Finite Group Databasebeta

← #13137915

Subgroups — #13137915

Order 1728.

400 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 73 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#64106⋊C3C211yesmaximal, characteristic
2#11721129C211yesmaximal, characteristic
3#11720529C211yesmaximal, characteristic, complemented
4#63634⋊C3C211yesmaximal, characteristic, complemented
5#11720317C211yesmaximal, characteristic
6#11720313C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10590444—31nomaximal
9#10590440—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#64106D311yescharacteristic
22C3×#897—31no—
23#63900—31no—
24D3×C4⋊Dic3—31no—
25#63634D311yescharacteristic, complemented
26C3×#807—31no—
27C3×#807—31no—
28C3×#807—31no—
29C3×#807—61no—
30#63567—31no—
31#63567—31no—
32#63566—31no—
33#63565—31no—
34#63557—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4×(C3⋊D3)⋊C3—21no—
44C4×(C3⋊D3)⋊C3—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47((C3⋊D3)⋊C3)⋊C4—42no—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5044—91no—
51#4884—91no—
52#4884—91no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634C2×D311yescharacteristic
55C2×C3×C4×D3—31no—
56C2×C3×Dic6—31no—
57C2×#624—61no—
58C2×D3×Dic3—61no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C4⋊C4C2×D311yescharacteristic
62#3752—21no—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C3×(C2×D3)⋊C4—61no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—61no—
71C3×C4⋊Dic3—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76#3724—61no—
77Dic3⋊Dic3—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3Q1612yes—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85#1084—41no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89C3×C22⋊Q8—91no—
90C2×C4⋊Dic3—91no—
91#897—31no—
92#897—91no—
93D3×C4⋊C4—91no—
94D3×C4⋊C4—91no—
95C3×#107—91no—
96C3×#107—91no—
97#835—91no—
98#835—91no—
99#830—91no—
100(C2×D3)⋊C8—91no—
101(C2×D3)⋊C8—91no—
102#816—91no—
103#816—91no—
104#810—91no—
105#810—91no—
106#807—31no—
107#807—31no—
108#807—31no—
109#807—61no—
110#807—91no—
111C22×C3⋊D3#6711yescharacteristic
112C22×C3×D3—31no—
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
122C4×C3⋊D3—21no—
123C4×C3⋊D3—21no—
124#634—21no—
125#634—21no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—121no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136#624—122no—
137D3×Dic3—122no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—62no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143#484—271noSylow
144C2×H(3,𝔽3)#12311yescharacteristic
145C2×H(3,𝔽3)C2×Q1611yescharacteristic
146C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
147(C3⋊D3)⋊C3—22no—
148C2×C3×Q8—91no—
149C22×C3×C4—91no—
150C22×Dic3—181no—
151C2×C4×D3—31no—
152C2×C4×D3—91no—
153C2×C4×D3—181no—
154C2×C4×D3—181no—
155C2×Dic6—31no—
156C2×Dic6—91no—
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×C22⋊C4—181no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—61no—
173C4⋊Dic3—61no—
174C4⋊Dic3—91no—
175C4⋊Dic3—181no—
176C4⋊Dic3—181no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—61no—
183C2×C3⋊C8—91no—
184C22×C32—21no—
185C22×C32D3×D411yescharacteristic
186C22×C32C2×#6711yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3#21612yes—
190C2×C3×D3—32no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—41no—
196C32×C4—41no—
197C32×C4—81no—
198C32×C4#23511yescharacteristic
199C32×C4#23711yescharacteristic
200C32×C4#23711yescharacteristic
201C32×C4#23911yescharacteristic
202C3⋊Dic3—21no—
203C3⋊Dic3—41no—
