Finite Group Databasebeta

← #13140898

Subgroups — #13140898

Order 1728.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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
1Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2#11721116C211yesmaximal, characteristic
3#11720583C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#11720315C211yesmaximal, characteristic
6#11720311C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic
8#10591631—31nomaximal
9#10591546—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6196—21no—
12C4⋊#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21Q8×C3⋊Dic3D311yescharacteristic
22C3×Q8×Dic3—31no—
23#63919—31no—
24#63918—31no—
25#63662D311yescharacteristic
26C3×#835—31no—
27C3×#835—31no—
28C3×#835—31no—
29C3×#835—61no—
30#63576—31no—
31#63575—31no—
32#63575—31no—
33#63575—31no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)Q812yescomplemented
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#6196—42no—
46#6180—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#5021—91no—
49#4793—91no—
50#4793—91no—
51C2×C32×Q8—21no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C32×Q8C2×D311yescharacteristic
54C2×#627—61no—
55C2×#627—61no—
56C4⋊(C3⋊Dic3)—21no—
57C4⋊(C3⋊Dic3)C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D311yescharacteristic
59C4×C3⋊Dic3—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—31no—
66C3×C4⋊Dic3—61no—
67C3×C4⋊Dic3—61no—
68C3×C4×Dic3—31no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—61no—
74#3724—61no—
75Dic3⋊Dic3—61no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)—22no—
78C4×H(3,𝔽3)C2×Q811yescharacteristic
79C4×H(3,𝔽3)#4111yescharacteristic
80#1084—41no—
81(C3⋊Dic3)⋊C3—21no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84(C3⋊Dic3)⋊C3Q1612yes—
85C3×C4×Q8—91no—
86Q8×Dic3—31no—
87Q8×Dic3—91no—
88#888—91no—
89#888—91no—
90#869—91no—
91C3×#103—91no—
92C3×#103—91no—
93#835—31no—
94#835—31no—
95#835—31no—
96#835—61no—
97#835—91no—
98#818—91no—
99#818—91no—
100Dic3⋊C8—91no—
101Dic3⋊C8—91no—
102#807—91no—
103#807—91no—
104#807—91no—
105#807—91no—
106C4⋊(C3⋊C8)—91no—
107C32×Q8—21no—
108C32×Q8—21no—
109C32×Q8—22no—
110C32×Q8—41no—
111C32×Q8Dic612yescomplemented
112C32×Q8Dic612yescomplemented
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3⋊Dic3#6711yescharacteristic
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—121no—
130#627—122no—
131#627—122no—
132C3⋊(C3⋊C8)—21no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8—61no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—121no—
137#477—271noSylow
138C2×H(3,𝔽3)C22⋊Q811yescharacteristic
139C2×H(3,𝔽3)C2×Q1611yescharacteristic
140C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
141C2×C3×Q8—31no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8D3211yescharacteristic
145C2×Dic6—181no—
146C2×Dic6—181no—
147C2×Dic6—181no—
148C2×C3×C8—91no—
149C2×C3×C8—91no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4—181no—
154C3×C42—91no—
155C3×C42—181no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—61no—
161C4⋊Dic3—61no—
162C4⋊Dic3—91no—
163C4⋊Dic3—91no—
164C4⋊Dic3—181no—
165C4⋊Dic3—181no—
166Dic3⋊C4—181no—
167Dic3⋊C4—181no—
168C4×Dic3—31no—
169C4×Dic3—61no—
170C4×Dic3—91no—
171C4×Dic3—181no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—61no—
176C2×C3⋊C8—91no—
177C22×C32—21no—
178C22×C32D3×D411yescharacteristic
179C22×C32C2×#6711yescharacteristic
180C32×C4—22no—
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—42no—
