Finite Group Databasebeta

← #10695735

Subgroups — #10695735

Order 768.

830 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 111 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#82824C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic, complemented
3C2×#4738C211yesmaximal, characteristic
4C2×C3×#328C211yesmaximal, characteristic, complemented
5#69424C211yesmaximal, characteristic, complemented
6C2×#4451C211yesmaximal, characteristic
7#67674C211yesmaximal, characteristic, complemented
8#9965—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×#216—21no—
12C2×#930C2211yescharacteristic
13C22×#204—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5231—21no—
17C2×C3⋊#95—22no—
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#816—21no—
21#5050—22no—
22#5050—22no—
23#4992—21no—
24#4991—21no—
25#4986—21no—
26C2×#810—21no—
27#4958—22no—
28#4738—22no—
29#4738—22no—
30C3×#328—22no—
31#4451—22no—
32C2×#451—31no—
33C2×#448—31no—
34#2639—31no—
35C2×#328D311yescharacteristic, complemented
36C2×#327—31no—
37#1446—31no—
38#1280—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C23×Dic3—21no—
44C23×Dic3D412yes—
45C22×D12D411yescharacteristic
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—21no—
50C22×Dic6—21no—
51C22×Dic6—44no—
52C22×Dic6D412yes—
53C22×C3×C8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×#216—24no—
62C2×#216—44no—
63#930—21no—
64#930—21no—
65#930—21no—
66#930—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3D411yescharacteristic
71C22×C3⋊C8—21no—
72C2×#204—24no—
73C2×#204—44no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76C3×#95—21no—
77C3×#95—21no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—44no—
83#830—21no—
84#830D412yes—
85#816—24no—
86#810—24no—
87C23×Q8—31no—
88C22×Q16—61no—
89C22×SD16—61no—
90C2×C22⋊Q8C2×D311yescharacteristic
91C2×#121—31no—
92#451—62no—
93#451—62no—
94#448—62no—
95#448—62no—
96C2×#103—61no—
97C2×#103—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8C2×D311yescharacteristic
100#400—61no—
101#397—61no—
102#394—61no—
103#393—61no—
104#386—62no—
105C2×#95—31no—
106C2×#95—62no—
107C2×#95C2×D311yescharacteristic
108#328—22no—
109#327—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—42no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—42no—
119C22×C3×C4—42no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—42no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—42no—
132C2×#67—42no—
133C22×Dic3—24no—
134C22×Dic3—22no—
135C22×Dic3—21no—
136C22×Dic3—22no—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—42no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—42no—
143C22×Dic3—42no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12—42no—
147C2×Dic6—24no—
148C2×Dic6—24no—
149C2×Dic6—22no—
150C2×Dic6—22no—
151C2×Dic6—44no—
152C2×Dic6—44no—
153C2×Dic6—44no—
154C2×Dic6—44no—
155C2×Dic6—42no—
156C2×Dic6—42no—
157C2×Dic6—41no—
158C2×Dic6—41no—
159C2×C3×C8—22no—
160C2×C3×C8—42no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167#216—48no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—82no—
171(C2×D3)⋊C4—82no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—42no—
176#204—48no—
177C22×Q8—32no—
178C22×Q8—61no—
179C22×Q8—61no—
180C22×Q8—61no—
181C22×Q8—61no—
182C22×Q8—61no—
183C22×Q8—124no—
184C22×Q8D1211yescharacteristic
185C22×D4—31no—
186C22×D4—61no—
187C23×C4—21no—
188C23×C4—32no—
189C23×C4—61no—
190C23×C4C22×D311yescharacteristic
191C2×Q16—64no—
192C2×Q16—124no—
193C2×SD16—64no—
194C2×SD16—124no—
195C22×C8—21no—
196C22×C8—61no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201#121—61no—
202#121—61no—
203#121—61no—
204#121—61no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4D1211yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212#103—64no—
213#103—64no—
214C22⋊C8—21no—
215C22⋊C8—32no—
216C22⋊C8—61no—
217C22⋊C8#6712yes—
218#95—21no—
219#95—21no—
220#95—62no—
221#95—62no—
222#95—61no—
223#95—61no—
224#95—124no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—22no—
228C23×C3#12011yescharacteristic
229C23×C3#12111yescharacteristic
230C23×C3#12711yescharacteristic
231C23×C3#13512yes—
232C23×C3#13712yes—
233C22×D3—41no—
234C22×D3—42no—
235C22×D3—41no—
236C22×D3—41no—
237C22×D3—42no—
238C3×Q8—42no—