204C3⋊Dic3—41no—
205C3×Dic3—61no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C3×Dic3—241no—
211C22⋊Q8—91no—
212C2×C4⋊C4—271no—
213#107—91no—
214#107—91no—
215#103—271no—
216#103—271no—
217C22⋊C8—271no—
218H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—31no—
221C22×D3—91no—
222C3×Q8—181no—
223C3×Q8—181no—
224C2×C3×C4—21no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—91no—
231C2×C3×C4—121no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4C2×D3211yescharacteristic
238C2×Dic3—31no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—91no—
244C2×Dic3—121no—
245C2×Dic3—181no—
246C2×Dic3—181no—
247C2×Dic3—362no—
248C4×D3—61no—
249C4×D3—61no—
250C4×D3—181no—
251C4×D3—181no—
252C4×D3—362no—
253C4×D3—362no—
254Dic6—61no—
255Dic6—61no—
256Dic6—181no—
257Dic6—181no—
258C3×C8—181no—
259C3×C8—181no—
260C3⋊C8—61no—
261C3⋊C8—61no—
262C3⋊C8—62no—
263C3⋊C8—61no—
264C3⋊C8—181no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32#89411yescharacteristic
269C2×C32#91111yescharacteristic
270C2×C32D3×Q1611yescharacteristic
271C2×C32#93511yescharacteristic
272C2×C32C2×#21611yescharacteristic
273C2×C32#94911yescharacteristic
274C3⋊D3—22no—
275C3×D3—62no—
276C2×Q8—91no—
277C22×C4—91no—
278C22×C4—541no—
279C2×C8—91no—
280C2×C8—91no—
281C4⋊C4—91no—
282C4⋊C4—91no—
283C4⋊C4—181no—
284C4⋊C4—541no—
285C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C22⋊C4—181no—
287C22×C3—31no—
288C22×C3—31no—
289C22×C3—61no—
290C22×C3—92no—
291C22×C3—181no—
292C22×C3—181no—
293C22×C3D3×#6711yescharacteristic
294C2×D3—32no—
295C2×D3—61no—
296C2×D3—61no—
297C2×D3—92no—
298C2×D3—181no—
299C2×D3—181no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—41no—
305C3×C4—61no—
306C3×C4—61no—
307C3×C4—121no—
308C3×C4—121no—
309C3×C4—181no—
310C3×C4—241no—
311C3×C4—361no—
312C3×C4—361no—
313C3×C4—361no—
314C3×C4#379511yescharacteristic
315C3×C4#379611yescharacteristic
316Dic3—61no—
317Dic3—121no—
318Dic3—121no—
319Dic3—121no—
320Dic3—121no—
321Dic3—181no—
322Dic3—241no—
323Dic3—361no—
324Dic3—361no—
325C32—21no—
326C32#488411yescharacteristic, complemented
327C32#504411yescharacteristic, complemented
328C23—91no—
329Q8—181no—
330Q8—181no—
331C2×C4—21no—
332C2×C4—91no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—1082no—
339C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
340C8—181no—
341C8—181no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—61no—
349C2×C3—61no—
350C2×C3—61no—
351C2×C3—182no—
352C2×C3#6388411yescharacteristic
353C2×C3#6392511yescharacteristic
354C2×C3D3×#21611yescharacteristic
355D3—62no—
356D3—182no—
357C22—92no—
358C22—181no—
359C22—181no—
360C22#8867611yescharacteristic
361C4—41no—
362C4—181no—
363C4—361no—
364C4—361no—
365C4—361no—
366C4#8864811yescharacteristic
367C4#8864911yescharacteristic
368C3—31no—
369C3—31no—
370C3—61no—
371C3#1059044011yescharacteristic
372C2—182no—
373C2#1172112511yescharacteristic
374C2#1172119511yescharacteristic
375C2#1172123611yescharacteristic
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 radicalC22×(C3⋊D3)⋊C3D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#484
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
3. #11720529C2semidirect
4. #63634⋊C3C2semidirect
6. #11720313C2semidirect
7. #11720309C2semidirect
25. #63634D3semidirect
218. H(3,𝔽3)#484semidirect
326. C32#4884semidirect
327. C32#5044semidirect

Maximal subgroups

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

subgroupindex
#64106⋊C32
#117211292
#117205292
#63634⋊C32
#117203172
#117203132
#117203092
#105904443
#105904403