Finite Group Databasebeta

← #10695852

Subgroups — #10695852

Order 768.

878 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 89 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×C3×#447C211yesmaximal, characteristic, complemented
2C2×#5217C211yesmaximal, characteristic, complemented
3#82627C211yesmaximal, characteristic, complemented
4#72129C211yesmaximal, characteristic, complemented
5#71676C211yesmaximal, characteristic, complemented
6#69921C211yesmaximal, characteristic, complemented
7#67680C211yesmaximal, characteristic, complemented
8#9986—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×D8—21no—
11C2×C3×#121C2211yescharacteristic
12C22×#213—21no—
13C2×#930C2211yescharacteristic
14C3×#447—22no—
15C3×#447—22no—
16C2×C3×#102—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5217—22no—
19#5217—21no—
20#5217—22no—
21C2×#833—22no—
22C2×#832—21no—
23#5207C2211yescharacteristic
24#5207C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5039—22no—
27#5036—21no—
28#5034—21no—
29#4968—21no—
30#4525—22no—
31#4468—22no—
32#3192—31no—
33C2×#447—31no—
34C2×#447D311yescharacteristic, complemented
35#1783—31no—
36#1471—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—21no—
46C22×C3×D4—44no—
47C22×C3×D4D411yescharacteristic
48C22×C3×D4D412yes—
49C23×C3×C4C2311yescharacteristic
50C22×#67—21no—
51C22×D12D411yescharacteristic
52C2×C3×D8—24no—
53C2×C3×D8—44no—
54C22×C3×C8—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×#213—24no—
62C2×#213—22no—
63C2×#213—44no—
64#930—21no—
65#930—21no—
66#930—21no—
67#930—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×C4⋊Dic3D411yescharacteristic
70C2×#208—21no—
71C2×#208D412yes—
72C22×C3⋊C8—21no—
73C3×#102—24no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76#833—44no—
77#832—24no—
78#832—22no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#809—22no—
86#808—22no—
87C23×D4C2×D311yescharacteristic
88C22×D8—21no—
89C22×D8—61no—
90C2×#121—31no—
91C2×#121C2×D311yescharacteristic
92#481—61no—
93#481—61no—
94#469—62no—
95#447—22no—
96#447—22no—
97#447—62no—
98#447—61no—
99#447—61no—
100#447—62no—
101C2×#102—21no—
102C2×#102—61no—
103C2×#100—62no—
104#407—31no—
105#407—31no—
106C2×C22⋊C8—31no—
107C2×C22⋊C8C2×D311yescharacteristic
108#323—62no—
109#323—62no—
110C24×C3—22no—
111C24×C3—21no—
112C24×C3—44no—
113C24×C3—41no—
114C24×C3—41no—
115C24×C3—82no—
116C24×C3—82no—
117C24×C3#4111yescharacteristic
118C23×D3—21no—
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4—42no—
129C2×C3×D4—42no—
130C2×C3×D4—44no—
131C2×C3×D4—44no—
132C2×C3×D4—44no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×C3×D4—82no—
138C22×C3×C4—21no—
139C22×C3×C4—42no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C2×#67—41no—
147C2×#67—41no—
148C2×#67—82no—
149C22×Dic3—21no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—42no—
153C3×D8—48no—
154C2×C3×C8—22no—
155C2×C3×C8—42no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160#213—48no—
161#213—82no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166#208—42no—
167#208—42no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—42no—
171C2×C3⋊C8—41no—
172C25—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—31no—
180C22×D4—44no—
181C22×D4—61no—
182C22×D4#6711yescharacteristic
183C22×D4#6712yes—
184C23×C4C22×D311yescharacteristic
185C2×D8—24no—
186C2×D8—44no—
187C2×D8—64no—
188C2×D8—62no—
189C2×D8—124no—
190C2×M16—32no—
191C2×M16—61no—
192C22×C8—21no—
193C22×C8—61no—
194#121—21no—
195#121—21no—
196#121—21no—
197#121—21no—
198#121—61no—
199#121—61no—
200#121—61no—
201#121—61no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4#6711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—61no—
206#106—62no—
207#102—24no—
208#102—62no—
209#102—64no—
210#102—62no—
211#100—124no—
212C22⋊C8—21no—
213C22⋊C8—32no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C22⋊C8#6712yes—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—24no—
223C23×C3—41no—
224C23×C3—44no—
225C23×C3—44no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—41no—
229C23×C3—42no—
230C23×C3—82no—
231C23×C3—82no—
