Finite Group Databasebeta

← #10697672

Subgroups — #10697672

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#79059C211yesmaximal, characteristic, complemented
2#79055C211yesmaximal, characteristic
3#79027C211yesmaximal, characteristic, complemented
4#70501C211yesmaximal, characteristic, complemented
5#70498C211yesmaximal, characteristic, complemented
6#70470C211yesmaximal, characteristic, complemented
7C3×#1292C211yesmaximal, characteristic, complemented
8#10256—31nomaximal, Sylow
9C2×#908—21no—
10#5477C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16#5133—21no—
17#5131—21no—
18#5121—21no—
19#5114—22no—
20C2×(C2×D3)⋊C8—22no—
21C2×C4×C3⋊C8C2211yescharacteristic
22#4851—21no—
23#4848—21no—
24#4847—22no—
25#4846—21no—
26#4844—22no—
27#4843—21no—
28#4842—21no—
29#4842—21no—
30#4841—21no—
31#4841—21no—
32#4484—22no—
33#4483—22no—
34#3246—31no—
35#3077—31no—
36#3036—31no—
37#1954—31no—
38#1493—31no—
39#1492—31no—
40#1292D311yescharacteristic, complemented
41C2×#235—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—21no—
45C2×C3×M16—21no—
46C22×C3×C8D412yes—
47C2×C3×C42C2311yescharacteristic
48#930—21no—
49#930—21no—
50C2×(C2×D3)⋊C4—21no—
51#926—21no—
52C2×C4⋊Dic3D411yescharacteristic
53#924—21no—
54#924—21no—
55C22×C3⋊C8—21no—
56#908—42no—
57#908—42no—
58C2×D24—22no—
59C2×#204—22no—
60C2×#204—22no—
61#875—21no—
62#874D411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
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×C4⋊C8—21no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8D412yes—
80C3×C4⋊C8D412yes—
81#821—22no—
82#821—22no—
83#821—22no—
84#821—22no—
85(C2×D3)⋊C8—44no—
86#818—22no—
87#817—22no—
88#817—22no—
89#816—22no—
90#816—22no—
91C4×C3⋊C8—21no—
92C4×C3⋊C8—21no—
93C2×AGL(1,ℤ/8)—61no—
94#533—31no—
95#532—31no—
96C2×C4×D4—31no—
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#423C2×D311yescharacteristic
110#423C2×D311yescharacteristic
111C2×C4⋊C8C2×D311yescharacteristic
112#418—61no—
113#417—61no—
114C2×C22⋊C8—62no—
115C2×C4×C8—31no—
116Q8⋊C8—62no—
117D4⋊C8—62no—
118C23×D3—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—41no—
123C22×Dic3—21no—
124#235—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12SD1612yescomplemented
137C2×C4×D3—42no—
138C2×C4×D3—42no—
139C2×C4×D3—41no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C2×Dic6—21no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C3×M16—41no—
146C3×M16—41no—
147C3×M16—41no—
148C3×M16—41no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—42no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157#217—41no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172C4×Dic3—41no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176D24—44no—
177#204—44no—
178#204—82no—
179C2×#41—61no—
180C22×D4—31no—
181C23×C4—61no—
182AGL(1,ℤ/8)—122no—
183AGL(1,ℤ/8)—122no—
184C2×SD16—62no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×M16—21no—
188C2×M16—21no—
189C22×C8—61no—
190C22×C8D1212yes—
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—61no—
204C4×D4—61no—
205C4×D4—61no—
206C4×D4—61no—
207C4×D4—61no—
208C4×D4—61no—
209#117—61no—
210C2×C4⋊C4—31no—
211C2×C22⋊C4—61no—
212C2×C42C22×D311yescharacteristic
213#107—62no—
214#106—62no—
215#106—62no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8D1212yes—
221C4⋊C8D1212yes—
222#103—62no—
223#103—62no—
224#102—62no—
225#102—62no—
226#102—62no—
227#102—62no—
228C22⋊C8—124no—
229C4×C8—61no—
230C4×C8—61no—
231C23×C3#12111yescharacteristic
232C22×D3—22no—
233C22×D3—42no—
234C22×D3—42no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—41no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—41no—
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12711yescharacteristic
247C2×C3×C4C2×SD1611yescharacteristic
248C2×C3×C4#13511yescharacteristic
249C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
250C2×C3×C4#13711yescharacteristic
251#67—81no—
