Finite Group Databasebeta

← #10695861

Subgroups — #10695861

Order 768.

852 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#5218C211yesmaximal, characteristic, complemented
2C2×#5217C211yesmaximal, characteristic
3#78642C211yesmaximal, characteristic, complemented
4C3×#1783C211yesmaximal, characteristic, complemented
5#69972C211yesmaximal, characteristic, complemented
6#69970C211yesmaximal, characteristic, complemented
7#67688C211yesmaximal, characteristic, complemented
8#9987—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×#214—21no—
11C22×#213—21no—
12C2×#930C2211yescharacteristic
13C2×#924C2211yescharacteristic
14C2×C3×#100—22no—
15C3×#407C2211yescharacteristic
16C3×#407C2211yescharacteristic
17#5218—22no—
18#5218—22no—
19#5217—22no—
20#5217—22no—
21C2×#832—21no—
22C2×#832—21no—
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5134—22no—
26#4738—21no—
27#4736—21no—
28#4731—21no—
29#4722—21no—
30#4469—22no—
31#4468—22no—
32#3193—31no—
33C2×#450—31no—
34C2×#447—31no—
35#1783D311yescharacteristic, complemented
36#1472—31no—
37#1471—31no—
38#1298—31no—
39C25×C3—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—44no—
46C22×C3×D4D412yes—
47C23×C3×C4C2311yescharacteristic
48C22×#67—21no—
49C22×D12D411yescharacteristic
50C22×Dic6D411yescharacteristic
51C2×C3×M16—21no—
52C2×C3×M16D412yes—
53C2×#217—21no—
54C2×#214—24no—
55C2×#214—44no—
56C2×#213—24no—
57C2×#213—44no—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65#924—21no—
66#924—21no—
67#924—21no—
68#924—21no—
69C2×Dic3⋊C4—21no—
70C22×C3⋊C8—21no—
71C22×C3⋊C8—21no—
72C2×#204—22no—
73C3×#100—44no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78#832—24no—
79#832—24no—
80#830—21no—
81#830—21no—
82#830D412yes—
83#830D412yes—
84#821—22no—
85#818—22no—
86#816—22no—
87C23×D4C2×D311yescharacteristic
88C22×SD16—61no—
89C22×D8—61no—
90C2×C22⋊Q8—31no—
91C2×#121—31no—
92#484—61no—
93#480—61no—
94#468—62no—
95#450—61no—
96#450—62no—
97#450—62no—
98#448—61no—
99#447—62no—
100#447—62no—
101C2×#102—61no—
102C2×#102—61no—
103C2×#100—22no—
104#407C2×D311yescharacteristic
105#407C2×D311yescharacteristic
106C2×C22⋊C8—31no—
107C2×C22⋊C8—31no—
108#324—62no—
109#323—62no—
110C24×C3—22no—
111C24×C3—44no—
112C24×C3—41no—
113C24×C3—41no—
114C24×C3—82no—
115C24×C3—82no—
116C24×C3#4111yescharacteristic
117C23×D3—21no—
118C2×C3×D4—24no—
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—44no—
127C2×C3×D4—44no—
128C2×C3×D4—44no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C22×C3×C4—42no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C2×#67—41no—
139C2×#67—41no—
140C2×#67—82no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12—42no—
147C2×Dic6—21no—
148C2×Dic6—21no—
149C2×Dic6—42no—
150C3×M16—42no—
151C3×M16—42no—
152C2×C3×C8—22no—
153C2×C3×C8—41no—
154#217—41no—
155#217—41no—
156#214—48no—
157#213—48no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—42no—
170#204—82no—
171C25—21no—
172C22×Q8—31no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—31no—
179C22×D4—44no—
180C22×D4—61no—
181C22×D4D1212yes—
182C23×C4C22×D311yescharacteristic
183C2×SD16—62no—
184C2×SD16—64no—
185C2×SD16—124no—
186C2×D8—64no—
187C2×D8—124no—
188C2×M16—21no—
189C2×M16#6712yes—
190C22×C8—61no—
191C22×C8—61no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196#121—61no—
197#121—61no—
198#121—61no—
199#121—61no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205#107—62no—
206#103—62no—
207#102—62no—
208#102—64no—
209#102—64no—
210#100—44no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—32no—
216C22⋊C8—32no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—24no—
222C23×C3—41no—
223C23×C3—44no—
224C23×C3—44no—
225C23×C3—42no—
226C23×C3—42no—
227C23×C3—82no—
228C23×C3—82no—
229C23×C3—82no—
230C23×C3—82no—
231C23×C3—816no—
232C23×C3—82no—
233C23×C3—82no—
