Finite Group Databasebeta

← #13138843

Subgroups — #13138843

Order 1728.

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

normal subgroups only

This group has 414 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
1#11721132C211yesmaximal, characteristic
2#11721116C211yesmaximal, characteristic, complemented
3#11720551C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic, complemented
8#10590778—31nomaximal
9#10590764—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4—21no—
12C2×#6196—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88448—21no—
21#63919—31no—
22#63918—31no—
23#63898—31no—
24#63894—31no—
25#63648D311yescharacteristic
26#63641D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#818—31no—
29C3×#818—31no—
30C3×#818—61no—
31C3×Dic3⋊C8—31no—
32#63576—31no—
33#63575—31no—
34#63564—31no—
35#63562—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)Q812yescomplemented
45((C3⋊D3)⋊C3)⋊C4—42no—
46#6196—42no—
47C8×H(3,𝔽3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#4890—91no—
50#4846—91no—
51#4759—91no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C3×D12—31no—
54C2×#627—61no—
55C2×#627—61no—
56C2×#624—61no—
57C2×D3×Dic3—61no—
58C2×C32×C8—21no—
59C2×C32×C8C2×D311yescharacteristic
60C2×C32×C8C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×C4⋊Dic3—61no—
71C3×C4×Dic3—31no—
72C2×C3×C3⋊C8—31no—
73#3724—61no—
74Dic3⋊Dic3—61no—
75#3722—61no—
76(C2×D3)⋊Dic3—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—22no—
79C2×(C3⋊D3)⋊C3—41no—
80C2×(C3⋊D3)⋊C3—41no—
81C4×H(3,𝔽3)C2×Q811yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—41no—
84#1084—41no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3D812yes—
88D4×Dic3—91no—
89#893—91no—
90#888—91no—
91#888—91no—
92C4×D12—91no—
93#869—91no—
94C3×#107—91no—
95C3×C4⋊C8—91no—
96C3×#102—91no—
97#832—91no—
98#821—31no—
99#821—91no—
100#821—91no—
101#818—31no—
102#818—31no—
103#818—61no—
104#818—91no—
105Dic3⋊C8—31no—
106Dic3⋊C8—91no—
107#809—91no—
108#807—91no—
109#807—91no—
110C22×C3⋊D3—21no—
111C22×C3×D3—61no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3D1211yescharacteristic
117C2×C3⋊Dic3D1211yescharacteristic
118C4⋊(C3⋊D3)—21no—
119C4⋊(C3⋊D3)Dic612yescomplemented
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—121no—
128C2×C3×Dic3—121no—
129C3×D12—32no—
130C3×D12—61no—
131#627—122no—
132#627—122no—
133#624—122no—
134D3×Dic3—122no—
135C32×C8—22no—
136C32×C8—21no—
137C32×C8—21no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—61no—
140#474—271noSylow
141C2×H(3,𝔽3)C22⋊Q811yescharacteristic
142C2×H(3,𝔽3)C2×D811yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144(C3⋊D3)⋊C3—42no—
145C2×C3×D4—91no—
146C22×Dic3—181no—
147C2×D12—31no—
148C2×D12—91no—
149C2×C4×D3—181no—
150C2×C4×D3—181no—
151C2×Dic6—181no—
152C2×Dic6—181no—
153C2×Dic6—181no—
154C2×C3×C8—31no—
155C2×C3×C8—31no—
156C2×C3×C8—61no—
157C2×C3×C8—91no—
158C2×C3×C8D3211yescharacteristic
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—91no—
162C3×C42—91no—
163#217—181no—
164(C2×D3)⋊C4—181no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—61no—
172C4⋊Dic3—61no—
173C4⋊Dic3—91no—
174C4⋊Dic3—181no—
175Dic3⋊C4—181no—
176Dic3⋊C4—181no—
177C4×Dic3—31no—
178C4×Dic3—91no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—91no—
181C22×C32—21no—
182C22×C32C2×D1211yescharacteristic
183C22×C32D3×D411yescharacteristic
184C2×C3⋊D3—22no—
185C2×C3⋊D3—41no—
186C2×C3⋊D3—41no—
187C2×C3×D3—62no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4C2×Dic611yescharacteristic
