacteristic
377C2#1172119411yescharacteristic
378C2#1172123411yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊#272C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#474
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. #64097⋊C3C2semidirect
4. #63634⋊C3C2semidirect
6. #11720315C2semidirect
7. #11720311C2semidirect
21. #64097D3semidirect
25. #63634D3semidirect
40. C4⋊#272Q8semidirect
219. H(3,𝔽3)#474semidirect
329. C32#4890semidirect
330. C32#5087semidirect

Maximal subgroups

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

subgroupindex
#64097⋊C32
#117211492
#117205312
#63634⋊C32
#117203242
#117203152
#117203112
#105904713
#105904633