Finite Group Databasebeta

← #10697670

Subgroups — #10697670

Order 768.

580 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 103 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#82307C211yesmaximal, characteristic
2#82296C211yesmaximal, characteristic, complemented
3#79027C211yesmaximal, characteristic, complemented
4C3×#1954C211yesmaximal, characteristic, complemented
5#69738C211yesmaximal, characteristic, complemented
6#69735C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#10256—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#932—21no—
11#5477C2211yescharacteristic
12#5470C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C22⋊C8—22no—
15C2×C3×C4×C8C2211yescharacteristic
16#5215—21no—
17#5114—22no—
18#5019—21no—
19#5016—22no—
20#5015—21no—
21#5009—22no—
22#5008—21no—
23#5007—21no—
24#4966—21no—
25#4964—21no—
26#4924C2211yescharacteristic
27#4924C2211yescharacteristic
28#4851—21no—
29#4848—21no—
30#4842—21no—
31#4841—21no—
32#4459—22no—
33#4456—22no—
34#3246—31no—
35#3077—31no—
36#3036—31no—
37#1954D311yescharacteristic, complemented
38#1493—31no—
39#1492—31no—
40#1292—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C22×C3×C8—21no—
45C22×C3×C8D412yes—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#214—22no—
54#932—42no—
55#932—42no—
56C2×#213—22no—
57#930—21no—
58#930—21no—
59#926—21no—
60#924—21no—
61#924—21no—
62C2×#208—21no—
63C2×#208—21no—
64C2×#204—22no—
65#875—21no—
66#874D411yescharacteristic
67C4×D12—21no—
68C4×D12—21no—
69#870—21no—
70#869D411yescharacteristic
71#869D411yescharacteristic
72#869D411yescharacteristic
73C4×Dic6—21no—
74C4×Dic6—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×C22⋊C8—44no—
78C3×C4×C8—21no—
79C3×C4×C8—21no—
80#832—22no—
81#832—22no—
82#821—22no—
83#817—22no—
84#816—22no—
85#810—22no—
86#809—22no—
87#808—22no—
88#807—22no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)D412yes—
92C4⋊(C3⋊C8)D412yes—
93C2×AGL(1,ℤ/8)—61no—
94#533—31no—
95#532—31no—
96C2×C4×D4C2×D311yescharacteristic
97#478—62no—
98#476—61no—
99#476—61no—
100#475—61no—
101#474—61no—
102#465—62no—
103#463—62no—
104#461—61no—
105#461—61no—
106#460—61no—
107#459—61no—
108#428—61no—
109#423—31no—
110#423—31no—
111C2×C4⋊C8C2×D311yescharacteristic
112#418—61no—
113#417—61no—
114C2×C22⋊C8—22no—
115C2×C4×C8C2×D311yescharacteristic
116Q8⋊C8—62no—
117D4⋊C8—62no—
118C24×C3—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4SD1612yescomplemented
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4—42no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C2×#67—41no—
134#235—41no—
135#235—41no—
136#235—41no—
137#235—41no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×C4×D3—41no—
142C2×Dic6—21no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C2×C3×C8—42no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158#217—41no—
159#214—44no—
160#213—44no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171C4×Dic3—41no—
172#208—41no—
173#208—41no—
174#208—41no—
175#208—41no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
178#204—82no—
179C2×#41—61no—
180C22×D4D1211yescharacteristic
181C23×C4—21no—
182AGL(1,ℤ/8)—122no—
183AGL(1,ℤ/8)—122no—
184C2×SD16—62no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×M16—61no—
188C2×M16—61no—
189C22×C8—21no—
190C22×C8#6712yes—
191C4⋊Q8—31no—
192C4⋊Q8—31no—
193C4⋊Q8—31no—
194#127—31no—
195#125—61no—
196#124—61no—
197C22⋊Q8—61no—
198C22⋊Q8—61no—
199#121—61no—
200#121—61no—
201C4×Q8—61no—
202C4×Q8—61no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—61no—
208C4×D4—61no—
209#117—61no—
210C2×C4⋊C4D1211yescharacteristic
211C2×C22⋊C4—21no—
212C2×C42C22×D311yescharacteristic
213#107—62no—
214#106—62no—
215#106—62no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—32no—
219C4⋊C8—32no—
220C4⋊C8—61no—
221C4⋊C8—61no—
222#103—62no—
223#103—62no—
224#102—62no—
225#102—62no—
226#102—62no—
227#102—62no—
228C22⋊C8—44no—
229C4×C8—21no—
230C4×C8—21no—
231C23×C3—22no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3#12111yescharacteristic
237C22×D3—41no—
238C3×D4—22no—
239C3×D4—22no—
240C3×D4—41no—
