Finite Group Databasebeta

← #11713928

Subgroups — #11713928

Order 768.

357 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 51 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
1C4⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic
2#73064C211yesmaximal, characteristic
3#73063C211yesmaximal, characteristic
4#73061C211yesmaximal, characteristic
5#73061C211yesmaximal, characteristic
6#73061C211yesmaximal, characteristic
7#73059C211yesmaximal, characteristic
8#27570—31nomaximal, Sylow
9C22×#226C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16#4985—41nomaximal
17#2514—31no—
18Q8×C4⋊C4D311yescharacteristic
19#2101—31no—
20#2100—31no—
21#2072—31no—
22#2066—31no—
23#1935—31no—
24#1919—31no—
25#1917—31no—
26#1878—31no—
27#1878—31no—
28#1878—31no—
29#1865—31no—
30#1805—31no—
31#1763—31no—
32C22×SL(2,𝔽3)C2311yescharacteristic
33C2×#226—21no—
34C2×#226—21no—
35C2×#226—21no—
36C2×C3×C4⋊C4—41no—
37C2×Dic3⋊C4—41no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)—21no—
41#859—21no—
42#859—21no—
43#859—21no—
44C3⋊#95—41no—
45C3⋊#95—41no—
46C3⋊#95—41no—
47C3⋊#95—41no—
48C3⋊#95—41no—
49C22×Q16—31no—
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×Q8C2×D311yescharacteristic
52C2×C4×Q8C2×D311yescharacteristic
53C2×#106—31no—
54C2×C4⋊C8—31no—
55C2×#103—31no—
56C2×#103—31no—
57C2×#103—31no—
58C2×#103—31no—
59C2×#103—31no—
60C2×#103—31no—
61C2×#97—31no—
62#401—61no—
63#400—61no—
64#391—31no—
65#391—31no—
66#391—31no—
67#389—31no—
68#389—31no—
69#389—31no—
70#387—61no—
71#384—31no—
72#383—61no—
73#382—31no—
74C4×C4⋊C4—31no—
75C4×C4⋊C4—31no—
76#340—31no—
77#340—31no—
78#340—31no—
79#340—31no—
80#340—31no—
81#340—31no—
82#340—31no—
83#340—31no—
84#336—31no—
85#336—31no—
86#336—31no—
87#336—31no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×Dic3—41no—
92C22×Dic3—41no—
93C22×Dic3—41no—
94C22×Dic3—41no—
95C2×SL(2,𝔽3)#4111yescharacteristic
96C2×SL(2,𝔽3)#4111yescharacteristic
97C2×SL(2,𝔽3)#4111yescharacteristic
98#226—41no—
99C3×C4⋊C4—82no—
100Dic3⋊C4—82no—
101C22×Q8—31no—
102C22×Q8C22×D311yescharacteristic
103C2×Q16—61no—
104C2×Q16—61no—
105C2×Q16—61no—
106C2×Q16—61no—
107C2×Q16—61no—
108C2×Q16—61no—
109C22×C8—31no—
110C22×C8—31no—
111C22×C8—31no—
112C22×C8—31no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×Q8—21no—
116C4×Q8—61no—
117C4×Q8—61no—
118C4×Q8—61no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4S411yescharacteristic
129C2×C42—31no—
130C2×C42—31no—
131C2×C42—31no—
132C2×C42—61no—
133#106—62no—
134C4⋊C8—121no—
135#103—62no—
136#103—61no—
137#103—61no—
138#103—61no—
139#103—61no—
140#103—61no—
141#103—61no—
142#103—121no—
143#103—121no—
144#97—121no—
145#95—31no—
146#95—31no—
147#95—61no—
148#95—61no—
149#95—61no—
150#95—61no—
151#95—61no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157C23×C3—41no—
158C2×C3×C4—42no—
159C2×C3×C4—81no—
160C2×C3×C4—81no—
161C2×C3×C4—81no—
162C2×C3×C4—81no—
163C2×C3×C4—81no—
164C2×C3×C4—81no—
165C2×C3×C4—81no—
166C2×C3×C4—81no—
167C2×Dic3—42no—
168C2×Dic3—81no—
169C2×Dic3—81no—
170C2×Dic3—81no—
171C2×Dic3—81no—
172C2×Dic3—81no—
173C2×Dic3—81no—
174C2×Dic3—81no—
175C2×Dic3—81no—
176C2×Dic3—81no—
177C2×Dic3—81no—
178C2×Dic3—81no—
179SL(2,𝔽3)#12611yescharacteristic
180C2×Q8—31no—
181C2×Q8—31no—
182C2×Q8—31no—
183C2×Q8—61no—
184C2×Q8—61no—
185C2×Q8—61no—
186C2×Q8—61no—
187C2×Q8—61no—
188C2×Q8—61no—
189C2×Q8#23511yescharacteristic
190C2×Q8#23711yescharacteristic
191C2×Q8#23911yescharacteristic
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4C2×S411yescharacteristic
201C22×C4C2×S411yescharacteristic
202C22×C4C2×S411yescharacteristic
203Q16—121no—
204Q16—121no—
205Q16—121no—
206Q16—121no—
207C2×C8—61no—
208C2×C8—61no—
209C2×C8—61no—
210C2×C8—61no—
211C2×C8—61no—
212C2×C8—61no—
213C2×C8—61no—
214C2×C8—61no—
215C2×C8—61no—
216C2×C8—61no—
217C2×C8—61no—
218C2×C8—61no—
219C4⋊C4—22no—
220C4⋊C4—32no—
221C4⋊C4—61no—
222C4⋊C4—61no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—121no—
233C4⋊C4—122no—
234C4⋊C4—241no—
235C42—61no—
236C42—61no—
237C42—61no—
238C42—61no—
239C42—61no—
240C42—61no—
241C42—122no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C3×C4—82no—
250C3×C4—161no—
251C3×C4—161no—
252Dic3—82no—
253Dic3—161no—
254Dic3—161no—
255Dic3—161no—
256C23C22×S411yescharacteristic
257Q8—31no—
258Q8—31no—
259Q8—31no—
260Q8—61no—
261Q8—121no—
262Q8—121no—
263Q8—121no—
264Q8—121no—
265Q8#89811yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4#98312yes—
302C8—121no—
303C8—121no—
304C8—121no—
305C8—121no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C22#591211yescharacteristic
314C22#591411yescharacteristic
315C22C2×#98311yescharacteristic
316C22#591911yescharacteristic
317C22#592311yescharacteristic
318C22#592411yescharacteristic
319C22#592711yescharacteristic
320C4—22no—
321C4—31no—
322C4—31no—
323C4—31no—
324C4—31no—
325C4—41no—
326C4—41no—
327C4—121no—
328C4—121no—
329C4—121no—
330C4—241no—
331C4—241no—
332C4—241no—
333C4—241no—
334C3—41noSylow
335C2#8531311yescharacteristic
336C2#8533011yescharacteristic
337C2#8536211yescharacteristic
338C2#8539511yescharacteristic
339C2#8540511yescharacteristic
340C2#8542511yescharacteristic
341C2#8543211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C4⋊C4D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Maximal subgroups

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

subgroupindex
C4⋊C4×SL(2,𝔽3)2
#730642
#730632
#730612
#730592
#275703
#49854