Finite Group Databasebeta

← #10692813

Subgroups — #10692813

Order 768.

576 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 89 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#83289C211yesmaximal, characteristic
2#73395C211yesmaximal, characteristic, complemented
3C3×#1505C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic, complemented
5#70808C211yesmaximal, characteristic, complemented
6#67748C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#9441—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#951—21no—
11C2×#949—21no—
12C2×#875C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×#376C2211yescharacteristic
15#5086C412yescomplemented
16#5074C412yescomplemented
17#5059—22no—
18#4940—21no—
19C4×(C2×D3)⋊C4—21no—
20#4932C2211yescharacteristic
21#4926C2211yescharacteristic
22C4×#208C2211yescharacteristic
23#4669—21no—
24#4666—21no—
25#4665—21no—
26#4646—21no—
27C3×#327—21no—
28C3×#327—21no—
29#4537—21no—
30#4537—21no—
31#4464—22no—
32#3153—31no—
33#2372—31no—
34#1533—31no—
35#1505—31no—
36#1505D311yescharacteristic, complemented
37#1300—31no—
38#1300—31no—
39C2×C3×#41—21no—
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C22×Dic6D411yescharacteristic
43C3×C4⋊Q8C2×C411yescharacteristic, complemented
44C3×#127C2×C411yescharacteristic, complemented
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52#951—42no—
53#951—42no—
54#949—42no—
55#949—42no—
56C2×#216—22no—
57C2×#215—22no—
58C2×#214—22no—
59C2×#213—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C2×C4×Dic3—21no—
68C2×#208—21no—
69#875—21no—
70#875—21no—
71#875C2×C412yes—
72#875D412yes—
73C3×C4⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76C3×#95—21no—
77C3×#95D411yescharacteristic
78C3×#95D411yescharacteristic
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95D411yescharacteristic
83C3⋊#95D411yescharacteristic
84#830—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803C2×C412yescomplemented
88C4×C3⋊C8—21no—
89C2×#137—61no—
90C2×AGL(1,ℤ/8)—61no—
91#532C2×D311yescharacteristic
92C2×#124—31no—
93#497—32no—
94#496—32no—
95#495—62no—
96#432—31no—
97#432C2×D311yescharacteristic
98C4×M16—31no—
99#391—61no—
100#388—61no—
101#388—61no—
102#386—61no—
103C4×C22⋊C4—61no—
104C4×C22⋊C4—61no—
105#376—31no—
106#376C2×D311yescharacteristic
107#329—62no—
108#327—21no—
109#327—21no—
110#327—61no—
111#327—61no—
112C23×D3—21no—
113C3×#41—41no—
114C3×#41—41no—
115C3×#41—41no—
116C3×#41—41no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—41no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C2×D12—21no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12C22⋊C412yes—
135C2×C4×D3—42no—
136C2×C4×D3—42no—
137C2×C4×D3—81no—
138C2×Dic6—21no—
139C2×Dic6—41no—
140C2×Dic6—41no—
141C2×Dic6C22⋊C412yes—
142C2×C3×C8—41no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42—42no—
148C3×C42C22×C411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151C3×C42#4111yescharacteristic
152#216—44no—
153#215—44no—
154#214—44no—
155#213—44no—
156(C2×D3)⋊C4—22no—
157(C2×D3)⋊C4—22no—
158(C2×D3)⋊C4—22no—
159(C2×D3)⋊C4—22no—
160(C2×D3)⋊C4—42no—
161(C2×D3)⋊C4—42no—
162(C2×D3)⋊C4—81no—
163Dic3⋊C4—81no—
164C4×Dic3—42no—
165C4×Dic3—81no—
166#208—22no—
167#208—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—41no—
170C2×#41—21no—
171C22×Q8—31no—
172C22×D4—31no—
173C23×C4—61no—
174#137—122no—
175#137—122no—
176AGL(1,ℤ/8)—122no—
177AGL(1,ℤ/8)—122no—
178C2×Q16—62no—
179C2×SD16—62no—
180C2×SD16—62no—
181C2×D8—62no—
182C2×M16—61no—
183C4⋊Q8C4×D311yescharacteristic, complemented
184#127C4×D311yescharacteristic, complemented
185#124—21no—
186#124—32no—
187#124—32no—
188#124—61no—
189#124—61no—
190#121—21no—
191#121—21no—
192C4×D4—21no—
193C4×D4—21no—
194C2×C4⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C42—21no—
201C2×C42—61no—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204C4⋊C8—21no—
205C4⋊C8—61no—
206C22⋊C8—21no—
207C22⋊C8—61no—
208#97—21no—
209#97—32no—
