Finite Group Databasebeta

← #13138791

Subgroups — #13138791

Order 1728.

395 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721136C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic, complemented
3#11720547C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic
7#11720315C211yesmaximal, characteristic, complemented
8#10590772—31nomaximal
9#10590769—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4×#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#88450—21no—
19#88448—21no—
20#88448—21no—
21#63894—31no—
22Dic3×D12—31no—
23#63869—31no—
24#63869—31no—
25#63648D311yescharacteristic
26#63644D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#817—31no—
29C3×Dic3⋊C8—31no—
30C3×Dic3⋊C8—31no—
31C3×Dic3⋊C8—61no—
32#63575—31no—
33#63575—31no—
34#63564—31no—
35#63563—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084D411yescharacteristic
39C2×#1084D411yescharacteristic
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)Q812yes—
45((C3⋊D3)⋊C3)⋊C4—42no—
46C8×H(3,𝔽3)—21no—
47#6180—21no—
48#4871—91no—
49#4847—91no—
50#4763—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⋊Dic3)C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)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—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—61no—
70#3724—61no—
71#3724—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—21no—
83#1084—21no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87D4×Dic3—91no—
88#893—91no—
89#890—91no—
90#890—91no—
91C4×D12—91no—
92#870—91no—
93C3×#106—91no—
94C3×C4⋊C8—91no—
95C3×#102—91no—
96#832—91no—
97#821—31no—
98#821—91no—
99#821—91no—
100#818—91no—
101#817—31no—
102#817—91no—
103Dic3⋊C8—31no—
104Dic3⋊C8—31no—
105Dic3⋊C8—61no—
106#809—91no—
107#807—91no—
108#807—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⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C4⋊(C3⋊D3)—21no—
118C4⋊(C3⋊D3)Dic612yes—
119C2×C3×Dic3—31no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—32no—
129C3×D12—61no—
130#624—122no—
131D3×Dic3—122no—
132C32×C8—21no—
133C32×C8—21no—
134C32×C8—41no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—61no—
137C3×C3⋊C8—121no—
138#478—271noSylow
139C2×H(3,𝔽3)C22⋊Q811yescharacteristic
140C2×H(3,𝔽3)#13511yescharacteristic
141C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
142(C3⋊D3)⋊C3—42no—
143C2×C3×D4—91no—
144C22×Dic3—181no—
145C2×D12—31no—
146C2×D12—91no—
147C2×C4×D3—181no—
148C2×C4×D3—181no—
149C2×C3×C8—31no—
150C2×C3×C8—31no—
151C2×C3×C8—61no—
152C2×C3×C8—91no—
153C2×C3×C8D3211yescharacteristic
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4—91no—
157C3×C42—91no—
158#217—181no—
159(C2×D3)⋊C4—181no—
160(C2×D3)⋊C4—181no—
161C4⋊Dic3—31no—
162C4⋊Dic3—31no—
163C4⋊Dic3—31no—
164C4⋊Dic3—91no—
165C4⋊Dic3—91no—
166C4⋊Dic3—91no—
167C4⋊Dic3—181no—
168C4⋊Dic3—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173C4×Dic3—31no—
174C4×Dic3—31no—
175C4×Dic3—61no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—61no—
179C22×C32—21no—
180C22×C32D3×D411yescharacteristic
181C22×C32D3×D411yescharacteristic
182C2×C3⋊D3—22no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—62no—
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4C2×Dic611yescharacteristic
191C32×C4#23511yescharacteristic
192C32×C4#23711yescharacteristic
193C32×C4D3×Q811yescharacteristic
194C3⋊Dic3—41no—
195C3⋊Dic3—41no—
196C3⋊Dic3—41no—
197C3×Dic3—62no—
198C3×Dic3—61no—
199C3×Dic3—61no—
200C3×Dic3—61no—
201C3×Dic3—61no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206#125—271no—
207C4×D4—271no—
208#107—271no—
209#106—91no—
210C4⋊C8—91no—
211#102—91no—
212#102—271no—
213H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—61no—
216C22×D3—181no—
217C3×D4—92no—
218C3×D4—181no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—61no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4C2×D3211yescharacteristic
228C2×Dic3—31no—
229C2×Dic3—31no—
230C2×Dic3—31no—
231C2×Dic3—31no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—181no—
238C2×Dic3—181no—
239C2×Dic3—181no—
240C2×Dic3—362no—
241D12—32no—
242D12—61no—
243D12—92no—
244D12—181no—
245C4×D3—362no—
246C4×D3—362no—
247C3×C8—21no—
248C3×C8—61no—
249C3×C8—61no—
250C3×C8—121no—
251C3×C8—181no—
252C3⋊C8—61no—
253C3⋊C8—61no—
254C3⋊C8—121no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32#87811yescharacteristic
259C2×C32#89611yescharacteristic
260C2×C32#91111yescharacteristic
261C2×C32#91411yescharacteristic
262C2×C32#91611yescharacteristic
263C2×C32#91611yescharacteristic
264C3⋊D3—42no—
265C3×D3—122no—
266C2×D4—91no—
267C22×C4—541no—
268C2×C8—91no—
269C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C4⋊C4—91no—
271C4⋊C4—91no—
272C4⋊C4—91no—
273C4⋊C4—541no—
274C4⋊C4—541no—
275C22⋊C4—541no—
276C42—91no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—61no—
280C22×C3—182no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3#380211yescharacteristic
284C2×D3—62no—
285C2×D3—121no—
286C2×D3—121no—
287C2×D3—182no—
288C2×D3—361no—
289C2×D3—361no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—31no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—181no—
297C3×C4—181no—
298C3×C4—361no—
299C3×C4—361no—
300C3×C4—361no—
301C3×C4D3×Dic611yescharacteristic
302C3×C4#380411yescharacteristic
303Dic3—61no—
304Dic3—61no—
305Dic3—61no—
306Dic3—61no—
307Dic3—62no—
308Dic3—121no—
309Dic3—121no—
310Dic3—121no—
311Dic3—121no—
312Dic3—361no—
313Dic3—361no—
314Dic3—361no—
315C32—21no—
316C32#476311yescharacteristic, complemented
317C32#487111yescharacteristic, complemented
318C23—181no—
319D4—92no—
320D4—181no—
321C2×C4—91no—
322C2×C4—91no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4—181no—
326C2×C4—1082no—
327C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
328C8—21no—
329C8—181no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—61no—
337C2×C3—61no—
338C2×C3—61no—
339C2×C3—362no—
340C2×C3#6379811yescharacteristic
341C2×C3#6380911yescharacteristic
342C2×C3#6389911yescharacteristic
343D3—122no—
344D3—362no—
345C22—182no—
346C22—361no—
347C22—361no—
348C22#8865511yescharacteristic
349C4—181no—
350C4—181no—
351C4—361no—
352C4—361no—
353C4—361no—
354C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
355C4#8867111yescharacteristic
356C3—31no—
357C3—31no—
358C3—61no—
359C3#1059078211yescharacteristic
360C2—362no—
361C2#1172083911yescharacteristic
362C2#1172084311yescharacteristic
363C2#1172113411yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
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. #11721095C2semidirect
3. #11720547C2semidirect
5. #63644⋊C3C2semidirect
7. #11720315C2semidirect
26. #63644D3semidirect
213. H(3,𝔽3)#478semidirect
316. C32#4763semidirect
317. C32#4871semidirect

Maximal subgroups

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

subgroupindex
#117211362
#117210952
#117205472
#63648⋊C32
#63644⋊C32
#117203182
#117203152
#105907723
#105907693