Finite Group Databasebeta

← #13138845

Subgroups — #13138845

Order 1728.

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

normal subgroups only

This group has 393 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#11721132C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic, complemented
3#11720551C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic
7#11720307C211yesmaximal, characteristic, complemented
8#10590780—31nomaximal
9#10590774—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88450—21no—
19#88449—21no—
20#88448—21no—
21#63898—31no—
22#63894—31no—
23#63872—31no—
24#63869—31no—
25#63648D311yescharacteristic
26#63641D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#818—31no—
29C3×#818—31no—
30C3×#818—61no—
31C3×Dic3⋊C8—31no—
32#63574—31no—
33#63573—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)Q812yes—
45((C3⋊D3)⋊C3)⋊C4—42no—
46C8×H(3,𝔽3)—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#4891—91no—
49#4847—91no—
50#4766—91no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C3×D12—31no—
53C2×#624—61no—
54C2×D3×Dic3—61no—
55C2×C32×C8—21no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C4×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×C4⋊Dic3—61no—
68C3×C4×Dic3—31no—
69C2×C3×C3⋊C8—31no—
70#3724—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74#3722—61no—
75(C2×D3)⋊Dic3—61no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C2×(C3⋊D3)⋊C3—22no—
78C2×(C3⋊D3)⋊C3—41no—
79C2×(C3⋊D3)⋊C3—41no—
80C4×H(3,𝔽3)C2×Q811yescharacteristic
81C4×H(3,𝔽3)#4111yescharacteristic
82#1084—41no—
83#1084—41no—
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87D4×Dic3—91no—
88#893—91no—
89#890—91no—
90#890—91no—
91C4×D12—91no—
92#870—91no—
93C3×#107—91no—
94C3×C4⋊C8—91no—
95C3×#102—91no—
96#832—91no—
97#821—31no—
98#821—91no—
99#821—91no—
100#818—31no—
101#818—31no—
102#818—61no—
103#817—91no—
104Dic3⋊C8—31no—
105Dic3⋊C8—91no—
106#809—91no—
107#808—91no—
108#808—91no—
109C22×C3⋊D3—21no—
110C22×C3×D3—61no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3D1211yescharacteristic
116C2×C3⋊Dic3D1211yescharacteristic
117C4⋊(C3⋊D3)—21no—
118C4⋊(C3⋊D3)Dic612yes—
119C2×C3×Dic3—31no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—121no—
127C2×C3×Dic3—121no—
128C3×D12—32no—
129C3×D12—61no—
130#624—122no—
131D3×Dic3—122no—
132C32×C8—22no—
133C32×C8—21no—
134C32×C8—21no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—61no—
137#478—271noSylow
138C2×H(3,𝔽3)C22⋊Q811yescharacteristic
139C2×H(3,𝔽3)#13511yescharacteristic
140C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
141(C3⋊D3)⋊C3—42no—
142C2×C3×D4—91no—
143C22×Dic3—181no—
144C2×D12—31no—
145C2×D12—91no—
146C2×C4×D3—181no—
147C2×C4×D3—181no—
148C2×C3×C8—31no—
149C2×C3×C8—31no—
150C2×C3×C8—61no—
151C2×C3×C8—91no—
152C2×C3×C8D3211yescharacteristic
153C3×C4⋊C4—91no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156C3×C42—91no—
157#217—181no—
158(C2×D3)⋊C4—181no—
159(C2×D3)⋊C4—181no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4⋊Dic3—31no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—61no—
166C4⋊Dic3—61no—
167C4⋊Dic3—91no—
168C4⋊Dic3—181no—
169C4⋊Dic3—181no—
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174C4×Dic3—31no—
175C4×Dic3—91no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—91no—
178C22×C32—21no—
179C22×C32C2×D1211yescharacteristic
180C22×C32D3×D411yescharacteristic
181C2×C3⋊D3—22no—
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3×D3—62no—
185C2×C3×D3—121no—
186C2×C3×D3—121no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4C2×Dic611yescharacteristic
