Finite Group Databasebeta

← #61903⋊C3 #11710491

Subgroups — #61903⋊C3 #11710491

Order 768.

474 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 65 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#3497⋊C3C211yesmaximal, characteristic, complemented
2#3480⋊C3C211yesmaximal, characteristic, complemented
3#3392⋊C3C211yesmaximal, characteristic, complemented
4#3391⋊C3C211yesmaximal, characteristic, complemented
5#3359⋊C3C211yesmaximal, characteristic, complemented
6#3358⋊C3C211yesmaximal, characteristic, complemented
7#2984⋊C3C211yesmaximal, characteristic, complemented
8#61903C311yesmaximal, characteristic, complemented, Sylow
9C2×2−1+4⋊C3—21no—
10#557⋊C3C2211yescharacteristic
11C4⋊(#41⋊C3)C2211yescharacteristic
12C4⋊(#41⋊C3)C2211yescharacteristic
13D4×SL(2,𝔽3)C2211yescharacteristic
14#548⋊C3—21no—
15#548⋊C3—21no—
16C4×#41⋊C3—21no—
17C2×C4×SL(2,𝔽3)C2211yescharacteristic
18C3×#482—41nomaximal
19C8×SL(2,𝔽3)C2211yescharacteristic
20C8×SL(2,𝔽3)C2211yescharacteristic
21#3497C2×C311yescharacteristic
22#3480C2×C311yescharacteristic, complemented
23#3399—31no—
24#3397—31no—
25#3392C2×C311yescharacteristic, complemented
26#3391C2×C311yescharacteristic, complemented
27#3373—31no—
28#3367—31no—
29#3359C2×C311yescharacteristic, complemented
30#3358C2×C311yescharacteristic, complemented
31#3214—31no—
32#3207—31no—
33#3115—31no—
34#3106—31no—
35#2984C2×C311yescharacteristic, complemented
362−1+4⋊C3—42no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3—21no—
39C22×SL(2,𝔽3)—21no—
40C22×SL(2,𝔽3)D411yescharacteristic
41C3×#121—41no—
42C3×#117—41no—
43C4×SL(2,𝔽3)—21no—
44C4×SL(2,𝔽3)D411yescharacteristic
45C4×SL(2,𝔽3)C2311yescharacteristic
46C3×#107—41no—
47C3×#106—41no—
48C3×#102—41no—
49C3×#102—41no—
50C3×C22⋊C8—41no—
51C2×2−1+4—21no—
52#559—31no—
53#557C22×C311yescharacteristic
54#556—31no—
55#553—61no—
56#550—31no—
57#550C22×C311yescharacteristic
58#550C22×C311yescharacteristic
59D4×Q8—31no—
60D4×Q8C22×C311yescharacteristic
61#548—21no—
62#548—21no—
63#548—61no—
64#537—61no—
65#532—31no—
66#532—31no—
67#532—31no—
68#529—61no—
69C4×#41—21no—
70C4×#41—31no—
71C2×C4×Q8C22×C311yescharacteristic
72#496—32no—
73#493—31no—
74#492—31no—
75#489—32no—
76#487—31no—
77#486—31no—
78#482—31no—
79#482A411yescharacteristic
80#477—31no—
81#476—31no—
82#475—31no—
83#474—31no—
84#463—32no—
85#463—32no—
86#461—31no—
87#461—31no—
88#460—31no—
89#460—31no—
90#459—31no—
91#459—31no—
92#459—31no—
93#459—31no—
94#449—32no—
95#449—31no—
96#449—31no—
97#446—31no—
98#446—31no—
99C8×Q8C22×C311yescharacteristic
100C8×Q8C22×C311yescharacteristic
101#442—32no—
102#440—32no—
103C4×SD16—31no—
104C4×SD16—31no—
105C4×D8—31no—
106C4×D8—31no—
107#432—31no—
108#431—31no—
109C2×C3×D4—41no—
110C2×C3×D4—81no—
111C22×C3×C4—41no—
112#41⋊C3—41no—
113#41⋊C3—41no—
114C2×SL(2,𝔽3)—21no—
115C2×SL(2,𝔽3)—41no—
116C2×SL(2,𝔽3)C2×D411yescharacteristic
117C2×C3×C8—41no—
118C2×C3×C8—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C22⋊C4—81no—
123C3×C22⋊C4—81no—
124C3×C42—81no—
1252−1+4—42no—
1262−1+4—122no—
127C2×#41—21no—
128C2×#41—21no—
129C2×#41—31no—
130C2×#41—61no—
131C2×#41—61no—
132C2×#41—61no—
133C22×Q8—21no—
134C22×Q8—61no—
135C22×Q8C3×D411yescharacteristic
136C2×SD16—32no—
137C2×SD16—31no—
138C2×SD16—31no—
139C2×D8—32no—
140C2×D8—31no—
141C2×D8—31no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144C4⋊Q8—31no—
145#127—31no—
146#127—31no—
147#127—31no—
148#126—62no—
149#125—61no—
150#124—32no—
151#124—61no—
152#124—61no—
153#124—61no—
154#123—62no—
155C22⋊Q8—62no—
156C22⋊Q8—61no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159#121—31no—
160#121—61no—
161#121—62no—
162#121—61no—
163#121—61no—
164#121—61no—
165#121—61no—
166#121—61no—
167#121—61no—
168#121C2×A411yescharacteristic
169C4×Q8—21no—
170C4×Q8—31no—
171C4×Q8—31no—
172C4×Q8—31no—
173C4×Q8—31no—
174C4×Q8C3×D411yescharacteristic
175C4×Q8C23×C311yescharacteristic
176C4×D4—32no—
177C4×D4—31no—
178C4×D4—31no—
179C4×D4—31no—
180C4×D4—31no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C4×D4—61no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188#117—61no—
189#117—61no—
190#117C2×A411yescharacteristic
