Finite Group Databasebeta

← #10692926

Subgroups — #10692926

Order 768.

1034 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2308C211yesmaximal, characteristic, complemented
2#84049C211yesmaximal, characteristic
3#72129C211yesmaximal, characteristic, complemented
4#69531C212yesmaximal, complemented
5#67628C212yesmaximal, complemented
6#9463—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C22⋊C4—21no—
9C2×#953—21no—
10C2×#953—21no—
11C2×#940C412yescomplemented
12C2×#930C2211yescharacteristic
13C3×#386—24no—
14C3×#381C2212yes—
15C3×#379—21no—
16#5223—22no—
17#5221C412yescomplemented
18C2×#833—22no—
19C4×#217C2211yescharacteristic
20#5207C2212yes—
21#4451—24no—
22#2433—31no—
23#2308D311yescharacteristic, complemented
24#1783—31no—
25#1510—32no—
26#1284—32no—
27C25×C3D411yescharacteristic
28C25×C3D411yescharacteristic
29C22×C3×D4—22no—
30C22×C3×D4—22no—
31C22×C3×D4—44no—
32C22×C3×D4C2×C412yes—
33C22×C3×D4D412yes—
34C23×C3×C4—22no—
35C23×C3×C4C2311yescharacteristic
36C22×#67—21no—
37C22×#67—22no—
38C22×#67—22no—
39C22×D12C2×C411yescharacteristic, complemented
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—44no—
45#953—44no—
46#953—44no—
47C2×#217—21no—
48C2×#217—22no—
49C2×#217—22no—
50#940—44no—
51#930—22no—
52#930—22no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3C2×C411yescharacteristic, complemented
55C2×C4×Dic3C2×C412yescomplemented
56C2×#208—22no—
57C3×#95—22no—
58C3×#95D414yescomplemented
59#833—44no—
60C3⋊#95—21no—
61#830—22no—
62#830—22no—
63C23×D4C2×D311yescharacteristic
64C2×#127—32no—
65C2×#121—31no—
66C2×#120—61no—
67C2×#120—61no—
68C22×C22⋊C4—21no—
69C2×#100—62no—
70#407—32no—
71#392—62no—
72#390—32no—
73#386—24no—
74#381C2×D312yes—
75#379—21no—
76C4×C22⋊C4—31no—
77#327—64no—
78C24×C3—21no—
79C24×C3—24no—
80C24×C3—21no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3—42no—
84C24×C3—42no—
85C24×C3—42no—
86C24×C3—81no—
87C24×C3—81no—
88C24×C3C22⋊C412yes—
89C24×C3C22⋊C412yes—
90C24×C3C2×D411yescharacteristic
91C23×D3—21no—
92C2×C3×D4—24no—
93C2×C3×D4—24no—
94C2×C3×D4—24no—
95C2×C3×D4—24no—
96C2×C3×D4—42no—
97C2×C3×D4—42no—
98C2×C3×D4—44no—
99C2×C3×D4—44no—
100C2×C3×D4—48no—
101C22×C3×C4—24no—
102C22×C3×C4—42no—
103C22×C3×C4—44no—
104C22×C3×C4—82no—
105C22×C3×C4—82no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D412yes—
109C22×C3×C4#4112yes—
110C2×#67—44no—
111C2×#67—44no—
112C2×#67—44no—
113C2×#67—42no—
114C2×#67—42no—
115C2×#67—44no—
116C2×#67—41no—
117C2×#67—41no—
118C2×#67—82no—
119C2×#67—82no—
120C22×Dic3—22no—
121C22×Dic3—21no—
122C2×D12—22no—
123C2×D12—42no—
124C3×C22⋊C4—44no—
125C3×C22⋊C4—44no—
126C3×C22⋊C4—82no—
127C3×C22⋊C4—82no—
128#217—24no—
129#217—44no—
130#217—44no—
131(C2×D3)⋊C4—42no—
132C4⋊Dic3—22no—
133C4×Dic3—42no—
134#208—44no—
135C2×C3⋊C8—42no—
136C25#6711yescharacteristic
137C25#6711yescharacteristic
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—31no—
141C22×D4—44no—
142C22×D4—61no—
143C22×D4—62no—
144C22×D4—62no—
145C22×D4C4×D312yes—
146C22×D4D1212yes—
147C23×C4—22no—
148C23×C4C22×D311yescharacteristic
149C2×M16—62no—
150#127—124no—
151#121—62no—
152#121—62no—
153#120—124no—
154#120—124no—
155C2×C4⋊C4—31no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—62no—
163C2×C22⋊C4—62no—
164C2×C22⋊C4—61no—
165C2×C42—32no—
166#100—124no—
167C22⋊C8—62no—
168C22⋊C8—62no—
169#95—22no—
170#95—61no—
171#95#6714yescomplemented
172C23×C3—22no—
173C23×C3—22no—
174C23×C3—24no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—22no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—41no—
183C23×C3—42no—
184C23×C3—42no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—42no—
192C23×C3—42no—
193C23×C3—81no—
194C23×C3—81no—
195C23×C3—81no—
196C23×C3—81no—
197C23×C3—88no—
198C23×C3—88no—
199C23×C3#9914yes—
200C23×C3C2×C22⋊C411yescharacteristic
201C23×C3#12011yescharacteristic
202C23×C3#12311yescharacteristic
203C22×D3—41no—
204C22×D3—41no—
205C22×D3—42no—
206C22×D3—41no—
