Finite Group Databasebeta

← #10695756

Subgroups — #10695756

Order 768.

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

normal subgroups only

This group has 658 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
1C3×#2639C211yesmaximal, characteristic
2C2×#5044C211yesmaximal, characteristic, complemented
3C2×#5034C211yesmaximal, characteristic
4C2×#4538C211yesmaximal, characteristic, complemented
5C2×#4537C211yesmaximal, characteristic
6#69423C211yesmaximal, characteristic, complemented
7#67613C211yesmaximal, characteristic, complemented
8#9969—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C3×#400—21no—
13C3×#397—21no—
14C3×#394—21no—
15C3×#393—21no—
16C3×#386—22no—
17C2×C3×#95C2211yescharacteristic
18C2×#835—21no—
19C2×#832—21no—
20C2×C3⋊#95—22no—
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5044—22no—
25#5044—22no—
26#5034—22no—
27#5034—22no—
28C2×#808—21no—
29C2×#807—21no—
30#4538—22no—
31#4537—22no—
32C2×#484—31no—
33C2×#481—31no—
34#2639D311yescharacteristic
35C2×#328—31no—
36C2×#327—31no—
37#1445—31no—
38#1280—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4D412yes—
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C23×Dic3—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D412yes—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C3×#95—22no—
70C3×#95—22no—
71C3×#95—21no—
72C3×#95—21no—
73#835—24no—
74#832—24no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—44no—
78#830—21no—
79#830—21no—
80#830D412yes—
81#830D412yes—
82#808—24no—
83#807—24no—
84C2×C22⋊Q8C2×D311yescharacteristic
85C2×#121C2×D311yescharacteristic
86C22×C4⋊C4—31no—
87#484—62no—
88#484—62no—
89#481—62no—
90#481—62no—
91C2×#107—61no—
92C2×#106—61no—
93C2×#103—61no—
94C2×#102—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8—31no—
97#400—21no—
98#397—21no—
99#394—21no—
100#393—21no—
101#386—22no—
102C2×#95—31no—
103C2×#95—62no—
104C2×#95C2×D311yescharacteristic
105#328—62no—
106#327—62no—
107C24×C3—21no—
108C24×C3C2×D411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—42no—
126C22×C3×C4—42no—
127C22×C3×C4—42no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—42no—
137C22×Dic3—41no—
138C22×Dic3—42no—
139C22×Dic3—42no—
140C22×Dic3—41no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3—42no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—82no—
155C3×C22⋊C4—82no—
156C4⋊Dic3—22no—
157C4⋊Dic3—22no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—42no—
163C2×C3⋊C8—42no—
164C22×Q8#6711yescharacteristic
165C22×D4—21no—
166C22×D4#6711yescharacteristic
167C23×C4—61no—
168C23×C4—61no—
169C23×C4#6712yes—
170C23×C4C22×D311yescharacteristic
171C22×C8—61no—
172C22×C8—61no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177#121—21no—
178#121—21no—
179#121—21no—
180#121—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4#6711yescharacteristic
189C2×C4⋊C4#6711yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194#107—64no—
195#106—64no—
196#103—64no—
197#102—64no—
198C22⋊C8—32no—
199C22⋊C8—32no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202#95—22no—
203#95—22no—
204#95—21no—
205#95—21no—
206#95—61no—
207#95—61no—
208#95—124no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—22no—
212C23×C3—41no—
213C23×C3—42no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—42no—
217C23×C3#12011yescharacteristic
218C23×C3#12111yescharacteristic
219C23×C3#12711yescharacteristic
220C23×C3#13512yes—
221C23×C3#13712yes—
222C3×Q8—42no—
223C3×Q8—81no—
224C3×D4—42no—
225C3×D4—81no—
226C3×D4—82no—
227C3×D4—81no—
228C3×D4—81no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—44no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—82no—
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×Dic3—44no—
252C2×Dic3—42no—
253C2×Dic3—44no—
254C2×Dic3—42no—
255C2×Dic3—82no—
256C2×Dic3—81no—
257C2×Dic3—82no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261C3⋊C8—42no—
262C3⋊C8—42no—
263C24—21no—
264C24C2×#6711yescharacteristic
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—42no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—122no—
287C22×C4—122no—
288C22×C4—121no—
289C22×C4—122no—
290C22×C4—121no—
291C22×C4—121no—
292C22×C4—122no—
293C22×C4—121no—
294C22×C4D3×D411yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4#23711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299C22×C4C2×#6711yescharacteristic
300C2×C8—62no—
301C2×C8—62no—
302C2×C8—122no—
303C2×C8—122no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—62no—
311C4⋊C4—62no—
312C4⋊C4—121no—
313C4⋊C4—121no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3#39211yescharacteristic
334C22×C3#45212yes—
335C22×C3#45311yescharacteristic
336C22×C3#45411yescharacteristic
337C22×C3#45812yes—
338C22×C3#46412yes—
339C22×C3#49511yescharacteristic
340C22×C3#49711yescharacteristic
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—82no—
346C3×C4—81no—
347C3×C4—81no—
348C3×C4—81no—
349Dic3—82no—
350Dic3—82no—
351C23—21no—
352C23—22no—
353C23—21no—
354C23—41no—
355C23—42no—
356C23—41no—
357C23—42no—
358C23—41no—
359C23#93811yescharacteristic
360C23#94011yescharacteristic
361C23#95012yes—
362C23#95112yes—
363C23#95311yescharacteristic
364Q8—42no—
365Q8—81no—
366D4—42no—
367D4—81no—
368D4—82no—
369D4—81no—
370D4—81no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—82no—
389C2×C4—122no—
390C2×C4—124no—
391C2×C4—124no—
392C2×C4—122no—
393C2×C4—241no—
394C2×C4—242no—
395C2×C4—242no—
396C2×C4—241no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4#93711yescharacteristic
400C2×C4#93711yescharacteristic
401C2×C4#93911yescharacteristic
402C2×C4#93911yescharacteristic
403C8—122no—
404C8—122no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—21no—
408C2×C3—161no—
409C2×C3#161512yes—
410C2×C3#161812yes—
411C2×C3#200611yescharacteristic
412C2×C3#201511yescharacteristic
413C2×C3#202211yescharacteristic
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—82no—
426C22—82no—
427C22#505412yes—
428C22#505911yescharacteristic
429C22#506011yescharacteristic
430C22#522311yescharacteristic
431C22#523912yes—
432C22#524111yescharacteristic
433C22#524312yes—
434C22#524511yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—82no—
443C4—242no—
444C4—242no—
445C3#996911yescharacteristic, complemented, Sylow
446C2—21no—
447C2—22no—
448C2—21no—
449C2—161no—
450C2#7172911yescharacteristic
451C2#7184911yescharacteristic
452C2#7187911yescharacteristic
453C2#7188312yes—
454C2#7188712yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2639C2
socleC23×C3#120
prime radicalC2×C3×#121C22
hypercenter#2639D3
hypocenterC3#9969
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2639

Complements

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

NH ≅ G/Ntype
2. C2×#5044C2semidirect
4. C2×#4538C2semidirect
6. #69423C2semidirect
7. #67613C2semidirect
445. C3#9969semidirect

Maximal subgroups

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

subgroupindex
C3×#26392
C2×#50442
C2×#50342
C2×#45382
C2×#45372
#694232
#676132
#99693