185C32×C4C2×Dic611yescharacteristic
186C32×C4C2×Dic611yescharacteristic
187C32×C4#23511yescharacteristic
188C32×C4#23711yescharacteristic
189C3⋊Dic3—21no—
190C3⋊Dic3—41no—
191C3⋊Dic3—41no—
192C3⋊Dic3#21612yes—
193C3×Dic3—32no—
194C3×Dic3—61no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197C3×Dic3—121no—
198C3×Dic3—121no—
199C3×Dic3—241no—
200C4⋊Q8—271no—
201C4×Q8—91no—
202#107—271no—
203#107—271no—
204C4⋊C8—271no—
205#103—91no—
206#103—91no—
207H(3,𝔽3)#47711yescharacteristic, complemented, Sylow
208C3×Q8—21no—
209C3×Q8—32no—
210C3×Q8—32no—
211C3×Q8—61no—
212C3×Q8—61no—
213C3×Q8—62no—
214C3×Q8—121no—
215C3×Q8#62712yes—
216C2×C3×C4—21no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—121no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4C2×D3211yescharacteristic
229C2×Dic3—31no—
230C2×Dic3—31no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—91no—
236C2×Dic3—91no—
237C2×Dic3—121no—
238C2×Dic3—181no—
239C2×Dic3—181no—
240Dic6—362no—
241Dic6—362no—
242Dic6—362no—
243C3×C8—181no—
244C3×C8—181no—
245C3⋊C8—61no—
246C3⋊C8—61no—
247C3⋊C8—61no—
248C3⋊C8—121no—
249C3⋊C8—181no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32#87811yescharacteristic
254C2×C32#91411yescharacteristic
255C2×C32D3×Q1611yescharacteristic
256C2×C32#92411yescharacteristic
257C2×C32C2×#21611yescharacteristic
258C2×C32#94911yescharacteristic
259C2×Q8—541no—
260C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
261C2×C8—91no—
262C2×C8—91no—
263C4⋊C4—91no—
264C4⋊C4—91no—
265C4⋊C4—91no—
266C4⋊C4—181no—
267C4⋊C4—541no—
268C42—91no—
269C42—181no—
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—61no—
273C22×C3D3×#6711yescharacteristic
274C3×C4—22no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—62no—
280C3×C4—62no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—92no—
284C3×C4—122no—
285C3×C4—181no—
286C3×C4—361no—
287C3×C4—361no—
288C3×C4—361no—
289C3×C4#380411yescharacteristic
290C3×C4C2×#62711yescharacteristic
291Dic3—32no—
292Dic3—61no—
293Dic3—92no—
294Dic3—121no—
295Dic3—121no—
296Dic3—121no—
297Dic3—121no—
298Dic3—181no—
299Dic3—241no—
300Dic3—361no—
301Dic3—361no—
302C32—21no—
303C32#479311yescharacteristic, complemented
304C32#502111yescharacteristic, complemented
305Q8—21no—
306Q8—1082no—
307Q8#619612yescomplemented
308C2×C4—21no—
309C2×C4—91no—
310C2×C4—91no—
311C2×C4—181no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C8—181no—
316C8—181no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—61no—
324C2×C3—61no—
325C2×C3—61no—
326C2×C3#6393911yescharacteristic
327C2×C3D3×#21611yescharacteristic
328C2×C3#6396411yescharacteristic
329C22#8867611yescharacteristic
330C4—22no—
331C4—92no—
332C4—181no—
333C4—361no—
334C4—361no—
335C4—361no—
336C4C2×#619611yescharacteristic
337C4#8867111yescharacteristic
338C3—31no—
339C3—31no—
340C3—61no—
341C3#1059154611yescharacteristic
342C2#1172123311yescharacteristic
343C2#1172123611yescharacteristic
344C2#1172129911yescharacteristic
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)#477
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
37. Q8×H(3,𝔽3)Q8semidirect
111. C32×Q8Dic6semidirect
112. C32×Q8Dic6semidirect
207. H(3,𝔽3)#477semidirect
303. C32#4793semidirect
304. C32#5021semidirect
307. Q8#6196semidirect

Maximal subgroups

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

subgroupindex
Q8×(C3⋊Dic3)⋊C32
#117211162
#117205832
#63662⋊C32
#117203152
#117203112
#117203062
#105916313
#105915463