Finite Group Databasebeta

← #11713868

Subgroups — #11713868

Order 768.

521 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 53 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×#5427C211yesmaximal, characteristic
2C2×#5386C211yesmaximal, characteristic, complemented
3#2341⋊C3C211yesmaximal, characteristic, complemented
4#73151C211yesmaximal, characteristic, complemented
5#73065C211yesmaximal, characteristic, complemented
6#73065C211yesmaximal, characteristic, complemented
7#73063C211yesmaximal, characteristic, complemented
8#27250—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C22×GL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13#5427—22no—
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17#5386—22no—
18#4673—41nomaximal
19C2×#475—31no—
20C2×#449—31no—
21#2491—31no—
22#2341D311yescharacteristic, complemented
23#2077—31no—
24#2054—31no—
25#2034—31no—
26#2023—31no—
27#1972—31no—
28#1926—31no—
29#1924—31no—
30#1880—31no—
31#1880—31no—
32#1788—31no—
33#1763—31no—
34C22×Dic6—41no—
35C2×#41⋊C3—21no—
36C2×#41⋊C3—21no—
37C2×#41⋊C3—21no—
38C22×SL(2,𝔽3)C2311yescharacteristic
39C2×GL(2,𝔽3)—21no—
40C2×GL(2,𝔽3)—21no—
41C2×GL(2,𝔽3)—21no—
42C2×(C2×D3)⋊C4—41no—
43C2×(C2×D3)⋊C4—41no—
44C2×(C2×D3)⋊C4—41no—
45C4×SL(2,𝔽3)—21no—
46C4×SL(2,𝔽3)—21no—
47#859—21no—
48#859—21no—
49#859D412yes—
50C3×#95—41no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C22×#41C2×D311yescharacteristic
54C22×SD16—31no—
55C22×D8—31no—
56C2×C4⋊Q8—31no—
57C2×#124—61no—
58C2×#121—31no—
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×Q8C2×D311yescharacteristic
61#475—62no—
62#475—62no—
63#449—62no—
64#449—62no—
65C2×#106—31no—
66C2×#106—31no—
67C2×C4⋊C8—31no—
68C2×#103—31no—
69C2×#103—31no—
70C2×#103—31no—
71C2×#102—31no—
72C2×#102—31no—
73C2×#102—31no—
74C2×#102—31no—
75C2×#102—31no—
76C2×C22⋊C8—31no—
77C2×C22⋊C8—31no—
78C2×#97—31no—
79#394—61no—
80#393—61no—
81#390—31no—
82#390—31no—
83#390—31no—
84#390—31no—
85#388—61no—
86#380—31no—
87#380—31no—
88C4×C4⋊C4—31no—
89C4×C22⋊C4—31no—
90#340—31no—
91#340—31no—
92#340—31no—
93#340—31no—
94#336—31no—
95#336—31no—
96C23×D3—41no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C2×Dic6—82no—
104C2×Dic6—82no—
105C2×Dic6—82no—
106#41⋊C3—41no—
107C2×SL(2,𝔽3)C2×D411yescharacteristic
108C2×SL(2,𝔽3)#4111yescharacteristic
109C2×SL(2,𝔽3)#4111yescharacteristic
110GL(2,𝔽3)—41no—
111(C2×D3)⋊C4—82no—
112(C2×D3)⋊C4—82no—
113(C2×D3)⋊C4—82no—
114C2×#41—21no—
115C2×#41—21no—
116C2×#41—21no—
117C2×#41—61no—
118C2×#41—61no—
119C2×#41—61no—
120C22×Q8—61no—
121C22×Q8C22×D311yescharacteristic
122C22×D4—31no—
123C22×D4—31no—
124C22×D4—61no—
125C23×C4—31no—
126C2×SD16—61no—
127C2×SD16—61no—
128C2×SD16—61no—
129C2×SD16—61no—
130C2×SD16—61no—
131C2×SD16—61no—
132C2×D8—62no—
133C2×D8—62no—
134C2×D8—121no—
135C22×C8—31no—
136C22×C8—31no—
137C22×C8—31no—
138C22×C8—31no—
139C4⋊Q8—62no—
140C4⋊Q8—121no—
141#124—124no—
142#121—62no—
143#121—121no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—61no—
147C4×Q8—61no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—61no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C42—31no—
164C2×C42—31no—
165C2×C42—61no—
166#106—62no—
167#106—62no—
168C4⋊C8—62no—
169#103—32no—
170#103—32no—
171#103—61no—
172#103—61no—
173#103—61no—
174#103—61no—
175#102—62no—
176#102—62no—
177#102—121no—
178#102—121no—
179#102—121no—
180C22⋊C8—62no—
181C22⋊C8—121no—
182#97—121no—
183#95—31no—
184#95—61no—
185#95—61no—
