Finite Group Databasebeta

← #10695645

Subgroups — #10695645

Order 768.

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

normal subgroups only

This group has 610 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
1#78612C211yesmaximal, characteristic
2C2×#4735C211yesmaximal, characteristic
3#73517C211yesmaximal, characteristic, complemented
4#69946C211yesmaximal, characteristic, complemented
5#69925C211yesmaximal, characteristic, complemented
6#69456C211yesmaximal, characteristic
7C3×#1282C211yesmaximal, characteristic, complemented
8#9951—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15#5212C2211yescharacteristic
16#5114—22no—
17C2×#821—21no—
18C2×#817—21no—
19C2×#806—22no—
20#4950—22no—
21#4935—21no—
22#4929—21no—
23#4738—21no—
24#4735—21no—
25#4735—22no—
26#4735—22no—
27#4731—21no—
28#4721—21no—
29#4661—22no—
30#4551—22no—
31#4467—22no—
32#3187—31no—
33C2×#481—31no—
34#2502—31no—
35#1964—31no—
36#1824—31no—
37#1520—31no—
38#1282D311yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3—21no—
42C22×D12D411yescharacteristic
43C22×Dic6D411yescharacteristic
44C22×C3×C8—21no—
45C22×C3×C8D412yes—
46C2×C3×C22⋊C4D412yes—
47C2×C3×C42—21no—
48C2×#217—21no—
49C2×#217—21no—
50#930—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—24no—
62C2×C4⋊Dic3D412yes—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65#924—21no—
66#924—21no—
67#924—21no—
68#924—21no—
69C2×Dic3⋊C4—21no—
70C2×#204—22no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yes—
75C3×#95—21no—
76C3⋊#95—24no—
77C3⋊#95—21no—
78#821—22no—
79#821—24no—
80#817—22no—
81#817—24no—
82#816—22no—
83#806—44no—
84C2×C22⋊Q8—31no—
85C2×#121—31no—
86C22×C4⋊C4—31no—
87#483—61no—
88#481—62no—
89#481—61no—
90#481—62no—
91#465—62no—
92#450—61no—
93#448—61no—
94C2×#106—61no—
95C2×#102—61no—
96C2×#99—62no—
97C2×C22⋊C8C2×D311yescharacteristic
98C2×C22⋊C8C2×D311yescharacteristic
99#398—62no—
100#391—61no—
101#385—62no—
102#384—61no—
103#380—31no—
104C4×C22⋊C4C2×D311yescharacteristic
105#340—62no—
106#328—62no—
107C24×C3C2×D411yescharacteristic
108C23×D3—21no—
109C22×C3×C4—22no—
110C22×C3×C4—42no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C2×#67—41no—
118C2×#67—41no—
119C2×#67—42no—
120C2×#67—42no—
121C22×Dic3—21no—
122C22×Dic3—22no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—41no—
126C22×Dic3—41no—
127C22×Dic3—41no—
128C22×Dic3—82no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—42no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—42no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—42no—
138C2×C3×C8—42no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4SD1614yescomplemented
141C3×C42—42no—
142#217—41no—
143#217—41no—
144#217—42no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—22no—
151C4⋊Dic3—21no—
152C4⋊Dic3—22no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155C4⋊Dic3—42no—
156C4⋊Dic3—84no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159#204—82no—
160C22×Q8—31no—
161C22×D4—31no—
162C22×D4—61no—
163C23×C4—61no—
164C23×C4C22×D311yescharacteristic
165C2×SD16—62no—
166C22×C8—21no—
167C22×C8D1212yes—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172#121—61no—
173#121—61no—
174#121—61no—
175#121—61no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—64no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4D1212yes—
191C2×C42—21no—
192#106—64no—
193#106—62no—
194#103—62no—
195#102—64no—
196#102—62no—
197#99—124no—
198C22⋊C8—21no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8D1212yes—
202#95—21no—
203#95—64no—
204#95—61no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3#12011yescharacteristic
208C23×C3#12111yescharacteristic
209C23×C3#12711yescharacteristic
210C23×C3C2×SD1612yes—
211C23×C3#13712yes—
212C22×D3—41no—
213C22×D3—42no—
214C22×D3—81no—
215C22×D3—81no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229#67—82no—
230#67—81no—
231#67—81no—
232C2×Dic3—42no—
233C2×Dic3—44no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—82no—
241C2×Dic3—82no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245D12—42no—
246D12—42no—
247Dic6—42no—
248Dic6—81no—
249C3×C8—42no—
250C3×C8—42no—
251C24—61no—
252C24C2×D1211yescharacteristic
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8—122no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—122no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—122no—
262C2×D4—122no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—61no—
266C22×C4—62no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—242no—
273C22×C4C2×D1211yescharacteristic
274C22×C4C2×D1211yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4#23911yescharacteristic
279SD16—242no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—62no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—62no—
289C4⋊C4—61no—
290C4⋊C4—122no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—244no—
296C22⋊C4—22no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C22⋊C4—122no—
302C22⋊C4#20414yescomplemented
303C42—42no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3#39211yescharacteristic
313C22×C3#44812yes—
314C22×C3#46112yes—
315C22×C3#49211yescharacteristic
316C22×C3#49711yescharacteristic
317C2×D3—81no—
318C2×D3—81no—
319C2×D3—81no—
320C2×D3—162no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—44no—
325C3×C4#45812yes—
326Dic3—81no—
327Dic3—82no—
328Dic3—81no—
329Dic3—81no—
330Dic3—161no—
331C23—41no—
332C23—41no—
333C23—121no—
334C23—122no—
335C23—241no—
336C23—241no—
337C23#87411yescharacteristic
338C23#88211yescharacteristic
339C23#89511yescharacteristic
340C23C2×#20412yes—
341C23#90912yes—
342Q8—122no—
343Q8—241no—
344D4—122no—
345D4—122no—
346D4—241no—
347D4—241no—
348D4—242no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—122no—
359C2×C4—124no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—242no—
367C2×C4—242no—
368C2×C4—241no—
369C2×C4—241no—
370C2×C4—241no—
371C2×C4#88211yescharacteristic
372C2×C4#88211yescharacteristic
373C2×C4#89511yescharacteristic
374C2×C4#89511yescharacteristic
375C8—42no—
376C8—42no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—22no—
380C2×C3—21no—
381C2×C3#200811yescharacteristic
382C2×C3#201311yescharacteristic
383C2×C3#202211yescharacteristic
384D3—161no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—241no—
394C22—241no—
395C22—241no—
396C22—482no—
397C22#467111yescharacteristic
398C22#469311yescharacteristic
399C22#471511yescharacteristic
400C22#473812yes—
401C22#484112yes—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—44no—
406C4—241no—
407C4—241no—
408C4—242no—
409C4—241no—
410C4—481no—
411C4#474212yes—
412C3#995111yescharacteristic, complemented, Sylow
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—481no—
418C2#6868711yescharacteristic
419C2#6896611yescharacteristic
420C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1282D3
hypocenterC3#9951
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
3. #73517C2semidirect
4. #69946C2semidirect
5. #69925C2semidirect
7. C3×#1282C2semidirect
38. #1282D3semidirect
140. C3×C22⋊C4SD16semidirect
302. C22⋊C4#204semidirect
412. C3#9951semidirect

Maximal subgroups

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

subgroupindex
#786122
C2×#47352
#735172
#699462
#699252
#694562
C3×#12822
#99513