Finite Group Databasebeta

← #10695473

Subgroups — #10695473

Order 768.

654 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 113 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#82788C211yesmaximal, characteristic
2C2×#5043C211yesmaximal, characteristic
3C2×#4661C211yesmaximal, characteristic
4C2×C3×#328C211yesmaximal, characteristic
5#69924C211yesmaximal, characteristic
6C2×#4469C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#9927—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C4⋊Dic3—21no—
11C22×C4⋊Dic3C2211yescharacteristic
12C22×Dic3⋊C4C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5230—21no—
16C2×#835—21no—
17#5212C2211yescharacteristic
18C2×C3⋊#95—21no—
19C2×#830C2211yescharacteristic
20#5043—22no—
21#5043—22no—
22#4985—21no—
23#4982—22no—
24C2×#808—21no—
25#4946—21no—
26#4943—21no—
27#4661—22no—
28#4661—22no—
29C3×#328—22no—
30#4551—22no—
31#4469—22no—
32C2×#483—31no—
33#2627—31no—
34C2×#398—31no—
35#1823—31no—
36C2×#328D311yescharacteristic
37C2×#324—31no—
38#1294—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C23×Dic3D411yescharacteristic
44C23×Dic3D411yescharacteristic
45C22×C3×C8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×#217—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—42no—
61C2×C4⋊Dic3—42no—
62C2×C4⋊Dic3D412yes—
63C2×C4⋊Dic3D412yes—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—22no—
71C22×C3⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×#95—21no—
75C3×#95D412yes—
76#835—24no—
77C3⋊#95—24no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82#830—21no—
83#830D412yes—
84#808—24no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C22×C4⋊C4—31no—
87C22×C4⋊C4—31no—
88C22×C4⋊C4—61no—
89#483—62no—
90#483—62no—
91C2×#106—61no—
92C2×#103—61no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8C2×D311yescharacteristic
95#400—61no—
96#398—62no—
97#398—62no—
98#398—62no—
99#395—61no—
100#385—61no—
101#381—61no—
102#380—31no—
103C2×#95—61no—
104C2×#95C2×D311yescharacteristic
105#340—62no—
106#328—22no—
107#324—62no—
108C24×C3C2×D411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C22×C3×C4—24no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—42no—
117C22×C3×C4—42no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—22no—
134C22×Dic3—21no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—42no—
138C22×Dic3—41no—
139C22×Dic3—41no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3SD1612yes—
145C2×C3×C8—42no—
146C2×C3×C8—41no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153#217—41no—
154#217—41no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4⋊Dic3—48no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C4⋊Dic3—42no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164Dic3⋊C4—41no—
165Dic3⋊C4—82no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—42no—
168C22×Q8D1211yescharacteristic
169C23×C4—21no—
170C23×C4—31no—
171C23×C4—31no—
172C23×C4—61no—
173C23×C4C22×D311yescharacteristic
174C22×C8—21no—
175C22×C8—61no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—32no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—64no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4—122no—
197C2×C4⋊C4—122no—
198C2×C4⋊C4D1211yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—61no—
201#106—64no—
202#103—64no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—32no—
206C22⋊C8—61no—
207#95—21no—
208#95—64no—
209#95—62no—
210#95—62no—
211#95—62no—
212#95—61no—
213#95#6712yes—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3#12011yescharacteristic
217C23×C3#12011yescharacteristic
218C23×C3#12111yescharacteristic
219C23×C3C2×SD1611yescharacteristic
220C23×C3#13511yescharacteristic
221C23×C3AGL(1,ℤ/8)11yescharacteristic
222C23×C3#13711yescharacteristic
223C3×Q8—42no—
224C3×Q8—81no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—24no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—44no—
233C2×C3×C4—44no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12711yescharacteristic
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—81no—
261C2×Dic3—81no—
262C2×Dic3—81no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265C3×C8—81no—
266C3⋊C8—42no—
267C24D3×D411yescharacteristic
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—42no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—32no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—62no—
290C22×C4—61no—
291C22×C4—122no—
292C22×C4—121no—
293C22×C4—121no—
294C22×C4—121no—
295C22×C4—121no—
296C22×C4—121no—
297C22×C4—121no—
298C22×C4C2×D1211yescharacteristic
299C22×C4#23511yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4D3×D411yescharacteristic
302C22×C4D3×D411yescharacteristic
303C22×C4C2×#6711yescharacteristic
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—62no—
307C2×C8—122no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—122no—
315C4⋊C4—128no—
316C4⋊C4—121no—
317C4⋊C4—121no—
318C4⋊C4—121no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C4⋊C4—242no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3#39211yescharacteristic
336C22×C3#44811yescharacteristic
337C22×C3#44911yescharacteristic
338C22×C3#45011yescharacteristic
339C22×C3#45211yescharacteristic
340C22×C3#45612yes—
341C22×C3#45812yes—
342C22×C3#46511yescharacteristic
343C22×C3#46811yescharacteristic
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—44no—
349C3×C4—81no—
350C3×C4—81no—
351Dic3—42no—
352Dic3—82no—
353Dic3—82no—
354Dic3—81no—
355C23—22no—
356C23—22no—
357C23#88211yescharacteristic
358C23#88411yescharacteristic
359C23D3×SD1611yescharacteristic
360C23#91411yescharacteristic
361C23#91811yescharacteristic
362C23#91911yescharacteristic
363C23#93711yescharacteristic
364Q8—42no—
365Q8—81no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—121no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4—241no—
402C2×C4#88211yescharacteristic
403C2×C4#88411yescharacteristic
404C2×C4#93011yescharacteristic
405C2×C4#94011yescharacteristic
406C8—81no—
407C8—122no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3#160912yes—
413C2×C3#161112yes—
414C2×C3#200011yescharacteristic
415C2×C3#200511yescharacteristic
416C2×C3#202211yescharacteristic
417C22—21no—
418C22—21no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—21no—
424C22—21no—
425C22#474212yes—
426C22#480611yescharacteristic
427C22#480711yescharacteristic
428C22#481011yescharacteristic
429C22#481211yescharacteristic
430C22#482612yes—
431C22#496011yescharacteristic
432C22#516911yescharacteristic
433C22#518711yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—44no—
439C4—81no—
440C4—81no—
441C4—122no—
442C4—242no—
443C4—241no—
444C4—242no—
445C3#992711yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2#6832812yes—
451C2#6832912yes—
452C2#7003111yescharacteristic
453C2#7027211yescharacteristic
454C2#7038911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenterC2×#328D3
hypocenterC3#9927
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
445. C3#9927semidirect

Maximal subgroups

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

subgroupindex
#827882
C2×#50432
C2×#46612
C2×C3×#3282
#699242
C2×#44692
#676122
#99273