252#67—81no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258D12—22no—
259D12—22no—
260D12—42no—
261D12—41no—
262D12—41no—
263D12—41no—
264D12—41no—
265D12—41no—
266D12—41no—
267C4×D3—82no—
268C4×D3—81no—
269C4×D3—81no—
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—61no—
282#41—121no—
283#41—121no—
284#41—121no—
285#41—121no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—61no—
289C2×D4—32no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—121no—
299C22×C4—61no—
300C22×C4—122no—
301C22×C4—122no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4C2×D1211yescharacteristic
306C22×C4D3×D411yescharacteristic
307C22×C4#23911yescharacteristic
308SD16—124no—
309SD16—242no—
310D8—124no—
311M16—41no—
312M16—41no—
313M16—41no—
314M16—41no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—42no—
319C2×C8—62no—
320C2×C8—121no—
321C2×C8—121no—
322C4⋊C4—61no—
323C4⋊C4—61no—
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—121no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C42—121no—
338C42C2×D1211yescharacteristic
339C42C2×D1211yescharacteristic
340C42D3×D411yescharacteristic
341C42D3×D411yescharacteristic
342C22×C3—21no—
343C22×C3—21no—
344C22×C3#39211yescharacteristic
345C22×C3#46511yescharacteristic
346C22×C3#46811yescharacteristic
347C2×D3—44no—
348C2×D3—42no—
349C2×D3—82no—
350C2×D3—81no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4#44811yescharacteristic
359C3×C4#44911yescharacteristic
360C3×C4#45011yescharacteristic
361C3×C4#45211yescharacteristic
362C3×C4#45311yescharacteristic
363C3×C4#45411yescharacteristic
364Dic3—81no—
365Dic3—81no—
366Dic3—81no—
367Dic3—81no—
368C23—62no—
369C23—122no—
370C23—122no—
371C23—121no—
372C23—121no—
373C23—121no—
374C23#89511yescharacteristic
375Q8—122no—
376Q8—121no—
377Q8—121no—
378Q8—121no—
379Q8—121no—
380D4—62no—
381D4—62no—
382D4—122no—
383D4—121no—
384D4—121no—
385D4—121no—
386D4—121no—
387D4—121no—
388D4—121no—
389D4—241no—
390D4—241no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—122no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—242no—
400C2×C4—241no—
401C2×C4—241no—
402C2×C4—241no—
403C2×C4—241no—
404C2×C4#87411yescharacteristic
405C2×C4#88211yescharacteristic
406C2×C4#88211yescharacteristic
407C2×C4#88211yescharacteristic
408C2×C4#89511yescharacteristic
409C2×C4#89511yescharacteristic
410C2×C4#91111yescharacteristic
411C2×C4D3×SD1611yescharacteristic
412C2×C4#91611yescharacteristic
413C2×C4#91811yescharacteristic
414C8—42no—
415C8—42no—
416C8—42no—
417C8—122no—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3#200011yescharacteristic
421C2×C3#200611yescharacteristic
422C2×C3#201711yescharacteristic
423D3—82no—
424D3—161no—
425C22—21no—
426C22—21no—
427C22—124no—
428C22—122no—
429C22—242no—
430C22—241no—
431C22—241no—
432C22—241no—
433C22—241no—
434C22—241no—
435C22#467111yescharacteristic
436C22#486411yescharacteristic
437C22#488611yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—241no—
442C4—241no—
443C4—241no—
444C4—241no—
445C4#474711yescharacteristic
446C4#475311yescharacteristic
447C4#477611yescharacteristic
448C4#478111yescharacteristic
449C4#480611yescharacteristic
450C4#480811yescharacteristic
451C3#1025611yescharacteristic, complemented, Sylow
452C2—21no—
453C2—21no—
454C2—242no—
455C2—481no—
456C2#6872711yescharacteristic
457C2#6888711yescharacteristic
458C2#6900811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#392
prime radical#79055C2
hypercenter#1292D3
hypocenterC3#10256
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
1. #79059C2semidirect
3. #79027C2semidirect
4. #70501C2semidirect
5. #70498C2semidirect
6. #70470C2semidirect
7. C3×#1292C2semidirect
40. #1292D3semidirect
136. C2×D12SD16semidirect
451. C3#10256semidirect

Maximal subgroups

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

subgroupindex
#790592
#790552
#790272
#705012
#704982
#704702
C3×#12922
#102563