Finite Group Databasebeta

← #10695628

Subgroups — #10695628

Order 768.

786 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C3×#2434C211yesmaximal, characteristic, complemented
2C2×#5227C211yesmaximal, characteristic
3#82736C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic, complemented
5#71690C211yesmaximal, characteristic, complemented
6#69456C211yesmaximal, characteristic
7#67617C211yesmaximal, characteristic, complemented
8#9949—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×C3×C4⋊Q8—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×#215—21no—
13C2×#930C2211yescharacteristic
14C3×#395—22no—
15C3×#391—21no—
16C3×#385—22no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5227—21no—
19#5227—22no—
20#5227—22no—
21C2×#835—21no—
22C2×#835—22no—
23#5212C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5048—22no—
27#5046—21no—
28#5043—21no—
29#4968—21no—
30C2×#806—22no—
31#4538—22no—
32#3187—31no—
33C2×#448—31no—
34#2434D311yescharacteristic, complemented
35#1964—31no—
36#1787—31no—
37#1520—31no—
38#1282—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—44no—
45C22×C3×Q8D412yes—
46C22×C3×Q8D411yescharacteristic
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C22×#67—21no—
50C22×D12D411yescharacteristic
51C3×C4⋊Q8—44no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4D412yes—
61C2×C3×C42—21no—
62C2×#217—21no—
63C2×#215—24no—
64C2×#215—22no—
65C2×#215—44no—
66#930—21no—
67#930—21no—
68#930—21no—
69#930—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D411yescharacteristic
73C22×C3⋊C8—21no—
74C22×C3⋊C8D412yes—
75C3×#95—24no—
76C3×#95—21no—
77#835—24no—
78#835—22no—
79#835—44no—
80C3⋊#95—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#809—22no—
86#808—22no—
87#806—44no—
88C23×Q8C2×D311yescharacteristic
89C22×SD16—61no—
90C2×C4⋊Q8—21no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C2×#121—31no—
93#483—61no—
94#481—61no—
95#465—62no—
96#450—61no—
97#448—62no—
98#448—61no—
99#448—62no—
100C2×#103—61no—
101C2×#103—62no—
102C2×#99—62no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8—31no—
105#395—22no—
106#391—21no—
107#385—22no—
108#380—31no—
109C4×C22⋊C4C2×D311yescharacteristic
110#328—62no—
111C24×C3C2×D411yescharacteristic
112C23×D3—21no—
113C2×C3×Q8—24no—
114C2×C3×Q8—24no—
115C2×C3×Q8—24no—
116C2×C3×Q8—22no—
117C2×C3×Q8—21no—
118C2×C3×Q8—22no—
119C2×C3×Q8—21no—
120C2×C3×Q8—44no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C2×C3×Q8—44no—
126C2×C3×Q8—44no—
127C2×C3×Q8—41no—
128C2×C3×Q8—41no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—42no—
134C22×C3×C4—44no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—42no—
146C2×#67—42no—
147C22×Dic3—21no—
148C22×Dic3—41no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12—42no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—48no—
155C3×C4⋊C4—41no—
156C3×C4⋊C4—41no—
157C3×C22⋊C4—22no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4SD1614yescomplemented
161C3×C42—42no—
162#217—42no—
163#215—48no—
164#215—82no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—42no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—42no—
173C2×C3⋊C8—42no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—44no—
180C22×Q8#6711yescharacteristic
181C22×Q8#6712yes—
182C22×D4—31no—
183C22×D4—61no—
184C23×C4—21no—
185C23×C4C22×D311yescharacteristic
186C2×SD16—64no—
187C2×SD16—62no—
188C2×SD16—124no—
189C22×C8—32no—
190C22×C8—61no—
191C4⋊Q8—44no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196#121—61no—
197#121—61no—
198#121—61no—
199#121—61no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4#6711yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4#6712yes—
209C2×C42—21no—
210#106—62no—
211#103—64no—
212#103—62no—
213#103—124no—
214#102—62no—
215#99—124no—
216C22⋊C8—32no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C22⋊C8—61no—
220#95—24no—
221#95—21no—
