Finite Group Databasebeta

← #11713917

Subgroups — #11713917

Order 768.

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

normal subgroups only

This group has 388 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
1#2341⋊C3C211yesmaximal, characteristic
2#73151C211yesmaximal, characteristic
3#73151C211yesmaximal, characteristic
4#73063C211yesmaximal, characteristic, complemented
5#73062C211yesmaximal, characteristic, complemented
6C4⋊#859C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#27548—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16#4660—41nomaximal
17#2497—31no—
18#2341D311yescharacteristic
19#2077—31no—
20#2075—31no—
21#2071—31no—
22#2065—31no—
23#1936—31no—
24#1925—31no—
25#1924—31no—
26#1865—31no—
27#1863—31no—
28#1803—31no—
29#1788—31no—
30#1788—31no—
31#1763—31no—
32C2×#41⋊C3—21no—
33C2×#41⋊C3—21no—
34C2×#41⋊C3—21no—
35C22×SL(2,𝔽3)C2311yescharacteristic
36C2×C4⋊Dic3—41no—
37C2×Dic3⋊C4—41no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)—21no—
40#859—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×#41C2×D311yescharacteristic
50C2×C4×Q8C2×D311yescharacteristic
51C2×C4×Q8C2×D311yescharacteristic
52C2×#107—31no—
53C2×#106—31no—
54C2×C4⋊C8—31no—
55C2×#103—31no—
56C2×#103—31no—
57C2×#103—31no—
58C2×#103—31no—
59C2×#102—31no—
60C2×#102—31no—
61C2×#102—31no—
62C2×#102—31no—
63C2×C22⋊C8—31no—
64C2×C22⋊C8—31no—
65C2×#97—31no—
66#400—61no—
67#396—61no—
68#390—31no—
69#390—31no—
70#389—31no—
71#389—31no—
72#388—61no—
73#387—61no—
74#384—31no—
75#380—31no—
76#380—31no—
77#380—31no—
78C4×C4⋊C4—31no—
79C4×C22⋊C4—31no—
80#340—31no—
81#340—31no—
82#340—31no—
83#340—31no—
84#340—31no—
85#340—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—
95#41⋊C3—41no—
96C2×SL(2,𝔽3)#4111yescharacteristic
97C2×SL(2,𝔽3)#4111yescharacteristic
98C2×SL(2,𝔽3)#4111yescharacteristic
99C4⋊Dic3—82no—
100Dic3⋊C4—82no—
101C2×#41—21no—
102C2×#41—21no—
103C2×#41—21no—
104C2×#41—61no—
105C2×#41—61no—
106C2×#41—61no—
107C22×Q8C22×D311yescharacteristic
108C22×D4—31no—
109C23×C4—31no—
110C22×C8—31no—
111C22×C8—31no—
112C22×C8—31no—
113C22×C8—31no—
114C4×Q8—21no—
115C4×Q8—21no—
116C4×Q8—61no—
117C4×Q8—61no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C22⋊C4—31no—
129C2×C22⋊C4—31no—
130C2×C22⋊C4—61no—
131C2×C22⋊C4—61no—
132C2×C22⋊C4—61no—
133C2×C42—31no—
134C2×C42—31no—
135C2×C42—61no—
136#107—121no—
137#106—121no—
138C4⋊C8—121no—
139#103—61no—
140#103—61no—
141#103—61no—
142#103—61no—
143#103—61no—
144#103—61no—
145#103—61no—
146#103—61no—
147#102—62no—
148#102—62no—
149#102—121no—
150#102—121no—
151C22⋊C8—121no—
152C22⋊C8—121no—
153#97—121no—
154#95—31no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95S411yescharacteristic
161C23×C3—41no—
162C2×C3×C4—42no—
163C2×C3×C4—81no—
164C2×C3×C4—81no—
165C2×C3×C4—81no—
166C2×C3×C4—81no—
167C2×C3×C4—81no—
168C2×C3×C4—81no—
169C2×C3×C4—81no—
170C2×C3×C4—81no—
171C2×Dic3—42no—
172C2×Dic3—81no—
173C2×Dic3—81no—
174C2×Dic3—81no—
175C2×Dic3—81no—
176C2×Dic3—81no—
177C2×Dic3—81no—
178C2×Dic3—81no—
179C2×Dic3—81no—
180C2×Dic3—81no—
181C2×Dic3—81no—
182C2×Dic3—81no—
183SL(2,𝔽3)#12611yescharacteristic
184C24—31no—
185#41—41no—
186#41—122no—
187#41—121no—
188#41—121no—
189#41—121no—
190C2×Q8—31no—
191C2×Q8—31no—
192C2×Q8—31no—
193C2×Q8#23511yescharacteristic
194C2×Q8#23711yescharacteristic
195C2×Q8#23711yescharacteristic
196C2×D4—32no—
197C2×D4—61no—
198C2×D4—61no—
199C2×D4—61no—
200C2×D4—61no—
201C2×D4—61no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4C2×S411yescharacteristic
216C22×C4C2×S411yescharacteristic
217C22×C4C2×S411yescharacteristic
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8—61no—
222C2×C8—61no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—61no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—122no—
243C4⋊C4—241no—
244C4⋊C4—241no—
245C4⋊C4—241no—
246C22⋊C4—62no—
247C22⋊C4—121no—
248C22⋊C4—241no—
249C22⋊C4—241no—
250C22⋊C4—241no—
251C42—61no—
252C42—61no—
253C42—61no—
254C42—61no—
255C42—241no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C3×C4—82no—
264C3×C4—161no—
265C3×C4—161no—
266Dic3—82no—
267Dic3—161no—
268Dic3—161no—
269Dic3—161no—
270C23—61no—
271C23—61no—
272C23—61no—
273C23—61no—
274C23—61no—
275C23—61no—
276C23—61no—
277C23C22×S411yescharacteristic
278Q8—31no—
279Q8—31no—
280Q8—31no—
281Q8—61no—
282Q8#87911yescharacteristic
283D4—62no—
284D4—62no—
285D4—121no—
286D4—121no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4#98412yes—
324C8—121no—
325C8—121no—
326C8—121no—
327C8—121no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—41no—
335C22—121no—
336C22—121no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22—121no—
341C22—121no—
342C22#591211yescharacteristic
343C22#591411yescharacteristic
344C22#591411yescharacteristic
345C22C2×#98411yescharacteristic
346C22#592311yescharacteristic
347C22#592511yescharacteristic
348C22#592611yescharacteristic
349C4—22no—
350C4—31no—
351C4—31no—
352C4—31no—
353C4—31no—
354C4—41no—
355C4—41no—
356C4—121no—
357C4—121no—
358C4—241no—
359C4—241no—
360C4—241no—
361C4—241no—
362C3—41noSylow
363C2—241no—
364C2#8529411yescharacteristic
365C2#8533811yescharacteristic
366C2#8536711yescharacteristic
367C2#8537711yescharacteristic
368C2#8538811yescharacteristic
369C2#8540611yescharacteristic
370C2#8541311yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
Fitting#2341D3
prime radicalC22×#41⋊C3C22
hypercenter#95S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. #73063C2semidirect
5. #73062C2semidirect
6. C4⋊#859C2semidirect
7. #73059C2semidirect

Maximal subgroups

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

subgroupindex
#2341⋊C32
#731512
#730632
#730622
C4⋊#8592
#730592
#275483
#46604