Finite Group Databasebeta

← #13137877

Subgroups — #13137877

Order 1728.

385 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 69 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#11721755C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic
3#63635⋊C3C211yesmaximal, characteristic
4#63634⋊C3C211yesmaximal, characteristic
5#11720326C211yesmaximal, characteristic
6#11720326C211yesmaximal, characteristic
7#11720304C211yesmaximal, characteristic
8#10590544—31nomaximal
9#10590543—31nomaximal
10C2×#6229C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12#88540—21no—
13C4×#1084—21no—
14C4×#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88448—21no—
21C3×#887—31no—
22C3×#887—31no—
23C3×#887—61no—
24#63869—31no—
25#63869—31no—
26#63635D311yescharacteristic
27#63634D311yescharacteristic
28C3×#808—31no—
29C3×#807—31no—
30Dic3⋊(C3⋊C8)—31no—
31Dic3⋊(C3⋊C8)—31no—
32#63566—31no—
33#63566—31no—
34#63565—31no—
35#63565—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084D411yescharacteristic
40C2×#1084D411yescharacteristic
41#6229—21no—
42#6229Q812yes—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45(C3⋊(C3⋊C8))⋊C3—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47#5022—91no—
48#4877—91no—
49#4857—91no—
50C2×C3×Dic6—31no—
51C2×C3×Dic6—31no—
52C2×C3×Dic6—61no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)C2×D311yescharacteristic
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—31no—
62C3×Dic3⋊C4—61no—
63C3×Dic3⋊C4—61no—
64C3×Dic3⋊C4—62no—
65C3×C4×Dic3—31no—
66C3×C4×Dic3—31no—
67C3×C4×Dic3—61no—
68C3×C4×Dic3—61no—
69C3×C4×Dic3—61no—
70C3×C4×Dic3—62no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73#3724—61no—
74#3724—61no—
75Dic3⋊Dic3—61no—
76Dic3⋊Dic3—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)C2×Q811yescharacteristic
80C4×H(3,𝔽3)#4111yescharacteristic
81#1084—22no—
82#1084—21no—
83#1084—21no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86C3×C4×Q8—91no—
87#890—91no—
88#890—91no—
89#887—31no—
90#887—31no—
91#887—61no—
92#870—91no—
93C3×#107—91no—
94C3×#106—91no—
95#835—91no—
96#835—91no—
97#816—91no—
98#816—91no—
99Dic3⋊C8—91no—
100Dic3⋊C8—91no—
101#810—91no—
102#810—91no—
103#808—31no—
104#808—91no—
105#807—31no—
106#807—91no—
107C4⋊(C3⋊C8)—91no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—21no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—31no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C3×Dic6—32no—
128C3×Dic6—32no—
129C3×Dic6—62no—
130C3×Dic6—62no—
131C3×Dic6—61no—
132C3×Dic6—61no—
133C3⋊(C3⋊C8)—21no—
134C3⋊(C3⋊C8)—21no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—61no—
137#479—271noSylow
138C2×H(3,𝔽3)C22⋊Q811yescharacteristic
139C2×H(3,𝔽3)#13511yescharacteristic
140C2×H(3,𝔽3)#13711yescharacteristic
141C2×C3×Q8—91no—
142C2×Dic6—31no—
143C2×Dic6—31no—
144C2×Dic6—61no—
145C2×C3×C8—91no—
146C2×C3×C8—91no—
147C3×C4⋊C4—31no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4D3211yescharacteristic
154C3×C42—91no—
155C3×C42—181no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—91no—
159C4⋊Dic3—91no—
160C4⋊Dic3—181no—
161C4⋊Dic3—181no—
162Dic3⋊C4—61no—
163Dic3⋊C4—61no—
164Dic3⋊C4—62no—
165Dic3⋊C4—181no—
166Dic3⋊C4—181no—
167Dic3⋊C4—181no—
168Dic3⋊C4—181no—
169C4×Dic3—31no—
170C4×Dic3—31no—
171C4×Dic3—61no—
172C4×Dic3—61no—
173C4×Dic3—61no—
174C4×Dic3—62no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—91no—
178C2×C3⋊C8—91no—
