Finite Group Databasebeta

← #10692229

Subgroups — #10692229

Order 768.

520 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2378C211yesmaximal, characteristic, complemented
2#82738C211yesmaximal, characteristic
3#71688C211yesmaximal, characteristic, complemented
4#69531C211yesmaximal, characteristic
5#69528C211yesmaximal, characteristic
6#67628C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9344—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#391—21no—
12C3×#389—21no—
13C3×#388—21no—
14C3×#388—21no—
15C3×#387—21no—
16C3×#387—21no—
17C3×#381C2211yescharacteristic
18C3×#380C2211yescharacteristic
19#5227—21no—
20C4×#217C2211yescharacteristic
21#5207C2211yescharacteristic
22#5207C2211yescharacteristic
23#5049—22no—
24#5046C412yescomplemented
25#5044C412yescomplemented
26#4967—21no—
27C2×#806—21no—
28C2×#806—21no—
29#4538—22no—
30#4451—21no—
31#4451—21no—
32#3193—31no—
33#2378D311yescharacteristic, complemented
34#1964—31no—
35#1510—31no—
36#1508—31no—
37#1305—31no—
38#1284—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×D12C2×C411yescharacteristic
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8C2×C412yes—
46C3×C22⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4D411yescharacteristic
55C2×C3×C22⋊C4D411yescharacteristic
56C2×C3×C42—21no—
57C2×C3×C42—21no—
58C2×#217—21no—
59C2×#215—22no—
60#930—21no—
61#930—21no—
62#930—21no—
63#930—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×C4⋊Dic3C2×C411yescharacteristic
66C2×C4×Dic3—21no—
67C2×#208—21no—
68C2×#208—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95D411yescharacteristic
76C3×#95D411yescharacteristic
77#835—22no—
78C3⋊#95—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830C2×C412yes—
83#809—22no—
84#807—22no—
85#806—42no—
86#806—42no—
87C2×C22⋊Q8C2×D311yescharacteristic
88C2×#121—31no—
89#484—32no—
90#480—61no—
91#468—62no—
92#450—32no—
93#448—61no—
94C2×#99—61no—
95C2×#99—61no—
96#407—31no—
97#407—31no—
98#391—21no—
99#389—21no—
100#388—21no—
101#388—21no—
102#387—21no—
103#387—21no—
104#381C2×D311yescharacteristic
105#380C2×D311yescharacteristic
106C4×C22⋊C4—31no—
107#328—62no—
108#327—61no—
109#327—61no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—81no—
114C2×C3×Q8C22⋊C412yes—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—81no—
122C22×C3×C4C22×C411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—41no—
131C2×#67—41no—
132C2×#67—41no—
133C2×#67—41no—
134C22×Dic3—21no—
135C22×Dic3—41no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—81no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—81no—
143C3×C4⋊C4—81no—
144C3×C4⋊C4—81no—
145C3×C4⋊C4C22⋊C412yes—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—81no—
151C3×C42—81no—
152C3×C42—81no—
153#217—22no—
154#217—41no—
155#215—82no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4×Dic3—81no—
161#208—81no—
162#208—81no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C22×Q8#6711yescharacteristic
166C22×D4—31no—
167C22×D4—61no—
168C23×C4C22×D311yescharacteristic
169C2×SD16—62no—
170C2×M16—61no—
171C2×M16—61no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8C4×D312yes—
175C22⋊Q8D1212yes—
176#121—61no—
177#121—61no—
178#121—61no—
179#121—61no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4#6711yescharacteristic
191C2×C22⋊C4#6711yescharacteristic
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—61no—
195#107—62no—
196#103—62no—
197#102—62no—
198#99—122no—
199#99—122no—
200C22⋊C8—32no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—61no—
211#95#6711yescharacteristic
212#95#6711yescharacteristic
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3C4×D411yescharacteristic
218C23×C3#12011yescharacteristic
219C23×C3#12111yescharacteristic
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3—81no—
224C22×D3—81no—
225C3×Q8—42no—
226C3×Q8—81no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4C2×C22⋊C411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12311yescharacteristic
255#67—81no—
256#67—81no—
257#67—81no—
258#67—81no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265D12—42no—
266D12—81no—
267C3⋊C8—42no—
268C3⋊C8—81no—
269C24—61no—
270C24C2×#6711yescharacteristic
271C2×Q8—21no—
272C2×Q8—81no—
273C2×Q8(C2×D3)⋊C412yes—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—241no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—61no—
290C22×C4—81no—
291C22×C4—121no—
292C22×C4C2×C4×D311yescharacteristic
293C22×C4C2×D1211yescharacteristic
294C22×C4#23511yescharacteristic
295C22×C4#23511yescharacteristic
296C22×C4C2×#6711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298SD16—242no—
299M16—241no—
300M16—241no—
301C2×C8—62no—
302C2×C8—121no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4(C2×D3)⋊C412yes—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—62no—
317C22⋊C4—81no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C42—81no—
322C42—81no—
323C42—241no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#38611yescharacteristic
335C22×C3#45311yescharacteristic
336C22×C3#45411yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—161no—
341C2×D3—161no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—42no—
347C3×C4—42no—
348C3×C4—81no—
349C3×C4—81no—
350C3×C4—81no—
351C3×C4—81no—
352Dic3—82no—
353Dic3—81no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—121no—
359C23—121no—
360C23—121no—
361C23—241no—
362C23—241no—
363C23C4×#6711yescharacteristic
364C23#93011yescharacteristic
365C23#95311yescharacteristic
366Q8—42no—
367Q8—81no—
368D4—122no—
369D4—241no—
370D4—241no—
371D4—241no—
372D4—241no—
373D4—241no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—122no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—241no—
403C2×C4—241no—
404C2×C4C2×(C2×D3)⋊C411yescharacteristic
405C2×C4C4×#6711yescharacteristic
406C2×C4#92911yescharacteristic
407C2×C4#93011yescharacteristic
408C8—122no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3#188311yescharacteristic
415C2×C3#190511yescharacteristic
416C2×C3#191311yescharacteristic
417D3—161no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—81no—
427C22—81no—
428C22—241no—
429C22—241no—
430C22—241no—
431C22—481no—
432C22—481no—
433C22#521311yescharacteristic
434C22#524111yescharacteristic
435C22#524511yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—42no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—242no—
447C4—241no—
448C3#934411yescharacteristic, complemented, Sylow
449C2—21no—
450C2—21no—
451C2—41no—
452C2—41no—
453C2—481no—
454C2#7165611yescharacteristic
455C2#7166211yescharacteristic
456C2#7166711yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2378C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#2378D3
hypocenterC3#9344
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2378

Complements

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

NH ≅ G/Ntype
1. C3×#2378C2semidirect
3. #71688C2semidirect
6. #67628C2semidirect
7. #67626C2semidirect
24. #5046C4semidirect
25. #5044C4semidirect
33. #2378D3semidirect
448. C3#9344semidirect

Maximal subgroups

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

subgroupindex
C3×#23782
#827382
#716882
#695312
#695282
#676282
#676262
#93443