190C32×C4#23711yescharacteristic
191C32×C4#23711yescharacteristic
192C32×C4D3×Q811yescharacteristic
193C3⋊Dic3—21no—
194C3⋊Dic3—21no—
195C3⋊Dic3—41no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—121no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C3×Dic3—241no—
204C3×Dic3—241no—
205#125—271no—
206C4×D4—271no—
207#107—91no—
208#106—271no—
209C4⋊C8—91no—
210#102—91no—
211#102—271no—
212H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
213C23×C3—181no—
214C22×D3—61no—
215C22×D3—181no—
216C3×D4—92no—
217C3×D4—181no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—61no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4C2×D3211yescharacteristic
227C2×Dic3—31no—
228C2×Dic3—31no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—91no—
235C2×Dic3—91no—
236C2×Dic3—121no—
237C2×Dic3—121no—
238C2×Dic3—181no—
239C2×Dic3—362no—
240D12—32no—
241D12—61no—
242D12—92no—
243D12—181no—
244C4×D3—362no—
245C4×D3—362no—
246C3×C8—21no—
247C3×C8—61no—
248C3×C8—61no—
249C3×C8—62no—
250C3×C8—181no—
251C3⋊C8—61no—
252C3⋊C8—181no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32#87811yescharacteristic
257C2×C32#89711yescharacteristic
258C2×C32#90411yescharacteristic
259C2×C32#90811yescharacteristic
260C2×C32#91211yescharacteristic
261C2×C32#91411yescharacteristic
262C3⋊D3—42no—
263C3×D3—122no—
264C2×D4—91no—
265C22×C4—541no—
266C2×C8—91no—
267C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
268C4⋊C4—91no—
269C4⋊C4—91no—
270C4⋊C4—91no—
271C4⋊C4—541no—
272C4⋊C4—541no—
273C22⋊C4—541no—
274C42—91no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—61no—
278C22×C3—182no—
279C22×C3—361no—
280C22×C3—361no—
281C22×C3D3×D1211yescharacteristic
282C2×D3—62no—
283C2×D3—121no—
284C2×D3—121no—
285C2×D3—182no—
286C2×D3—361no—
287C2×D3—361no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—61no—
293C3×C4—61no—
294C3×C4—181no—
295C3×C4—181no—
296C3×C4—361no—
297C3×C4—361no—
298C3×C4—361no—
299C3×C4D3×Dic611yescharacteristic
300C3×C4#380511yescharacteristic
301Dic3—61no—
302Dic3—61no—
303Dic3—121no—
304Dic3—121no—
305Dic3—121no—
306Dic3—121no—
307Dic3—121no—
308Dic3—181no—
309Dic3—181no—
310Dic3—241no—
311Dic3—241no—
312Dic3—361no—
313C32—21no—
314C32#476611yescharacteristic, complemented
315C32#484711yescharacteristic, complemented
316C23—181no—
317D4—92no—
318D4—181no—
319C2×C4—91no—
320C2×C4—91no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4—1082no—
325C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C8—21no—
327C8—181no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—61no—
337C2×C3—362no—
338C2×C3#6379411yescharacteristic
339C2×C3#6380711yescharacteristic
340C2×C3#6389311yescharacteristic
341D3—122no—
342D3—362no—
343C22—182no—
344C22—361no—
345C22—361no—
346C22#8865411yescharacteristic
347C4—181no—
348C4—181no—
349C4—361no—
350C4—361no—
351C4—361no—
352C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
353C4#8867211yescharacteristic
354C3—31no—
355C3—31no—
356C3—61no—
357C3#1059076711yescharacteristic
358C2—362no—
359C2#1172084811yescharacteristic
360C2#1172085511yescharacteristic
361C2#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)#478
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. #11721093C2semidirect
3. #11720551C2semidirect
5. #63641⋊C3C2semidirect
7. #11720307C2semidirect
26. #63641D3semidirect
212. H(3,𝔽3)#478semidirect
314. C32#4766semidirect
315. C32#4847semidirect

Maximal subgroups

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

subgroupindex
#117211322
#117210932
#117205512
#63648⋊C32
#63641⋊C32
#117203082
#117203072
#105907803
#105907743