Finite Group Databasebeta

← #11714055

Subgroups — #11714055

Order 768.

515 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C2×#519⋊C3C211yesmaximal, characteristic
2C2×#5419C211yesmaximal, characteristic
3#73148C211yesmaximal, characteristic
4#73062C211yesmaximal, characteristic, complemented
5#73061C211yesmaximal, characteristic, complemented
6#73061C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#33019—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C22×#226C2211yescharacteristic
11#519⋊C3—22no—
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5419—22no—
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18#4952—41nomaximal
19C2×#519D311yescharacteristic
20C2×#485—31no—
21C2×#451—31no—
22#2625—31no—
23#1878—31no—
24#1878—31no—
25#1878—31no—
26#1878—31no—
27#1865—31no—
28#1865—31no—
29#1863—31no—
30#1863—31no—
31#1825—31no—
32#1818—31no—
33#1787—31no—
34C22×SL(2,𝔽3)—21no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)D412yes—
37C22×SL(2,𝔽3)C2311yescharacteristic
38C2×#226—21no—
39C2×#226—21no—
40C2×#226—21no—
41C2×C3×C22⋊C4—41no—
42C2×#217—41no—
43C2×#217—41no—
44C2×Dic3⋊C4—41no—
45C4×SL(2,𝔽3)—21no—
46C4×SL(2,𝔽3)—21no—
47#859—21no—
48#859—21no—
49#859—21no—
50C3⋊#95—41no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C23×Q8C2×D311yescharacteristic
54C22×Q16—31no—
55C22×Q16—31no—
56C2×C22⋊Q8—31no—
57#519—22no—
58#519—62no—
59C2×C4×Q8—31no—
60C2×C4×Q8—31no—
61C2×C4×Q8C2×D311yescharacteristic
62C2×C4×Q8C2×D311yescharacteristic
63C2×#117—31no—
64C2×#117—31no—
65#485—64no—
66#451—62no—
67#451—62no—
68C2×#107—31no—
69C2×#106—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×#103—31no—
76C2×#103—31no—
77C2×#103—31no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8—31no—
80#400—61no—
81#398—61no—
82#397—61no—
83#397—61no—
84#391—31no—
85#391—31no—
86#389—31no—
87#389—31no—
88#380—31no—
89#340—31no—
90#340—31no—
91#340—31no—
92#340—31no—
93#336—31no—
94#336—31no—
95#336—31no—
96#336—31no—
97C24×C3—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104C2×SL(2,𝔽3)—22no—
105C2×SL(2,𝔽3)C2×D411yescharacteristic
106C2×SL(2,𝔽3)#4111yescharacteristic
107C2×SL(2,𝔽3)#4111yescharacteristic
108#226—41no—
109C3×C22⋊C4—82no—
110#217—82no—
111#217—82no—
112Dic3⋊C4—82no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—31no—
116C22×Q8—32no—
117C22×Q8—31no—
118C22×Q8—61no—
119C22×Q8—61no—
120C22×Q8#6712yes—
121C22×Q8C22×D311yescharacteristic
122C23×C4—31no—
123C2×Q16—61no—
124C2×Q16—61no—
125C2×Q16—62no—
126C2×Q16—62no—
127C2×Q16—62no—
128C2×Q16—61no—
129C2×Q16—61no—
130C2×Q16—61no—
131C2×Q16—61no—
132C22×C8—31no—
133C22×C8—31no—
134C22×C8—31no—
135C22×C8—31no—
136C22⋊Q8—62no—
137C22⋊Q8—62no—
138C4×Q8—21no—
139C4×Q8—21no—
140C4×Q8—62no—
141C4×Q8—62no—
142C4×Q8—62no—
143C4×Q8—62no—
144C4×Q8—61no—
145C4×Q8—61no—
146#117—34no—
147#117—62no—
148#117—62no—
149#117—62no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4S411yescharacteristic
164C2×C42—31no—
165C2×C42—31no—
166C2×C42—61no—
167#107—62no—
168#106—62no—
169#103—62no—
170#103—62no—
171#103—61no—
172#103—61no—
173#103—62no—
174#103—62no—
175#103—62no—
176#103—61no—
177#103—61no—
178#103—61no—
179#103—61no—
180C22⋊C8—62no—
181C22⋊C8—62no—
