Finite Group Databasebeta

← #10698103

Subgroups — #10698103

Order 768.

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

normal subgroups only

This group has 780 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
1C3×#2436C211yesmaximal, characteristic
2#84065C211yesmaximal, characteristic, complemented
3#82631C211yesmaximal, characteristic
4#72133C211yesmaximal, characteristic, complemented
5#71673C211yesmaximal, characteristic
6#69445C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#10330—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×C3×#124—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#944—21no—
13C2×#924C2211yescharacteristic
14C3×#393—22no—
15C3×#391—21no—
16C3×#386—22no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5230—22no—
19#5229—21no—
20C2×#836—22no—
21#5218—21no—
22#5210—22no—
23C4×#217C2211yescharacteristic
24#5207C2211yescharacteristic
25#5207C2211yescharacteristic
26#5042—22no—
27#5040—21no—
28#5033—21no—
29#4970—21no—
30#4537—22no—
31#4452—22no—
32#3193—31no—
33#2436D311yescharacteristic
34#2434—31no—
35#1784—31no—
36#1508—31no—
37#1507—31no—
38#1297—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—21no—
45C22×C3×Q8—44no—
46C22×C3×D4—21no—
47C22×C3×D4D411yescharacteristic
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C22×Dic6D411yescharacteristic
51C3×#124—44no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×#121—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—21no—
61C2×#217—21no—
62C2×#217—21no—
63#944—44no—
64C2×#214—22no—
65C2×C4⋊Dic3D411yescharacteristic
66#924—21no—
67#924—21no—
68#924—21no—
69#924—21no—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4—21no—
72C2×C4×Dic3—21no—
73C2×#208—21no—
74C2×#208D412yes—
75C3×#95—24no—
76C3×#95D412yes—
77#836—44no—
78#832—22no—
79C3⋊#95—24no—
80C3⋊#95D412yes—
81#830—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85#810—22no—
86#807—22no—
87C23×Q8C2×D311yescharacteristic
88C2×C4⋊Q8—61no—
89C2×#124—21no—
90C2×C22⋊Q8—31no—
91C2×#121C2×D311yescharacteristic
92#484—61no—
93#480—61no—
94#468—62no—
95#450—61no—
96#448—61no—
97C2×#101—62no—
98#407—31no—
99#407—31no—
100#395—62no—
101#393—22no—
102#391—21no—
103#391—61no—
104#386—22no—
105#385—62no—
106C4×C22⋊C4—31no—
107C4×C22⋊C4C2×D311yescharacteristic
108#328—62no—
109#327—62no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C2×C3×Q8—24no—
113C2×C3×Q8—24no—
114C2×C3×Q8—24no—
115C2×C3×Q8—42no—
116C2×C3×Q8—42no—
117C2×C3×Q8—42no—
118C2×C3×Q8—44no—
119C2×C3×Q8—44no—
120C2×C3×Q8—44no—
121C2×C3×Q8—41no—
122C2×C3×Q8—41no—
123C2×C3×Q8—41no—
124C2×C3×Q8—41no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—42no—
128C2×C3×D4—44no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—44no—
137C22×C3×C4—42no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×Dic3—22no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C2×Dic6—21no—
148C2×Dic6—21no—
149C2×Dic6—42no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C22⋊C4—48no—
153C3×C22⋊C4—42no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C42—42no—
157#217—24no—
158#217—41no—
159#217—41no—
160#214—82no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163Dic3⋊C4—48no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166C4×Dic3—42no—
167#208—42no—
168#208—42no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—41no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—31no—
178C22×Q8—44no—
179C22×D4—21no—
180C22×D4#6711yescharacteristic
181C23×C4#6712yes—
182C23×C4C22×D311yescharacteristic
183C2×SD16—62no—
184C2×M16—32no—
185C2×M16—61no—
186C4⋊Q8—124no—
187#124—44no—
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190C22⋊Q8—61no—
191C22⋊Q8—61no—
192#121—21no—
193#121—21no—
194#121—21no—
195#121—21no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—64no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4#6711yescharacteristic
200C2×C22⋊C4—24no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C42—21no—
206C2×C42—61no—
207#107—62no—
208#103—62no—
209#102—62no—
210#101—124no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215#95—24no—
216#95—32no—
217#95—64no—
218#95#6712yes—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—41no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—41no—
225C23×C3—42no—
226C23×C3#12011yescharacteristic
227C23×C3#12011yescharacteristic
228C23×C3#12111yescharacteristic
229C3×Q8—44no—
230C3×Q8—44no—
231C3×Q8—44no—
232C3×Q8—44no—
233C3×D4—42no—
234C3×D4—81no—
235C3×D4—82no—
236C3×D4—82no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12711yescharacteristic
259C2×Dic3—24no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—42no—
266Dic6—42no—
267Dic6—81no—
268C3⋊C8—42no—
269C3⋊C8—81no—
270C24—21no—
271C24C2×#6711yescharacteristic
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—61no—
286C2×Q8—61no—
287C2×Q8—122no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—42no—
291C2×D4—44no—
292C2×D4—41no—
293C2×D4—41no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—62no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4D3×D411yescharacteristic
305C22×C4D3×D411yescharacteristic
306C22×C4D3×D411yescharacteristic
307C22×C4#23711yescharacteristic
308C22×C4C2×#6711yescharacteristic
309C22×C4C2×#6711yescharacteristic
310SD16—242no—
311M16—122no—
312M16—122no—
313C2×C8—62no—
314C2×C8—121no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—128no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C22⋊C4—48no—
323C22⋊C4—42no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—64no—
327C22⋊C4—121no—
328C22⋊C4—121no—
329C42—42no—
330C42—122no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—44no—
339C22×C3—84no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3#39211yescharacteristic
344C22×C3#45412yes—
345C22×C3#45612yes—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—44no—
350C3×C4—82no—
351C3×C4—81no—
352C3×C4B(4,𝔽2)12yes—
353Dic3—44no—
354Dic3—81no—
355Dic3—81no—
356C23—22no—
357C23—22no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—42no—
363C23#93711yescharacteristic
364C23#93911yescharacteristic
365C23#95311yescharacteristic
366Q8—44no—
367Q8—44no—
368Q8—44no—
369Q8—44no—
370Q8—122no—
371Q8—241no—
372D4—42no—
373D4—81no—
374D4—82no—
375D4—82no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—64no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—122no—
401C2×C4#93711yescharacteristic
402C2×C4#93811yescharacteristic
403C2×C4#93911yescharacteristic
404C2×C4#94011yescharacteristic
405C8—122no—
406C8—241no—
407C2×C3—22no—
408C2×C3—22no—
409C2×C3—41no—
410C2×C3—161no—
411C2×C3#200311yescharacteristic
412C2×C3#200511yescharacteristic
413C2×C3#202111yescharacteristic
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—44no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22#506212yes—
427C22#522311yescharacteristic
428C22#524612yes—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—44no—
433C4—82no—
434C4—81no—
435C4—124no—
436C4—241no—
437C4—241no—
438C4#503212yes—
439C3#1033011yescharacteristic, complemented, Sylow
440C2—22no—
441C2—22no—
442C2—41no—
443C2—161no—
444C2#7167111yescharacteristic
445C2#7173011yescharacteristic
446C2#7186811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2436C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenter#2436D3
hypocenterC3#10330
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
2. #84065C2semidirect
4. #72133C2semidirect
6. #69445C2semidirect
7. #67618C2semidirect
439. C3#10330semidirect

Maximal subgroups

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

subgroupindex
C3×#24362
#840652
#826312
#721332
#716732
#694452
#676182
#103303