Finite Group Databasebeta

← #13138024

Subgroups — #13138024

Order 1728.

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

normal subgroups only

This group has 357 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#64099⋊C3C211yesmaximal, characteristic
2#11721090C211yesmaximal, characteristic
3#11720532C211yesmaximal, characteristic
4#63637⋊C3C211yesmaximal, characteristic
5#11720326C211yesmaximal, characteristic
6#11720313C211yesmaximal, characteristic
7#11720312C211yesmaximal, characteristic
8#10590568—31nomaximal
9#10590562—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16#3750⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88448—21no—
21#64099D311yescharacteristic
22C3×#890—31no—
23#63873—31no—
24#63866—31no—
25#63637D311yescharacteristic
26C3×#810—31no—
27C3×#810—31no—
28C3×#810—31no—
29C3×#810—61no—
30#63567—31no—
31#63566—31no—
32#63566—31no—
33#63565—31no—
34#63555—31no—
35#63554—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39#6229—21no—
40#6229—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#634⋊C3—21no—
46#634⋊C3—21no—
47#6180—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5069—91no—
50#4797—91no—
51#4797—91no—
52C2×#634C2×D311yescharacteristic
53C2×C3×Dic6—31no—
54C2×C3×Dic6—31no—
55C2×C3×Dic6—31no—
56C2×C3×Dic6—61no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—21no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×Dic3⋊C4—61no—
68C3×C4×Dic3—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—61no—
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)#4111yescharacteristic
80C4×H(3,𝔽3)#4111yescharacteristic
81#1084—41no—
82(C3⋊Dic3)⋊C3—21no—
83(C3⋊Dic3)⋊C3—21no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86C3×#125—91no—
87#944—91no—
88#944—91no—
89#890—31no—
90#890—91no—
91#869—91no—
92C3×#103—91no—
93C3×#103—91no—
94#835—91no—
95#835—91no—
96#816—91no—
97#816—91no—
98#815—91no—
99#815—91no—
100#810—31no—
101#810—31no—
102#810—31no—
103#810—61no—
104#810—91no—
105#810—91no—
106#810—91no—
107#803—91no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—41no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3#6711yescharacteristic
117C2×C3⋊Dic3#6711yescharacteristic
118#634—21no—
119#634—21no—
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—121no—
127C3×Dic6—62no—
128C3×Dic6—61no—
129C3×Dic6—61no—
130C3×Dic6—61no—
131C3×Dic6—61no—
132C3×Dic6—61no—
133C3×Dic6—61no—
134C3×Dic6—121no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—61no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—121no—
140#491—271noSylow
141C2×H(3,𝔽3)#12411yescharacteristic
142C2×H(3,𝔽3)#13711yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144C2×C3×Q8—91no—
145C2×C3×Q8—91no—
146C2×Dic6—31no—
147C2×Dic6—31no—
148C2×Dic6—31no—
149C2×Dic6—61no—
150C2×Dic6—91no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4—181no—
159C3×C4⋊C4D3211yescharacteristic
160C3×C42—91no—
161C4⋊Dic3—31no—
162C4⋊Dic3—61no—
163C4⋊Dic3—91no—
164C4⋊Dic3—181no—
165C4⋊Dic3—181no—
166C4⋊Dic3—181no—
167Dic3⋊C4—61no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
172Dic3⋊C4—181no—
173C4×Dic3—31no—
174C4×Dic3—91no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—61no—
179C2×C3⋊C8—91no—
180C22×C32—21no—
181C22×C32D3×D411yescharacteristic
182C22×C32C2×#6711yescharacteristic
183C32×C4—21no—
184C32×C4—21no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—81no—
188C32×C4#23511yescharacteristic
189C32×C4#23711yescharacteristic
190C32×C4#23911yescharacteristic
191C32×C4#23911yescharacteristic
192C3⋊Dic3—21no—
193C3⋊Dic3—21no—
194C3⋊Dic3—41no—
195C3⋊Dic3—41no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—122no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C4⋊Q8—271no—
204#125—91no—
205#103—91no—
206#103—91no—
207#103—271no—
208#103—271no—
209#97—271no—
210H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
211C3×Q8—181no—
212C3×Q8—181no—
213C3×Q8—181no—
214C3×Q8—181no—
215C2×C3×C4—21no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—121no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4C2×D3211yescharacteristic
228C2×Dic3—31no—
229C2×Dic3—31no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—91no—
235C2×Dic3—91no—
236C2×Dic3—121no—
237C2×Dic3—181no—
238C2×Dic3—181no—
239Dic6—61no—
240Dic6—61no—
241Dic6—61no—
242Dic6—61no—
243Dic6—62no—
244Dic6—61no—
245Dic6—61no—
246Dic6—121no—
247Dic6—181no—
248Dic6—181no—
249C3×C8—181no—
250C3×C8—181no—
251C3⋊C8—61no—
252C3⋊C8—61no—
253C3⋊C8—61no—
254C3⋊C8—121no—
255C3⋊C8—181no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32#88511yescharacteristic
260C2×C32#91511yescharacteristic
261C2×C32#91811yescharacteristic
262C2×C32#94711yescharacteristic
263C2×C32#95111yescharacteristic
264C2×C32#95111yescharacteristic
265C2×Q8—91no—
266C2×Q8—91no—
267C2×C8—91no—
268C2×C8—91no—
269C4⋊C4—91no—
270C4⋊C4—181no—
271C4⋊C4—181no—
272C4⋊C4—541no—
273C4⋊C4—541no—
274C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C42—91no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—61no—
279C22×C3D3×#6711yescharacteristic
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—41no—
285C3×C4—61no—
286C3×C4—61no—
287C3×C4—121no—
288C3×C4—121no—
289C3×C4—181no—
290C3×C4—181no—
291C3×C4—241no—
292C3×C4—361no—
293C3×C4—361no—
294C3×C4—361no—
295C3×C4#379611yescharacteristic
296C3×C4#379911yescharacteristic
297Dic3—61no—
298Dic3—61no—
299Dic3—121no—
300Dic3—121no—
301Dic3—122no—
302Dic3—121no—
303Dic3—121no—
304Dic3—181no—
305Dic3—181no—
306Dic3—361no—
307Dic3—361no—
308C32—21no—
309C32#479711yescharacteristic, complemented
310C32#506911yescharacteristic, complemented
311Q8—181no—
312Q8—181no—
313Q8—181no—
314Q8—181no—
315C2×C4—21no—
316C2×C4—91no—
317C2×C4—91no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
322C8—181no—
323C8—181no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—61no—
331C2×C3—61no—
332C2×C3—61no—
333C2×C3#6390711yescharacteristic
334C2×C3#6392911yescharacteristic
335C2×C3#6394311yescharacteristic
336C22#8867611yescharacteristic
337C4—41no—
338C4—181no—
339C4—181no—
340C4—361no—
341C4—361no—
342C4—361no—
343C4#8864911yescharacteristic
344C4#8865211yescharacteristic
345C3—31no—
346C3—31no—
347C3—61no—
348C3#1059056211yescharacteristic
349C2#1172115511yescharacteristic
350C2#1172119911yescharacteristic
351C2#1172123711yescharacteristic
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×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#491
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
210. H(3,𝔽3)#491semidirect
309. C32#4797semidirect
310. C32#5069semidirect

Maximal subgroups

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

subgroupindex
#64099⋊C32
#117210902
#117205322
#63637⋊C32
#117203262
#117203132
#117203122
#105905683
#105905623