191C2×C4⋊C4—31no—
192C2×C42—31no—
193C2×C42—61no—
194#107—31no—
195#107C2×A411yescharacteristic
196#106—31no—
197#106C2×A411yescharacteristic
198C4⋊C8—31no—
199C4⋊C8—31no—
200C4⋊C8—31no—
201C4⋊C8—31no—
202#103—32no—
203#103—31no—
204#103—31no—
205#102—32no—
206#102—31no—
207#102—31no—
208#102—31no—
209#102—31no—
210#102C2×A411yescharacteristic
211#102C2×A411yescharacteristic
212C22⋊C8—31no—
213C22⋊C8C2×A411yescharacteristic
214#97—32no—
215C4×C8—31no—
216C4×C8—31no—
217C23×C3—41no—
218C23×C3—81no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4—162no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229SL(2,𝔽3)AGL(1,ℤ/8)11yescharacteristic
230C3×C8—81no—
231C3×C8—81no—
232#41—41no—
233#41—41no—
234#41—121no—
235#41—121no—
236#41—122no—
237#41—122no—
238#41—122no—
239#41—122no—
240#41—122no—
241#41—122no—
242#41—121no—
243#41—121no—
244C2×Q8—21no—
245C2×Q8—31no—
246C2×Q8—31no—
247C2×Q8—41no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—122no—
252C2×Q8—122no—
253C2×Q8—121no—
254C2×Q8C2×C3×D411yescharacteristic
255C2×D4—21no—
256C2×D4—31no—
257C2×D4—31no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4C22×A411yescharacteristic
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—62no—
274C22×C4—61no—
275C22×C4C22×A411yescharacteristic
276SD16—121no—
277SD16—121no—
278SD16—122no—
279D8—121no—
280D8—121no—
281D8—122no—
282C2×C8—31no—
283C2×C8—31no—
284C2×C8C22×A411yescharacteristic
285C2×C8C22×A411yescharacteristic
286C4⋊C4—31no—
287C4⋊C4—31no—
288C4⋊C4—32no—
289C4⋊C4—31no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4C22×A411yescharacteristic
297C4⋊C4C22×A411yescharacteristic
298C4⋊C4C22×A411yescharacteristic
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—62no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C22⋊C4—61no—
309C42—21no—
310C42—31no—
311C42—31no—
312C42—31no—
313C42—31no—
314C42—61no—
315C42—61no—
316C22×C3—41no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—161no—
321C22×C3—161no—
322C22×C3—161no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—81no—
326C3×C4—161no—
327C3×C4—161no—
328C3×C4—161no—
329C23—21no—
330C23—61no—
331C23—61no—
332C23D4×A411yescharacteristic
333Q8—31no—
334Q8—61no—
335Q8—61no—
336Q8—61no—
337Q8—61no—
338Q8—121no—
339Q8—121no—
340Q8—122no—
341Q8C3×AGL(1,ℤ/8)11yescharacteristic
342D4—21no—
343D4—21no—
344D4—42no—
345D4—61no—
346D4—61no—
347D4—61no—
348D4—61no—
349D4—121no—
350D4—121no—
351D4—121no—
352D4—121no—
353D4—122no—
354D4—122no—
355D4—122no—
356D4—241no—
357D4—241no—
358D4—241no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—122no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—124no—
381C2×C4—241no—
382C2×C4D4×A411yescharacteristic
383C2×C4C23×A411yescharacteristic
384C8—21no—
385C8—21no—
386C8—61no—
387C8—61no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—161no—
392C2×C3—321no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—121no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22C2×D4×A411yescharacteristic
406C4—21no—
407C4—31no—
408C4—31no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—61no—
413C4—61no—
414C4—61no—
415C4—121no—
416C4—121no—
417C4#594511yescharacteristic
418C4#594511yescharacteristic
419C3—41noSylow
420C2—41no—
421C2—81no—
422C2—241no—
423C2—241no—
424C2#3487⋊C311yescharacteristic
425C2A4×AGL(1,ℤ/8)11yescharacteristic
426C2#3584⋊C311yescharacteristic
subgroupG/H
center, socleC22C2×D4×A4
commutator, cosocleC4×Q8C23×C3
FrattiniC2×C4C23×A4
Fitting#61903C3
hypercenter#482A4
hypocenterQ8C3×AGL(1,ℤ/8)
LCSG ▷ C4×Q8 ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #3497⋊C3C2semidirect
2. #3480⋊C3C2semidirect
3. #3392⋊C3C2semidirect
4. #3391⋊C3C2semidirect
5. #3359⋊C3C2semidirect
6. #3358⋊C3C2semidirect
7. #2984⋊C3C2semidirect
8. #61903C3semidirect
22. #3480C2×C3semidirect
25. #3392C2×C3semidirect
26. #3391C2×C3semidirect
29. #3359C2×C3semidirect
30. #3358C2×C3semidirect
35. #2984C2×C3semidirect

Maximal subgroups

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

subgroupindex
#3497⋊C32
#3480⋊C32
#3392⋊C32
#3391⋊C32
#3359⋊C32
#3358⋊C32
#2984⋊C32
#619033
C3×#4824