210#97—61no—
211C4×C8—61no—
212#95—21no—
213#95—31no—
214#95—31no—
215#95—61no—
216#95—61no—
217#95—61no—
218#95#6711yescharacteristic
219#95#6711yescharacteristic
220C23×C3—41no—
221C23×C3#12011yescharacteristic
222C22×D3—22no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C3×D4—81no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4C2×C22⋊C411yescharacteristic
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12311yescharacteristic
256C2×Dic3—22no—
257C2×Dic3—22no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265D12—42no—
266D12—81no—
267C4×D3—84no—
268Dic6—42no—
269Dic6—81no—
270C3×C8—81no—
271C3⋊C8—24no—
272C3⋊C8—81no—
273C24—61no—
274#41—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278C2×Q8—21no—
279C2×Q8—32no—
280C2×Q8—61no—
281C2×Q8—121no—
282C2×Q8—121no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—32no—
287C2×D4—41no—
288C2×D4—61no—
289C2×D4—121no—
290C2×D4—121no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—122no—
298C22×C4—122no—
299C22×C4—241no—
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302C22×C4C2×#6711yescharacteristic
303Q16—124no—
304SD16—124no—
305SD16—124no—
306D8—124no—
307M16—62no—
308M16—62no—
309C2×C8—41no—
310C2×C8—62no—
311C2×C8—121no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—241no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—62no—
320C22⋊C4—62no—
321C22⋊C4—122no—
322C22⋊C4—122no—
323C22⋊C4—241no—
324C42—42no—
325C42—122no—
326C42—241no—
327C42C2×C4×D311yescharacteristic
328C42#23511yescharacteristic
329C42D3×D411yescharacteristic
330C42#23711yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3#38611yescharacteristic
337C22×C3#45311yescharacteristic
338C22×C3#45311yescharacteristic
339C2×D3—42no—
340C2×D3—42no—
341C2×D3—42no—
342C2×D3—81no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—42no—
353C3×C4—81no—
354C3×C4—81no—
355Dic3—42no—
356Dic3—42no—
357Dic3—81no—
358C23—41no—
359C23—62no—
360C23—121no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23—121no—
366C23#93711yescharacteristic
367Q8—41no—
368Q8—41no—
369Q8—122no—
370Q8—241no—
371D4—41no—
372D4—41no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—122no—
378D4—241no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—244no—
402C2×C4D3×C22⋊C411yescharacteristic
403C2×C4#67⋊C411yescharacteristic
404C2×C4#88311yescharacteristic
405C2×C4#88411yescharacteristic
406C2×C4C4×#6711yescharacteristic
407C2×C4#93811yescharacteristic
408C8—64no—
409C8—81no—
410C8—241no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—161no—
414C2×C3#188711yescharacteristic
415C2×C3#190611yescharacteristic
416C2×C3#190611yescharacteristic
417D3—82no—
418C22—21no—
419C22—21no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—122no—
424C22—122no—
425C22—122no—
426C22—241no—
427C22—241no—
428C22—241no—
429C22—241no—
430C22#495611yescharacteristic
431C22#519811yescharacteristic
432C22#519811yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—122no—
443C4—122no—
444C4—241no—
445C3#944111yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—161no—
449C2—242no—
450C2#7005511yescharacteristic
451C2#7032311yescharacteristic
452C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1505C2
socleC22×C3#386
prime radical#83289C2
cosocleC2×C3×C42C23
hypercenter#1505D3
hypocenterC3#9441
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1505

Complements

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

NH ≅ G/Ntype
2. #73395C2semidirect
3. C3×#1505C2semidirect
4. #71812C2semidirect
5. #70808C2semidirect
6. #67748C2semidirect
7. #67615C2semidirect
15. #5086C4semidirect
16. #5074C4semidirect
36. #1505D3semidirect
43. C3×C4⋊Q8C2×C4semidirect
44. C3×#127C2×C4semidirect
87. #803C2×C4semidirect
183. C4⋊Q8C4×D3semidirect
184. #127C4×D3semidirect
445. C3#9441semidirect

Maximal subgroups

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

subgroupindex
#832892
#733952
C3×#15052
#718122
#708082
#677482
#676152
#94413