232C23×C3—82no—
233C23×C3—82no—
234C23×C3—81no—
235C23×C3—81no—
236C23×C3—816no—
237C23×C3—82no—
238C23×C3—82no—
239C23×C3#12111yescharacteristic
240C23×C3#12111yescharacteristic
241C23×C3#12111yescharacteristic
242C22×D3—41no—
243C22×D3—42no—
244C22×D3—81no—
245C22×D3—81no—
246C3×D4—24no—
247C3×D4—24no—
248C3×D4—42no—
249C3×D4—44no—
250C3×D4—42no—
251C3×D4—42no—
252C3×D4—44no—
253C3×D4—42no—
254C3×D4—82no—
255C3×D4—82no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—81no—
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12011yescharacteristic
266C2×C3×C4#12011yescharacteristic
267C2×C3×C4#12711yescharacteristic
268#67—82no—
269#67—82no—
270C2×Dic3—41no—
271C2×Dic3—81no—
272D12—42no—
273D12—42no—
274C3×C8—42no—
275C3⋊C8—42no—
276C3⋊C8—42no—
277C3⋊C8—81no—
278C24—22no—
279C24—21no—
280C24—44no—
281C24—41no—
282C24—41no—
283C24—61no—
284C24—82no—
285C24—82no—
286C24#23711yescharacteristic
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—61no—
306C2×D4—61no—
307C2×D4—82no—
308C2×D4—122no—
309C2×D4—121no—
310C2×D4—121no—
311C2×D4—242no—
312C22×C4—21no—
313C22×C4—42no—
314C22×C4—61no—
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4C2×#6711yescharacteristic
319C22×C4C2×#6711yescharacteristic
320C22×C4C2×#6711yescharacteristic
321D8—48no—
322D8—128no—
323D8—242no—
324M16—122no—
325M16—122no—
326C2×C8—22no—
327C2×C8—42no—
328C2×C8—62no—
329C2×C8—62no—
330C2×C8—122no—
331C2×C8—121no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—61no—
335C4⋊C4—61no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—121no—
339C22⋊C4—121no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—88no—
350C22×C3—88no—
351C22×C3—84no—
352C22×C3—84no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—162no—
359C22×C3#39211yescharacteristic
360C22×C3#47212yes—
361C22×C3#47212yes—
362C2×D3—81no—
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—162no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—81no—
370C3×C4B(4,𝔽2)12yes—
371Dic3—81no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—41no—
376C23—44no—
377C23—44no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—42no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—81no—
387C23—81no—
388C23—816no—
389C23—82no—
390C23—82no—
391C23—121no—
392C23—122no—
393C23—241no—
394C23—241no—
395C23#93811yescharacteristic
396C23#93911yescharacteristic
397C23#93911yescharacteristic
398D4—24no—
399D4—24no—
400D4—42no—
401D4—42no—
402D4—44no—
403D4—42no—
404D4—44no—
405D4—42no—
406D4—82no—
407D4—82no—
408D4—122no—
409D4—122no—
410D4—242no—
411D4—242no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—121no—
421C2×C4—241no—
422C2×C4#93711yescharacteristic
423C2×C4#93711yescharacteristic
424C2×C4#94011yescharacteristic
425C2×C4#95311yescharacteristic
426C8—42no—
427C8—122no—
428C8—122no—
429C8—241no—
430C2×C3—22no—
431C2×C3—22no—
432C2×C3—41no—
433C2×C3—84no—
434C2×C3—161no—
435C2×C3#201811yescharacteristic
436C2×C3#201811yescharacteristic
437C2×C3#202111yescharacteristic
438D3—161no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—44no—
447C22—44no—
448C22—88no—
449C22—88no—
450C22—84no—
451C22—84no—
452C22—82no—
453C22—82no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—162no—
458C22—241no—
459C22—241no—
460C22—241no—
461C22—482no—
462C22#516012yes—
463C22#519912yes—
464C22#522311yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—81no—
469C4—241no—
470C4#503212yes—
471C3#998611yescharacteristic, complemented, Sylow
472C2—22no—
473C2—22no—
474C2—41no—
475C2—84no—
476C2—161no—
477C2—481no—
478C2#7167111yescharacteristic
479C2#7204611yescharacteristic
480C2#7209511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#447C2
socleC22×C3#392
hypercenterC2×#447D3
hypocenterC3#9986
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
1. C2×C3×#447C2semidirect
2. C2×#5217C2semidirect
3. #82627C2semidirect
4. #72129C2semidirect
5. #71676C2semidirect
6. #69921C2semidirect
7. #67680C2semidirect
34. C2×#447D3semidirect
471. C3#9986semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4472
C2×#52172
#826272
#721292
#716762
#699212
#676802
#99863