241C3×D4—41no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—82no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12711yescharacteristic
256C2×C3×C4C2×SD1611yescharacteristic
257C2×C3×C4#13511yescharacteristic
258C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
259C2×C3×C4#13711yescharacteristic
260#67—81no—
261#67—81no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265D12—42no—
266D12—41no—
267D12—41no—
268D12—41no—
269D12—41no—
270C4×D3—81no—
271C4×D3—81no—
272Dic6—42no—
273Dic6—41no—
274Dic6—41no—
275Dic6—41no—
276Dic6—41no—
277C3×C8—42no—
278C3×C8—42no—
279C3⋊C8—42no—
280C3⋊C8—42no—
281C24—21no—
282#41—121no—
283#41—121no—
284#41—121no—
285#41—121no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—121no—
298C2×D4#20412yescomplemented
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—121no—
305C22×C4C2×D1211yescharacteristic
306C22×C4#23511yescharacteristic
307C22×C4C2×#6711yescharacteristic
308SD16—124no—
309SD16—242no—
310D8—124no—
311M16—121no—
312M16—121no—
313M16—121no—
314M16—121no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—62no—
321C2×C8—62no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—121no—
331C4⋊C4—121no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C42—121no—
338C42D3×D411yescharacteristic
339C42D3×D411yescharacteristic
340C42D3×D411yescharacteristic
341C42D3×D411yescharacteristic
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—44no—
345C22×C3—42no—
346C22×C3—82no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3#39211yescharacteristic
350C22×C3#46511yescharacteristic
351C22×C3#46811yescharacteristic
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—81no—
359C3×C4#44811yescharacteristic
360C3×C4#44911yescharacteristic
361C3×C4#45011yescharacteristic
362C3×C4#45211yescharacteristic
363C3×C4#45311yescharacteristic
364C3×C4#45411yescharacteristic
365Dic3—81no—
366Dic3—81no—
367Dic3—81no—
368C23—22no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—121no—
374C23#93011yescharacteristic
375Q8—122no—
376Q8—121no—
377Q8—121no—
378Q8—121no—
379Q8—121no—
380D4—22no—
381D4—22no—
382D4—41no—
383D4—41no—
384D4—122no—
385D4—121no—
386D4—121no—
387D4—121no—
388D4—121no—
389D4—241no—
390D4—241no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—241no—
403C2×C4—241no—
404C2×C4#88211yescharacteristic
405C2×C4#88211yescharacteristic
406C2×C4#88411yescharacteristic
407C2×C4#88411yescharacteristic
408C2×C4C2×#20411yescharacteristic
409C2×C4#90411yescharacteristic
410C2×C4#90811yescharacteristic
411C2×C4#90911yescharacteristic
412C2×C4#93711yescharacteristic
413C2×C4#94011yescharacteristic
414C8—42no—
415C8—42no—
416C8—122no—
417C8—122no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—82no—
421C2×C3#200011yescharacteristic
422C2×C3#200611yescharacteristic
423C2×C3#201711yescharacteristic
424D3—161no—
425C22—21no—
426C22—21no—
427C22—44no—
428C22—42no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22—241no—
433C22—241no—
434C22—241no—
435C22#496011yescharacteristic
436C22#511411yescharacteristic
437C22#513411yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—241no—
443C4—241no—
444C4—241no—
445C4#473111yescharacteristic
446C4#473311yescharacteristic
447C4#473411yescharacteristic
448C4#473811yescharacteristic
449C4#506211yescharacteristic
450C4#507711yescharacteristic
451C3#1025611yescharacteristic, complemented, Sylow
452C2—21no—
453C2—21no—
454C2—82no—
455C2—481no—
456C2#6966411yescharacteristic
457C2#6996211yescharacteristic
458C2#7001311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1954C2
socleC22×C3#392
prime radical#82307C2
hypercenter#1954D3
hypocenterC3#10256
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
2. #82296C2semidirect
3. #79027C2semidirect
4. C3×#1954C2semidirect
5. #69738C2semidirect
6. #69735C2semidirect
7. #67705C2semidirect
37. #1954D3semidirect
124. C2×C3×D4SD16semidirect
298. C2×D4#204semidirect
451. C3#10256semidirect

Maximal subgroups

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

subgroupindex
#823072
#822962
#790272
C3×#19542
#697382
#697352
#677052
#102563