239C3×Q8—81no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—44no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—82no—
254C2×C3×C4—81no—
255C2×C3×C4—81no—
256C2×C3×C4#12011yescharacteristic
257C2×C3×C4#12011yescharacteristic
258C2×C3×C4#12111yescharacteristic
259C2×C3×C4#12111yescharacteristic
260#67—82no—
261#67—81no—
262#67—81no—
263C2×Dic3—24no—
264C2×Dic3—42no—
265C2×Dic3—44no—
266C2×Dic3—42no—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—82no—
275D12—42no—
276D12—81no—
277Dic6—24no—
278Dic6—24no—
279Dic6—42no—
280Dic6—42no—
281Dic6—44no—
282Dic6—44no—
283C3×C8—42no—
284C3⋊C8—42no—
285C24—61no—
286C24D3×D411yescharacteristic
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—42no—
290C2×Q8—64no—
291C2×Q8—64no—
292C2×Q8—62no—
293C2×Q8—62no—
294C2×Q8—122no—
295C2×Q8—122no—
296C2×Q8—124no—
297C2×Q8—124no—
298C2×Q8—124no—
299C2×Q8—124no—
300C2×Q8—121no—
301C2×Q8—121no—
302C2×D4—61no—
303C2×D4—61no—
304C2×D4—122no—
305C2×D4—121no—
306C2×D4—122no—
307C2×D4—122no—
308C2×D4—121no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—64no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—61no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—122no—
324C22×C4—121no—
325C22×C4—121no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4C2×D1211yescharacteristic
329C22×C4#23511yescharacteristic
330C22×C4D3×D411yescharacteristic
331C22×C4D3×D411yescharacteristic
332C22×C4D3×D411yescharacteristic
333C22×C4C2×#6711yescharacteristic
334Q16—128no—
335SD16—128no—
336C2×C8—22no—
337C2×C8—42no—
338C2×C8—62no—
339C2×C8—122no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—121no—
349C22⋊C4—121no—
350C22⋊C4—242no—
351C22⋊C4—242no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3#39211yescharacteristic
361C22×C3#45212yes—
362C22×C3#45511yescharacteristic
363C22×C3#45611yescharacteristic
364C22×C3B(4,𝔽2)12yes—
365C22×C3#46412yes—
366C22×C3#49511yescharacteristic
367C22×C3#49711yescharacteristic
368C2×D3—82no—
369C2×D3—82no—
370C2×D3—81no—
371C2×D3—81no—
372C2×D3—81no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—82no—
378C3×C4—81no—
379C3×C4—81no—
380Dic3—44no—
381Dic3—82no—
382Dic3—81no—
383C23—22no—
384C23—21no—
385C23—21no—
386C23—121no—
387C23—122no—
388C23—122no—
389C23—121no—
390C23—121no—
391C23#88211yescharacteristic
392C23#88411yescharacteristic
393C23#91612yes—
394C23#91812yes—
395C23#94011yescharacteristic
396Q8—42no—
397Q8—64no—
398Q8—64no—
399Q8—81no—
400Q8—122no—
401Q8—122no—
402Q8—124no—
403Q8—124no—
404D4—122no—
405D4—241no—
406D4—241no—
407D4—241no—
408D4—242no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—64no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—122no—
427C2×C4—124no—
428C2×C4—122no—
429C2×C4—124no—
430C2×C4—124no—
431C2×C4—121no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—242no—
437C2×C4#88211yescharacteristic
438C2×C4#88411yescharacteristic
439C2×C4#93011yescharacteristic
440C2×C4#93711yescharacteristic
441C8—42no—
442C8—122no—
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3#160312yes—
447C2×C3#160612yes—
448C2×C3#200711yescharacteristic
449C2×C3#201511yescharacteristic
450C2×C3#202111yescharacteristic
451D3—161no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—241no—
461C22—241no—
462C22—242no—
463C22—242no—
464C22—241no—
465C22#474112yes—
466C22#479812yes—
467C22#480812yes—
468C22#482511yescharacteristic
469C22#482611yescharacteristic
470C22#496011yescharacteristic
471C22#516811yescharacteristic
472C22#519011yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—82no—
478C4—81no—
479C4—81no—
480C4—124no—
481C4—241no—
482C4—242no—
483C3#996511yescharacteristic, complemented, Sylow
484C2—21no—
485C2—21no—
486C2—22no—
487C2—481no—
488C2#6831912yes—
489C2#6832612yes—
490C2#7003711yescharacteristic
491C2#7027811yescharacteristic
492C2#7039311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3#120
prime radicalC2×#930C22
hypercenterC2×#328D3
hypocenterC3#9965
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. C2×#5050C2semidirect
4. C2×C3×#328C2semidirect
5. #69424C2semidirect
7. #67674C2semidirect
35. C2×#328D3semidirect
483. C3#9965semidirect

Maximal subgroups

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

subgroupindex
#828242
C2×#50502
C2×#47382
C2×C3×#3282
#694242
C2×#44512
#676742
#99653