179C22×C32—21no—
180C22×C32D3×D411yescharacteristic
181C22×C32D3×D411yescharacteristic
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—42no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4#23511yescharacteristic
188C32×C4#23511yescharacteristic
189C32×C4D3×Q811yescharacteristic
190C32×C4D3×Q811yescharacteristic
191C3⋊Dic3—41no—
192C3⋊Dic3—41no—
193C3×Dic3—64no—
194C3×Dic3—62no—
195C3×Dic3—62no—
196C3×Dic3—62no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C3×Dic3—61no—
200C3×Dic3—61no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204#125—271no—
205C4×Q8—91no—
206#107—91no—
207#106—91no—
208C4⋊C8—271no—
209#103—271no—
210#103—271no—
211H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
212C3×Q8—92no—
213C3×Q8—181no—
214C2×C3×C4—21no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—62no—
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—31no—
230C2×Dic3—31no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—62no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—181no—
239C2×Dic3—181no—
240Dic6—32no—
241Dic6—32no—
242Dic6—62no—
243Dic6—61no—
244Dic6—61no—
245Dic6—62no—
246C3×C8—181no—
247C3×C8—181no—
248C3⋊C8—61no—
249C3⋊C8—61no—
250C3⋊C8—181no—
251C3⋊C8—181no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32#89611yescharacteristic
256C2×C32#89611yescharacteristic
257C2×C32#91211yescharacteristic
258C2×C32#91511yescharacteristic
259C2×C32#91611yescharacteristic
260C2×C32#91811yescharacteristic
261C2×Q8—91no—
262C2×C8—91no—
263C2×C8—91no—
264C4⋊C4—91no—
265C4⋊C4—91no—
266C4⋊C4—181no—
267C4⋊C4—541no—
268C4⋊C4—541no—
269C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C42—91no—
271C42—181no—
272C22×C3—31no—
273C22×C3—31no—
274C22×C3—61no—
275C22×C3#381111yescharacteristic
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—41no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—121no—
284C3×C4—121no—
285C3×C4—122no—
286C3×C4—182no—
287C3×C4—181no—
288C3×C4—181no—
289C3×C4—361no—
290C3×C4—361no—
291C3×C4#379711yescharacteristic
292C3×C4#379811yescharacteristic
293Dic3—62no—
294Dic3—62no—
295Dic3—61no—
296Dic3—61no—
297Dic3—61no—
298Dic3—61no—
299Dic3—64no—
300Dic3—62no—
301Dic3—121no—
302Dic3—121no—
303Dic3—121no—
304Dic3—361no—
305Dic3—361no—
306C32—21no—
307C32#485711yescharacteristic, complemented
308C32#487711yescharacteristic, complemented
309Q8—92no—
310Q8—181no—
311C2×C4—21no—
312C2×C4—91no—
313C2×C4—91no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
318C8—181no—
319C8—181no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—61no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3#6386511yescharacteristic
330C2×C3#6393611yescharacteristic
331C2×C3#6395111yescharacteristic
332C22#8867711yescharacteristic
333C4—41no—
334C4—181no—
335C4—182no—
336C4—181no—
337C4—361no—
338C4—361no—
339C4#8865011yescharacteristic
340C4#8865111yescharacteristic
341C3—31no—
342C3—31no—
343C3—61no—
344C3#1059043411yescharacteristic
345C2#1172109611yescharacteristic
346C2#1172119211yescharacteristic
347C2#1172123111yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#479
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
211. H(3,𝔽3)#479semidirect
307. C32#4857semidirect
308. C32#4877semidirect

Maximal subgroups

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

subgroupindex
#117217552
#117210952
#63635⋊C32
#63634⋊C32
#117203262
#117203042
#105905443
#105905433