Finite Group Databasebeta

← #13140888

Subgroups — #13140888

Order 1728.

385 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 77 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×#1084C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic
3#63662⋊C3C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#11720307C211yesmaximal, characteristic
6#11720306C211yesmaximal, characteristic
7#11720304C211yesmaximal, characteristic
8#10591547—31nomaximal
9#10591545—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084—21no—
12C4⋊#1084C2211yescharacteristic
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×Q8×Dic3—31no—
22C3×Q8×Dic3—31no—
23C3×Q8×Dic3—61no—
24#63869—31no—
25#63869—31no—
26#63662D311yescharacteristic
27#63662D311yescharacteristic
28C3×#835—31no—
29C3×#835—31no—
30#63576—31no—
31#63575—31no—
32#63574—31no—
33#63573—31no—
34Dic3⋊(C3⋊C8)—31no—
35Dic3⋊(C3⋊C8)—31no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)Q812yes—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×#1084D411yescharacteristic
42C2×#1084D411yescharacteristic
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#4799—91no—
49#4796—91no—
50C2×C32×Q8—21no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×Q8C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)C2×D311yescharacteristic
55C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—61no—
62C3×C4⋊Dic3—61no—
63C3×C4⋊Dic3—61no—
64C3×C4⋊Dic3—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)—22no—
79C4×H(3,𝔽3)C2×Q811yescharacteristic
80C4×H(3,𝔽3)#4111yescharacteristic
81#1084—21no—
82#1084—21no—
83#1084—41no—
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86C3×C4×Q8—91no—
87Q8×Dic3—31no—
88Q8×Dic3—31no—
89Q8×Dic3—61no—
90#890—91no—
91#890—91no—
92#870—91no—
93C3×#103—91no—
94C3×#103—91no—
95#835—31no—
96#835—31no—
97#835—91no—
98#835—91no—
99#818—91no—
100#817—91no—
101Dic3⋊C8—91no—
102Dic3⋊C8—91no—
103#808—91no—
104#808—91no—
105#807—91no—
106#807—91no—
107C4⋊(C3⋊C8)—91no—
108C32×Q8—22no—
109C32×Q8—21no—
110C32×Q8—21no—
111C32×Q8—22no—
112C32×Q8Dic612yes—
113C32×Q8Dic612yes—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—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—31no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8D3211yescharacteristic
145C2×C3×C8—91no—
146C2×C3×C8—91no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C42—91no—
152C3×C42—181no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—61no—
158C4⋊Dic3—61no—
159C4⋊Dic3—61no—
160C4⋊Dic3—62no—
161C4⋊Dic3—91no—
162C4⋊Dic3—91no—
163C4⋊Dic3—181no—
164C4⋊Dic3—181no—
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—24no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—21no—
187C32×C4C2×Dic611yescharacteristic
188C32×C4C2×Dic611yescharacteristic
189C32×C4#23711yescharacteristic
190C32×C4#23711yescharacteristic
191C3⋊Dic3—41no—
192C3⋊Dic3—41no—
193C3×Dic3—62no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—121no—
199C3×Dic3—122no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204#125—271no—
205C4×Q8—91no—
206#107—271no—
207#106—271no—
208C4⋊C8—271no—
209#103—91no—
210#103—91no—
211H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
212C3×Q8—21no—
213C3×Q8—32no—
214C3×Q8—32no—
215C3×Q8—61no—
216C3×Q8—61no—
217C3×Q8—62no—
218C3×Q8—62no—
219C3×Q8#62712yes—
220C2×C3×C4—21no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—91no—
228C2×C3×C4—91no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—31no—
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—31no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—62no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—181no—
245C2×Dic3—181no—
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#87811yescharacteristic
256C2×C32#87811yescharacteristic
257C2×C32#91511yescharacteristic
258C2×C32#91611yescharacteristic
259C2×C32#91811yescharacteristic
260C2×C32#91911yescharacteristic
261C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
262C2×C8—91no—
263C2×C8—91no—
264C4⋊C4—91no—
265C4⋊C4—91no—
266C4⋊C4—91no—
267C4⋊C4—181no—
268C4⋊C4—541no—
269C4⋊C4—541no—
270C42—91no—
271C42—181no—
272C22×C3—31no—
273C22×C3—31no—
274C22×C3—61no—
275C22×C3#381111yescharacteristic
276C3×C4—22no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—62no—
282C3×C4—62no—
283C3×C4—64no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—181no—
287C3×C4—181no—
288C3×C4—361no—
289C3×C4—361no—
290C3×C4—361no—
291C3×C4#380511yescharacteristic
292C3×C4C2×#62711yescharacteristic
293Dic3—61no—
294Dic3—61no—
295Dic3—61no—
296Dic3—61no—
297Dic3—62no—
298Dic3—121no—
299Dic3—121no—
300Dic3—121no—
301Dic3—122no—
302Dic3—121no—
303Dic3—121no—
304Dic3—361no—
305Dic3—361no—
306C32—21no—
307C32#479611yescharacteristic, complemented
308C32#479911yescharacteristic, complemented
309Q8—21no—
310Q8#619612yes—
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#6395011yescharacteristic
330C2×C3#6395111yescharacteristic
331C2×C3#6398211yescharacteristic
332C22#8867711yescharacteristic
333C4—22no—
334C4—181no—
335C4—181no—
336C4—361no—
337C4—361no—
338C4—361no—
339C4C2×#619611yescharacteristic
340C4#8867211yescharacteristic
341C3—31no—
342C3—31no—
343C3—61no—
344C3#1059161511yescharacteristic
345C2#1172123011yescharacteristic
346C2#1172123111yescharacteristic
347C2#1172130911yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×Q8((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 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
211. H(3,𝔽3)#479semidirect
307. C32#4796semidirect
308. C32#4799semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
#117210952
#63662⋊C32
#117203072
#117203062
#117203042
#105915473
#105915453