234C23×C3#12111yescharacteristic
235C23×C3#12111yescharacteristic
236C23×C3#12111yescharacteristic
237C22×D3—41no—
238C22×D3—42no—
239C22×D3—81no—
240C22×D3—81no—
241C3×D4—24no—
242C3×D4—24no—
243C3×D4—42no—
244C3×D4—42no—
245C3×D4—44no—
246C3×D4—44no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12711yescharacteristic
257#67—82no—
258#67—82no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265D12—42no—
266D12—42no—
267Dic6—42no—
268Dic6—42no—
269C3×C8—42no—
270C3×C8—81no—
271C3⋊C8—42no—
272C3⋊C8—42no—
273C24—22no—
274C24—44no—
275C24—41no—
276C24—41no—
277C24—61no—
278C24—82no—
279C24—82no—
280C24#23511yescharacteristic
281C2×Q8—61no—
282C2×Q8—61no—
283C2×Q8—122no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—122no—
300C2×D4—121no—
301C2×D4—121no—
302C2×D4—242no—
303C22×C4—42no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4C2×D1211yescharacteristic
308C22×C4D3×D411yescharacteristic
309C22×C4D3×D411yescharacteristic
310C22×C4D3×D411yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4C2×#6711yescharacteristic
313SD16—128no—
314SD16—242no—
315D8—128no—
316M16—42no—
317M16—42no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—62no—
321C2×C8—62no—
322C2×C8—122no—
323C2×C8—122no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—121no—
329C4⋊C4—121no—
330C22⋊C4—121no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—44no—
342C22×C3—44no—
343C22×C3—88no—
344C22×C3—88no—
345C22×C3—84no—
346C22×C3—84no—
347C22×C3—82no—
348C22×C3—82no—
349C22×C3#39211yescharacteristic
350C22×C3#47112yes—
351C22×C3#47212yes—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C2×D3—162no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4B(4,𝔽2)12yes—
360Dic3—81no—
361Dic3—81no—
362Dic3—81no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—41no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—816no—
376C23—82no—
377C23—82no—
378C23—121no—
379C23—122no—
380C23—241no—
381C23—241no—
382C23#88411yescharacteristic
383C23#88411yescharacteristic
384C23#93011yescharacteristic
385Q8—122no—
386Q8—122no—
387D4—24no—
388D4—24no—
389D4—44no—
390D4—42no—
391D4—42no—
392D4—44no—
393D4—122no—
394D4—122no—
395D4—242no—
396D4—242no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—122no—
407C2×C4—241no—
408C2×C4—241no—
409C2×C4#88211yescharacteristic
410C2×C4#88211yescharacteristic
411C2×C4#93711yescharacteristic
412C2×C4#94011yescharacteristic
413C8—42no—
414C8—81no—
415C8—122no—
416C8—122no—
417C2×C3—22no—
418C2×C3—22no—
419C2×C3—41no—
420C2×C3—84no—
421C2×C3#201811yescharacteristic
422C2×C3#201911yescharacteristic
423C2×C3#202111yescharacteristic
424D3—161no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—44no—
433C22—44no—
434C22—88no—
435C22—88no—
436C22—84no—
437C22—84no—
438C22—82no—
439C22—82no—
440C22—241no—
441C22—241no—
442C22—241no—
443C22—482no—
444C22#474812yes—
445C22#474912yes—
446C22#496011yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—241no—
451C4—241no—
452C4—241no—
453C4#503212yes—
454C3#998711yescharacteristic, complemented, Sylow
455C2—22no—
456C2—22no—
457C2—41no—
458C2—84no—
459C2—481no—
460C2#6950011yescharacteristic
461C2#7009011yescharacteristic
462C2#7009111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1783C2
socleC22×C3#392
prime radicalC2×#5217C2
hypercenter#1783D3
hypocenterC3#9987
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1783

Complements

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

NH ≅ G/Ntype
1. C2×#5218C2semidirect
3. #78642C2semidirect
4. C3×#1783C2semidirect
5. #69972C2semidirect
6. #69970C2semidirect
7. #67688C2semidirect
35. #1783D3semidirect
454. C3#9987semidirect

Maximal subgroups

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

subgroupindex
C2×#52182
C2×#52172
#786422
C3×#17832
#699722
#699702
#676882
#99873