Finite Group Databasebeta

← #13140903

Subgroups — #13140903

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#64153⋊C3C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic
3#11720583C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#11720313C211yesmaximal, characteristic
6#11720312C211yesmaximal, characteristic
7#11720309C211yesmaximal, characteristic
8#10591616—31nomaximal
9#10591613—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15#3750⋊C3—21no—
16#3750⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88449—21no—
20#88448—21no—
21#64153D311yescharacteristic
22C3×#944—31no—
23#63872—31no—
24#63869—31no—
25#63662D311yescharacteristic
26C3×#835—31no—
27C3×#835—31no—
28C3×#835—31no—
29C3×#835—61no—
30#63567—31no—
31#63567—31no—
32#63566—31no—
33#63565—31no—
34#63555—31no—
35#63554—31no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—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#5082—91no—
50#4797—91no—
51#4797—91no—
52C2×C32×Q8—21no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C32×Q8C2×D311yescharacteristic
55C2×#634C2×D311yescharacteristic
56C2×C3×Dic6—31no—
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58#3750—21no—
59#3750—21no—
60C4×C3⋊Dic3C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—61no—
66C3×Dic3⋊C4—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—
74Dic3⋊Dic3—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×C4⋊Q8—91no—
87#944—31no—
88#944—91no—
89#890—91no—
90#890—91no—
91#870—91no—
92C3×#103—91no—
93C3×#103—91no—
94#835—31no—
95#835—31no—
96#835—31no—
97#835—61no—
98#835—91no—
99#835—91no—
100#835—91no—
101#816—91no—
102#816—91no—
103#815—91no—
104#815—91no—
105#810—91no—
106#810—91no—
107#803—91no—
108C32×Q8—22no—
109C32×Q8—21no—
110C32×Q8—21no—
111C32×Q8—21no—
112C32×Q8—21no—
113C32×Q8—41no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3#6711yescharacteristic
123C2×C3⋊Dic3#6711yescharacteristic
124#634—21no—
125#634—21no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—121no—
133C3×Dic6—61no—
134C3×Dic6—61no—
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—31no—
145C2×C3×Q8—31no—
146C2×C3×Q8—61no—
147C2×C3×Q8—91no—
148C2×C3×Q8D3211yescharacteristic
149C2×Dic6—31no—
150C2×Dic6—91no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—91no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—181no—
156C3×C4⋊C4—181no—
157C3×C42—91no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—31no—
161C4⋊Dic3—61no—
162C4⋊Dic3—91no—
163C4⋊Dic3—181no—
164C4⋊Dic3—181no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167Dic3⋊C4—181no—
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—42no—
186C32×C4—41no—
187C32×C4—41no—
188C32×C4#23511yescharacteristic
189C32×C4#23711yescharacteristic
190C32×C4#23711yescharacteristic
191C32×C4#23711yescharacteristic
192C3⋊Dic3—21no—
193C3⋊Dic3—21no—
194C3⋊Dic3—41no—
195C3⋊Dic3—41no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—121no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—241no—
203C4⋊Q8—91no—
204#125—271no—
205#103—91no—
206#103—91no—
207#103—271no—
208#103—271no—
209#97—271no—
210H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
211C3×Q8—21no—
212C3×Q8—21no—
213C3×Q8—61no—
214C3×Q8—61no—
215C3×Q8—61no—
216C3×Q8—61no—
217C3×Q8—62no—
218C3×Q8—121no—
219C3×Q8—181no—
220C3×Q8—181no—
221C2×C3×C4—21no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—91no—
228C2×C3×C4—91no—
229C2×C3×C4—121no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4C2×D3211yescharacteristic
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—91no—
241C2×Dic3—91no—
242C2×Dic3—121no—
243C2×Dic3—181no—
244C2×Dic3—181no—
245Dic6—61no—
246Dic6—61no—
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#93611yescharacteristic
263C2×C32#94211yescharacteristic
264C2×C32#94211yescharacteristic
265C2×Q8—91no—
266C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
267C2×C8—91no—
268C2×C8—91no—
269C4⋊C4—91no—
270C4⋊C4—91no—
271C4⋊C4—181no—
272C4⋊C4—181no—
273C4⋊C4—541no—
274C4⋊C4—541no—
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—122no—
290C3×C4—181no—
291C3×C4—181no—
292C3×C4—361no—
293C3×C4—361no—
294C3×C4—361no—
295C3×C4#380411yescharacteristic
296C3×C4#380511yescharacteristic
297Dic3—61no—
298Dic3—61no—
299Dic3—121no—
300Dic3—121no—
301Dic3—121no—
302Dic3—121no—
303Dic3—181no—
304Dic3—181no—
305Dic3—241no—
306Dic3—361no—
307Dic3—361no—
308C32—21no—
309C32#479711yescharacteristic, complemented
310C32#508211yescharacteristic, complemented
311Q8—21no—
312Q8—21no—
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#6394611yescharacteristic
334C2×C3#6394811yescharacteristic
335C2×C3#6395911yescharacteristic
336C22#8867611yescharacteristic
337C4—41no—
338C4—181no—
339C4—181no—
340C4—361no—
341C4—361no—
342C4—361no—
343C4#8867111yescharacteristic
344C4#8867211yescharacteristic
345C3—31no—
346C3—31no—
347C3—61no—
348C3#1059161311yescharacteristic
349C2#1172123411yescharacteristic
350C2#1172123811yescharacteristic
351C2#1172130411yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×Q8((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 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
210. H(3,𝔽3)#491semidirect
309. C32#4797semidirect
310. C32#5082semidirect

Maximal subgroups

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

subgroupindex
#64153⋊C32
#117210932
#117205832
#63662⋊C32
#117203132
#117203122
#117203092
#105916163
#105916133