222#95—61no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3#12011yescharacteristic
226C23×C3#12111yescharacteristic
227C23×C3#12711yescharacteristic
228C23×C3C2×SD1612yes—
229C23×C3#13712yes—
230C22×D3—41no—
231C22×D3—42no—
232C22×D3—81no—
233C22×D3—81no—
234C3×Q8—24no—
235C3×Q8—24no—
236C3×Q8—42no—
237C3×Q8—42no—
238C3×Q8—44no—
239C3×Q8—44no—
240C3×Q8—42no—
241C3×Q8—81no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—44no—
254C2×C3×C4—44no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4#12011yescharacteristic
261C2×C3×C4#12011yescharacteristic
262C2×C3×C4#12111yescharacteristic
263C2×C3×C4#12111yescharacteristic
264#67—82no—
265#67—81no—
266#67—81no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269C2×Dic3—81no—
270C2×Dic3—81no—
271C2×Dic3—81no—
272D12—42no—
273D12—42no—
274C3⋊C8—42no—
275C3⋊C8—42no—
276C24—61no—
277C24C2×#6711yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—21no—
285C2×Q8—44no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—122no—
297C2×D4—121no—
298C2×D4—122no—
299C2×D4—122no—
300C2×D4—121no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—61no—
310C22×C4—121no—
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4#23711yescharacteristic
315C22×C4C2×#6711yescharacteristic
316C22×C4C2×#6711yescharacteristic
317SD16—128no—
318SD16—242no—
319C2×C8—62no—
320C2×C8—62no—
321C2×C8—122no—
322C2×C8—122no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—48no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—122no—
331C22⋊C4—22no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—122no—
337C22⋊C4#21514yescomplemented
338C42—42no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—42no—
346C22×C3—42no—
347C22×C3#39211yescharacteristic
348C22×C3#44812yes—
349C22×C3#46112yes—
350C22×C3#49211yescharacteristic
351C22×C3#49711yescharacteristic
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C2×D3—162no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—44no—
360C3×C4—44no—
361C3×C4—81no—
362C3×C4—81no—
363C3×C4B(4,𝔽2)12yes—
364Dic3—81no—
365Dic3—161no—
366C23—41no—
367C23—41no—
368C23—121no—
369C23—122no—
370C23—241no—
371C23—241no—
372C23#93811yescharacteristic
373C23#94011yescharacteristic
374C23C2×#21512yes—
375C23#94212yes—
376C23#95311yescharacteristic
377Q8—24no—
378Q8—24no—
379Q8—42no—
380Q8—42no—
381Q8—44no—
382Q8—44no—
383Q8—42no—
384Q8—81no—
385D4—122no—
386D4—122no—
387D4—242no—
388D4—241no—
389D4—241no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—121no—
409C2×C4—122no—
410C2×C4—241no—
411C2×C4—241no—
412C2×C4—241no—
413C2×C4#93711yescharacteristic
414C2×C4#93711yescharacteristic
415C2×C4#93911yescharacteristic
416C2×C4#93911yescharacteristic
417C8—122no—
418C8—122no—
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3#200911yescharacteristic
424C2×C3#201311yescharacteristic
425C2×C3#202111yescharacteristic
426D3—161no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—241no—
436C22—241no—
437C22—241no—
438C22—482no—
439C22#508312yes—
440C22#508511yescharacteristic
441C22#508611yescharacteristic
442C22#522311yescharacteristic
443C22#522712yes—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—44no—
448C4—44no—
449C4—81no—
450C4—81no—
451C4—241no—
452C4—481no—
453C4#503212yes—
454C3#994911yescharacteristic, complemented, Sylow
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—481no—
460C2#7167111yescharacteristic
461C2#7179111yescharacteristic
462C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2434C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#2434D3
hypocenterC3#9949
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
1. C3×#2434C2semidirect
4. #72132C2semidirect
5. #71690C2semidirect
7. #67617C2semidirect
34. #2434D3semidirect
160. C3×C22⋊C4SD16semidirect
337. C22⋊C4#215semidirect
454. C3#9949semidirect

Maximal subgroups

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

subgroupindex
C3×#24342
C2×#52272
#827362
#721322
#716902
#694562
#676172
#99493