182#95—61no—
183#95—61no—
184#95—61no—
185#95—61no—
186#95—61no—
187#95—61no—
188#95—61no—
189C23×C3—42no—
190C23×C3—41no—
191C23×C3—81no—
192C23×C3—81no—
193C23×C3—81no—
194C23×C3—81no—
195C23×C3—81no—
196C23×C3—81no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×Dic3—42no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214C2×Dic3—81no—
215SL(2,𝔽3)#12311yescharacteristic
216C24C2×S411yescharacteristic
217C2×Q8—22no—
218C2×Q8—32no—
219C2×Q8—31no—
220C2×Q8—31no—
221C2×Q8—31no—
222C2×Q8—32no—
223C2×Q8—32no—
224C2×Q8—61no—
225C2×Q8—61no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—61no—
232C2×Q8—61no—
233C2×Q8—64no—
234C2×Q8—62no—
235C2×Q8—62no—
236C2×Q8—62no—
237C2×Q8#23511yescharacteristic
238C2×Q8#23511yescharacteristic
239C2×Q8C2×#6711yescharacteristic
240C22×C4—31no—
241C22×C4—32no—
242C22×C4—32no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4C2×S411yescharacteristic
255C22×C4C2×S411yescharacteristic
256Q16—121no—
257Q16—121no—
258Q16—121no—
259Q16—124no—
260Q16—121no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—61no—
265C2×C8—61no—
266C2×C8—61no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—61no—
273C4⋊C4—32no—
274C4⋊C4—32no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—122no—
286C4⋊C4—241no—
287C4⋊C4—241no—
288C22⋊C4—22no—
289C22⋊C4—62no—
290C22⋊C4—122no—
291C22⋊C4—241no—
292C42—61no—
293C42—61no—
294C42—61no—
295C42—61no—
296C42—64no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—82no—
305C22×C3—82no—
306C22×C3—82no—
307C22×C3—161no—
308C22×C3—161no—
309C22×C3—161no—
310C22×C3—161no—
311C3×C4—161no—
312C3×C4—161no—
313Dic3—82no—
314Dic3—161no—
315Dic3—161no—
316Dic3—161no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23GL(2,ℤ/4)12yes—
324C23C22×S411yescharacteristic
325Q8—31no—
326Q8—31no—
327Q8—31no—
328Q8—34no—
329Q8—61no—
330Q8—62no—
331Q8—62no—
332Q8—62no—
333Q8—121no—
334Q8—121no—
335Q8—121no—
336Q8—121no—
337Q8#92911yescharacteristic
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—31no—
350C2×C4—32no—
351C2×C4—62no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C8—121no—
376C8—121no—
377C8—121no—
378C8—121no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—41no—
385C2×C3—41no—
386C2×C3—162no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22#591211yescharacteristic
395C22#591211yescharacteristic
396C22#592611yescharacteristic
397C22#592711yescharacteristic
398C22#592711yescharacteristic
399C22#592711yescharacteristic
400C22C2×GL(2,ℤ/4)11yescharacteristic
401C4—31no—
402C4—31no—
403C4—31no—
404C4—31no—
405C4—41no—
406C4—41no—
407C4—62no—
408C4—121no—
409C4—121no—
410C4—241no—
411C4—241no—
412C4—241no—
413C4—241no—
414C3—41noSylow
415C2—42no—
416C2#8535211yescharacteristic
417C2#8535211yescharacteristic
418C2#8535611yescharacteristic
419C2#8535711yescharacteristic
420C2#8548511yescharacteristic
421C2#8553511yescharacteristic
422C2#8553911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#519D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #73062C2semidirect
5. #73061C2semidirect
6. #73061C2semidirect
7. #73059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#519⋊C32
C2×#54192
#731482
#730622
#730612
#730592
#330193
#49524