186#95S411yescharacteristic
187C23×C3—41no—
188C22×D3—42no—
189C22×D3—81no—
190C22×D3—81no—
191C22×D3—81no—
192C22×D3—81no—
193C22×D3—81no—
194C22×D3—81no—
195C2×C3×C4—42no—
196C2×C3×C4—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×C3×C4—81no—
204C2×Dic3—42no—
205C2×Dic3—42no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213Dic6—88no—
214SL(2,𝔽3)#12411yescharacteristic, complemented
215C24—31no—
216C24—61no—
217#41—41no—
218#41—122no—
219#41—121no—
220#41—121no—
221#41—121no—
222C2×Q8—31no—
223C2×Q8—31no—
224C2×Q8—31no—
225C2×Q8—122no—
226C2×Q8—122no—
227C2×Q8—241no—
228C2×Q8#23511yescharacteristic
229C2×Q8D3×D411yescharacteristic
230C2×Q8#23711yescharacteristic
231C2×D4—32no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—122no—
244C2×D4—122no—
245C2×D4—241no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4C2×S411yescharacteristic
259C22×C4C2×S411yescharacteristic
260C22×C4C2×S411yescharacteristic
261SD16—121no—
262SD16—121no—
263SD16—121no—
264SD16—121no—
265D8—124no—
266C2×C8—32no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—61no—
278C4⋊C4—32no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—122no—
289C22⋊C4—62no—
290C22⋊C4—121no—
291C22⋊C4—122no—
292C22⋊C4—122no—
293C22⋊C4—122no—
294C22⋊C4—122no—
295C22⋊C4—122no—
296C22⋊C4—241no—
297C22⋊C4—241no—
298C42—61no—
299C42—61no—
300C42—61no—
301C42—61no—
302C42—122no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C2×D3—82no—
311C2×D3—82no—
312C2×D3—82no—
313C2×D3—161no—
314C2×D3—161no—
315C2×D3—161no—
316C2×D3—161no—
317C3×C4—82no—
318C3×C4—161no—
319C3×C4—161no—
320Dic3—82no—
321Dic3—82no—
322Dic3—161no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23—61no—
327C23—61no—
328C23—61no—
329C23—61no—
330C23—121no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23C22×S411yescharacteristic
338Q8—31no—
339Q8—31no—
340Q8—31no—
341Q8—61no—
342Q8—244no—
343Q8#88511yescharacteristic
344D4—62no—
345D4—62no—
346D4—121no—
347D4—121no—
348D4—121no—
349D4—121no—
350D4—121no—
351D4—121no—
352D4—244no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4#98612yes—
387C8—62no—
388C8—121no—
389C8—121no—
390C8—121no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398D3—162no—
399C22—121no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22—241no—
412C22—241no—
413C22#591211yescharacteristic
414C22D4×S411yescharacteristic
415C22#591411yescharacteristic
416C22C2×#98611yescharacteristic
417C22#592311yescharacteristic
418C22#592411yescharacteristic
419C22#592611yescharacteristic
420C4—22no—
421C4—31no—
422C4—31no—
423C4—31no—
424C4—31no—
425C4—41no—
426C4—41no—
427C4—122no—
428C4—121no—
429C4—121no—
430C4—241no—
431C4—241no—
432C3—41noSylow
433C2—241no—
434C2—481no—
435C2#8530011yescharacteristic
436C2#8534811yescharacteristic
437C2#8536911yescharacteristic
438C2#8538211yescharacteristic
439C2#8538811yescharacteristic
440C2#8541111yescharacteristic
441C2#8541211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
Fitting#2341D3
prime radicalC2×#5427C2
hypercenter#95S4
hypocenterSL(2,𝔽3)#124
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
2. C2×#5386C2semidirect
3. #2341⋊C3C2semidirect
4. #73151C2semidirect
5. #73065C2semidirect
6. #73065C2semidirect
7. #73063C2semidirect
22. #2341D3semidirect
214. SL(2,𝔽3)#124semidirect

Maximal subgroups

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

subgroupindex
C2×#54272
C2×#53862
#2341⋊C32
#731512
#730652
#730632
#272503
#46734