Finite Group Databasebeta

← #10695633

Subgroups — #10695633

Order 768.

636 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3186C211yesmaximal, characteristic
2#82665C211yesmaximal, characteristic, complemented
3C2×#5034C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic, complemented
6#69925C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9950—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C3×#480—21no—
13C3×#480—21no—
14C3×#466—22no—
15C3×#447—21no—
16C3×#447—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5221—21no—
19#5220—22no—
20C2×#834—22no—
21C2×#832—21no—
22#5212C2211yescharacteristic
23C4×#217C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5034—22no—
26#5034—22no—
27#4977—21no—
28C2×#808—21no—
29#4953—22no—
30#4551—22no—
31#4537—22no—
32#3186D311yescharacteristic
33C2×#481—31no—
34#2504—31no—
35#1964—31no—
36#1824—31no—
37#1519—31no—
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C2×C3×D8—22no—
46C22×C3×C8D412yes—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217—24no—
61C2×#217D412yes—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D412yes—
69C2×C4×Dic3—21no—
70C22×C3⋊C8—21no—
71C3×#107—22no—
72C3×#102—22no—
73C3×#102—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76#834—44no—
77#832—24no—
78C3⋊#95—24no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830D412yes—
83#808—24no—
84C2×#121C2×D311yescharacteristic
85C2×#121C2×D311yescharacteristic
86C22×C4⋊C4—31no—
87#481—62no—
88#481—62no—
89#480—21no—
90#480—21no—
91#466—22no—
92#447—21no—
93#447—21no—
94C2×#106—61no—
95C2×#102—61no—
96C2×#99—62no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8C2×D311yescharacteristic
99#397—62no—
100#390—61no—
101#386—62no—
102#382—61no—
103#380—31no—
104C4×C22⋊C4—31no—
105#340—62no—
106#327—62no—
107C24×C3—21no—
108C24×C3—21no—
109C24×C3C2×D411yescharacteristic
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C2×C3×D4—42no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—42no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—41no—
136C22×Dic3—41no—
137C22×Dic3—41no—
138C22×Dic3—82no—
139C3×D8—82no—
140C2×C3×C8—22no—
141C2×C3×C8—42no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150#217—22no—
151#217—22no—
152#217—42no—
153#217—84no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158C4⋊Dic3—42no—
159C4×Dic3—42no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—42no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4#6711yescharacteristic
165C22×D4#6711yescharacteristic
166C23×C4—61no—
167C23×C4C22×D311yescharacteristic
168C2×D8—22no—
169C22×C8—61no—
170C22×C8#6712yes—
171#121—21no—
172#121—21no—
173#121—21no—
174#121—21no—
175#121—21no—
176#121—21no—
177#121—21no—
178#121—21no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6711yescharacteristic
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—32no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—64no—
193C2×C42—61no—
194#107—22no—
195#106—64no—
196#102—22no—
197#102—22no—
198#102—64no—
199#99—124no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C22⋊C8—61no—
204#95—64no—
205#95—61no—
206#95—61no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—81no—
216C23×C3—81no—
217C23×C3#12011yescharacteristic
218C23×C3#12111yescharacteristic
219C23×C3#12711yescharacteristic
220C23×C3#13512yes—
221C23×C3AGL(1,ℤ/8)12yes—
222C3×D4—42no—
223C3×D4—42no—
224C3×D4—42no—
225C3×D4—81no—
226C3×D4—82no—
227C3×D4—82no—
228C3×D4—81no—
229C3×D4—81no—
230C3×D4—81no—
231C3×D4—81no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—44no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—82no—
252C2×Dic3—82no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C3×C8—42no—
257C3⋊C8—42no—
258C24—21no—
259C24—21no—
260C24D3×D411yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—121no—
281C22×C4—121no—
282C22×C4—121no—
283C22×C4—242no—
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4#23711yescharacteristic
287C22×C4C2×#6711yescharacteristic
288C22×C4C2×#6711yescharacteristic
289C22×C4C2×#6711yescharacteristic
290D8—82no—
291C2×C8—22no—
292C2×C8—42no—
293C2×C8—62no—
294C2×C8—122no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—62no—
300C4⋊C4—62no—
301C4⋊C4—122no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—62no—
309C22⋊C4—62no—
310C22⋊C4—122no—
311C22⋊C4—244no—
312C42—122no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—84no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—162no—
329C22×C3#39211yescharacteristic
330C22×C3#44912yes—
331C22×C3#46312yes—
332C22×C3#49511yescharacteristic
333C22×C3#49611yescharacteristic
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—81no—
338C3×C4—81no—
339C3×C4#45812yes—
340Dic3—82no—
341Dic3—82no—
342Dic3—161no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23#91112yes—
354C23#91212yes—
355C23#93711yescharacteristic
356C23#93911yescharacteristic
357C23#94011yescharacteristic
358D4—42no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—82no—
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—82no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—124no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—242no—
384C2×C4—242no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4#93711yescharacteristic
389C2×C4#93811yescharacteristic
390C2×C4#93911yescharacteristic
391C2×C4#95311yescharacteristic
392C8—42no—
393C8—122no—
394C2×C3—22no—
395C2×C3—21no—
396C2×C3—22no—
397C2×C3—21no—
398C2×C3—161no—
399C2×C3—161no—
400C2×C3#200811yescharacteristic
401C2×C3#201211yescharacteristic
402C2×C3#202211yescharacteristic
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—42no—
410C22—42no—
411C22—84no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—162no—
419C22#515312yes—
420C22#515411yescharacteristic
421C22#515511yescharacteristic
422C22#515812yes—
423C22#522311yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—81no—
428C4—81no—
429C4—242no—
430C4—242no—
431C4—481no—
432C4#524312yes—
433C3#995011yescharacteristic, complemented, Sylow
434C2—22no—
435C2—21no—
436C2—22no—
437C2—21no—
438C2—161no—
439C2—161no—
440C2#7184911yescharacteristic
441C2#7204511yescharacteristic
442C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#3186C2
socleC22×C3#392
hypercenter#3186D3
hypocenterC3#9950
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
2. #82665C2semidirect
3. C2×#5034C2semidirect
4. #71686C2semidirect
5. #71675C2semidirect
6. #69925C2semidirect
7. #67677C2semidirect
433. C3#9950semidirect

Maximal subgroups

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

subgroupindex
C3×#31862
#826652
C2×#50342
#716862
#716752
#699252
#676772
#99503