207C22×D3—42no—
208C3×D4—48no—
209C3×D4—48no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—44no—
215C2×C3×C4—44no—
216C2×C3×C4—44no—
217C2×C3×C4—88no—
218C2×C3×C4C4×D412yes—
219C2×C3×C4#12112yes—
220#67—88no—
221#67—88no—
222#67—82no—
223#67—82no—
224C2×Dic3—24no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230D12—82no—
231C3⋊C8—82no—
232C24—21no—
233C24—24no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—61no—
241C24—81no—
242C24—81no—
243C24(C2×D3)⋊C412yes—
244C24(C2×D3)⋊C412yes—
245C24C2×#6711yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—48no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—62no—
256C2×D4—122no—
257C2×D4—124no—
258C2×D4—124no—
259C2×D4—122no—
260C2×D4—122no—
261C2×D4—124no—
262C2×D4—124no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—242no—
266C2×D4—242no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—62no—
271C22×C4—61no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4C2×C4×D311yescharacteristic
275C22×C4C2×D1211yescharacteristic
276C22×C4#23512yes—
277C22×C4C2×#6712yes—
278M16—124no—
279C2×C8—122no—
280C4⋊C4—62no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—64no—
284C22⋊C4—82no—
285C22⋊C4—82no—
286C22⋊C4—124no—
287C22⋊C4—124no—
288C22⋊C4—122no—
289C42—122no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—24no—
297C22×C3—21no—
298C22×C3—44no—
299C22×C3—44no—
300C22×C3—44no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—42no—
308C22×C3—82no—
309C22×C3—82no—
310C22×C3—84no—
311C22×C3—84no—
312C22×C3—84no—
313C22×C3#38611yescharacteristic
314C22×C3C2×#9912yes—
315C22×C3#45312yes—
316C2×D3—82no—
317C2×D3—82no—
318C2×D3—81no—
319C2×D3—81no—
320C2×D3—81no—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—84no—
324Dic3—44no—
325Dic3—81no—
326C23—22no—
327C23—22no—
328C23—24no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—41no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—88no—
352C23—88no—
353C23—121no—
354C23—121no—
355C23—122no—
356C23—122no—
357C23—121no—
358C23#80614yes—
359C23C2×(C2×D3)⋊C411yescharacteristic
360C23#92911yescharacteristic
361C23#95311yescharacteristic
362D4—48no—
363D4—48no—
364D4—242no—
365D4—248no—
366D4—248no—
367D4—242no—
368D4—242no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—64no—
377C2×C4—88no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4C4×#6712yes—
384C2×C4#93012yes—
385C8—242no—
386C2×C3—22no—
387C2×C3—22no—
388C2×C3—41no—
389C2×C3—44no—
390C2×C3—82no—
391C2×C3#188711yescharacteristic
392C2×C3#189811yescharacteristic
393C2×C3#191111yescharacteristic
394D3—161no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—24no—
401C22—21no—
402C22—21no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—82no—
414C22—82no—
415C22—84no—
416C22—84no—
417C22—84no—
418C22—242no—
419C22—242no—
420C22—241no—
421C22—241no—
422C22—241no—
423C22C2×#80612yes—
424C22#521311yescharacteristic
425C22#524112yes—
426C4—22no—
427C4—22no—
428C4—84no—
429C4—124no—
430C4—241no—
431C3#946311yescharacteristic, complemented, Sylow
432C2—22no—
433C2—22no—
434C2—41no—
435C2—44no—
436C2—82no—
437C2—481no—
438C2#7154311yescharacteristic
439C2#7165011yescharacteristic
440C2#7165211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2308C2
socleC22×C3#386
prime radical#84049C2
cosocleC23×C3×C4C23
hypercenter#2308D3
hypocenterC3#9463
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2308

Complements

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

NH ≅ G/Ntype
1. C3×#2308C2semidirect
3. #72129C2semidirect
4. #69531C2semidirect
5. #67628C2semidirect
11. C2×#940C4semidirect
17. #5221C4semidirect
23. #2308D3semidirect
39. C22×D12C2×C4semidirect
54. C2×C4⋊Dic3C2×C4semidirect
55. C2×C4×Dic3C2×C4semidirect
58. C3×#95D4semidirect
171. #95#67semidirect
431. C3#9463semidirect

Maximal subgroups

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

subgroupindex
C3×#23082
#840492
#721292
#695312
#676282
#94633