193C32×C4#23711yescharacteristic
194C32×C4#23711yescharacteristic
195C32×C4D3×Q811yescharacteristic
196C3⋊Dic3—21no—
197C3⋊Dic3—41no—
198C3⋊Dic3D2412yes—
199C3×Dic3—32no—
200C3×Dic3—61no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—241no—
207C3×Dic3—241no—
208C4⋊Q8—271no—
209C4×D4—271no—
210#107—91no—
211#107—271no—
212C4⋊C8—91no—
213#102—91no—
214#102—271no—
215H(3,𝔽3)#47411yescharacteristic, complemented, Sylow
216C23×C3—181no—
217C22×D3—61no—
218C22×D3—181no—
219C3×D4—92no—
220C3×D4—181no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—31no—
231C2×Dic3—31no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—91no—
238C2×Dic3—91no—
239C2×Dic3—121no—
240C2×Dic3—121no—
241C2×Dic3—181no—
242C2×Dic3—362no—
243D12—32no—
244D12—61no—
245D12—92no—
246D12—181no—
247C4×D3—362no—
248C4×D3—362no—
249Dic6—362no—
250Dic6—362no—
251Dic6—362no—
252C3×C8—21no—
253C3×C8—61no—
254C3×C8—61no—
255C3×C8—62no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32#87811yescharacteristic
263C2×C32#89711yescharacteristic
264C2×C32C2×D2411yescharacteristic
265C2×C32#90911yescharacteristic
266C2×C32D3×D811yescharacteristic
267C2×C32#91511yescharacteristic
268C3⋊D3—42no—
269C3×D3—122no—
270C2×Q8—541no—
271C2×D4—91no—
272C22×C4—541no—
273C2×C8—91no—
274C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—541no—
279C22⋊C4—541no—
280C42—91no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—61no—
284C22×C3—182no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3D3×D1211yescharacteristic
288C2×D3—62no—
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—182no—
292C2×D3—361no—
293C2×D3—361no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—61no—
299C3×C4—61no—
300C3×C4—92no—
301C3×C4—181no—
302C3×C4—361no—
303C3×C4—361no—
304C3×C4—361no—
305C3×C4D3×Dic611yescharacteristic
306C3×C4#380511yescharacteristic
307Dic3—32no—
308Dic3—61no—
309Dic3—92no—
310Dic3—121no—
311Dic3—121no—
312Dic3—121no—
313Dic3—121no—
314Dic3—121no—
315Dic3—181no—
316Dic3—241no—
317Dic3—241no—
318Dic3—361no—
319C32—21no—
320C32#475911yescharacteristic, complemented
321C32#484611yescharacteristic, complemented
322C23—181no—
323Q8—1082no—
324D4—92no—
325D4—181no—
326C2×C4—91no—
327C2×C4—91no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—1082no—
332C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C8—21no—
334C8—181no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—362no—
345C2×C3D3×D2411yescharacteristic
346C2×C3#6380011yescharacteristic
347C2×C3#6389311yescharacteristic
348D3—122no—
349D3—362no—
350C22—182no—
351C22—361no—
352C22—361no—
353C22#8865411yescharacteristic
354C4—92no—
355C4—181no—
356C4—361no—
357C4—361no—
358C4—361no—
359C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
360C4#8867211yescharacteristic
361C3—31no—
362C3—31no—
363C3—61no—
364C3#1059076211yescharacteristic
365C2—362no—
366C2#1172084711yescharacteristic
367C2#1172085211yescharacteristic
368C2#1172113311yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×C8((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 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721116C2semidirect
3. #11720551C2semidirect
5. #63641⋊C3C2semidirect
7. #11720306C2semidirect
26. #63641D3semidirect
44. C4⋊((C3⋊D3)⋊C3)Q8semidirect
119. C4⋊(C3⋊D3)Dic6semidirect
215. H(3,𝔽3)#474semidirect
320. C32#4759semidirect
321. C32#4846semidirect

Maximal subgroups

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

subgroupindex
#117211322
#117211162
#117205512
#63648⋊C32
#63641⋊C32
#117203082
#117203062
#105907783
#105907643