Finite Group Databasebeta

← #10695670

Subgroups — #10695670

Order 768.

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

normal subgroups only

This group has 888 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×#2435C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic
3#82741C211yesmaximal, characteristic, complemented
4#72128C211yesmaximal, characteristic
5#71690C211yesmaximal, characteristic, complemented
6#69457C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9955—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×#124—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C22×#214—21no—
13C2×#924C2211yescharacteristic
14C3×#394—22no—
15C3×#390—21no—
16C3×#379—22no—
17C3×C4×C22⋊C4C2211yescharacteristic
18#5228—21no—
19#5218—22no—
20#5218—22no—
21C2×#832—21no—
22C2×#832—22no—
23#5212C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×#830C2211yescharacteristic
26#5051—22no—
27#5050—21no—
28#5044—21no—
29#4970—21no—
30#4538—22no—
31#4527—22no—
32#3189—31no—
33C2×#450—31no—
34#2435D311yescharacteristic
35#1996—31no—
36#1786—31no—
37#1520—31no—
38#1282—31no—
39C25×C3—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—44no—
47C22×C3×D4D412yes—
48C23×C3×C4C2311yescharacteristic
49C22×Dic6D411yescharacteristic
50C3×#124—44no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—24no—
60C2×C3×C22⋊C4D412yes—
61C2×C3×C42—21no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#216—22no—
65C2×#214—24no—
66C2×#214—44no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69#924—21no—
70#924—21no—
71#924—21no—
72#924—21no—
73C2×Dic3⋊C4—21no—
74C22×C3⋊C8—21no—
75C22×C3⋊C8D412yes—
76C3×#95—21no—
77#835—22no—
78#832—24no—
79#832—44no—
80C3⋊#95—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830D412yes—
85#810—22no—
86#807—22no—
87C23×D4C2×D311yescharacteristic
88C22×SD16—61no—
89C2×#124—21no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92#484—61no—
93#484—61no—
94#467—62no—
95#451—61no—
96#451—61no—
97#450—62no—
98#450—62no—
99C2×#102—61no—
100C2×#102—62no—
101C2×C22⋊C8—31no—
102C2×C22⋊C8—31no—
103#394—22no—
104#390—21no—
105#380—31no—
106#379—22no—
107C4×C22⋊C4C2×D311yescharacteristic
108#342—62no—
109#328—62no—
110C24×C3—22no—
111C24×C3—44no—
112C24×C3—41no—
113C24×C3—41no—
114C24×C3—42no—
115C24×C3—42no—
116C24×C3—82no—
117C24×C3C2×D411yescharacteristic
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4—42no—
129C2×C3×D4—44no—
130C2×C3×D4—44no—
131C2×C3×D4—44no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
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—21no—
145C22×Dic3—21no—
146C22×Dic3—41no—
147C2×Dic6—21no—
148C2×Dic6—21no—
149C2×Dic6—42no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C22⋊C4—22no—
155C3×C22⋊C4—22no—
156C3×C22⋊C4—48no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—84no—
160C3×C42—42no—
161#217—41no—
162#217—41no—
163#217—42no—
164#216—82no—
165#214—48no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3—42no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8—42no—
175C25—21no—
176C22×Q8—31no—
177C22×Q8#6711yescharacteristic
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—44no—
184C22×D4#6712yes—
185C23×C4C22×D311yescharacteristic
186C2×Q16—62no—
187C2×SD16—64no—
188C2×SD16—124no—
189C22×C8—32no—
190C22×C8—61no—
191#124—44no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4#6711yescharacteristic
205C2×C22⋊C4—24no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4#6712yes—
211C2×C42—21no—
212#107—62no—
213#103—62no—
214#103—62no—
215#102—64no—
216#102—124no—
217C22⋊C8—32no—
218C22⋊C8—61no—
219C22⋊C8—61no—
220C22⋊C8—61no—
221#95—21no—
222#95—61no—
223C23×C3—24no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3—44no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3—44no—
232C23×C3—82no—
233C23×C3—82no—
234C23×C3—88no—
235C23×C3—88no—
236C23×C3—82no—
237C23×C3—81no—
238C23×C3—81no—
239C23×C3#12011yescharacteristic
240C23×C3#12111yescharacteristic
241C23×C3#12711yescharacteristic
242C23×C3#13512yes—
243C23×C3#13712yes—
244C3×Q8—42no—
245C3×Q8—81no—
246C3×D4—24no—
247C3×D4—24no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—44no—
251C3×D4—44no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4#12011yescharacteristic
266C2×C3×C4#12011yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—81no—
272C2×Dic3—81no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275C2×Dic3—81no—
276Dic6—42no—
277Dic6—42no—
278C3⋊C8—42no—
279C3⋊C8—42no—
280C24—22no—
281C24—44no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24—42no—
286C24—82no—
287C24C2×#6711yescharacteristic
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—42no—
291C2×Q8—61no—
292C2×Q8—61no—
293C2×Q8—122no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—121no—
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C22×C4#23711yescharacteristic
318C22×C4C2×#6711yescharacteristic
319C22×C4C2×#6711yescharacteristic
320Q16—242no—
321SD16—128no—
322C2×C8—62no—
323C2×C8—62no—
324C2×C8—122no—
325C2×C8—122no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—122no—
333C4⋊C4—121no—
334C4⋊C4—121no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—48no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—84no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C22⋊C4—122no—
344C42—42no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—48no—
354C22×C3—44no—
355C22×C3—84no—
356C22×C3—84no—
357C22×C3—88no—
358C22×C3—84no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3#39211yescharacteristic
362C22×C3#45212yes—
363C22×C3#46412yes—
364C22×C3#49511yescharacteristic
365C22×C3#49711yescharacteristic
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—82no—
370C3×C4—81no—
371C3×C4—81no—
372C3×C4B(4,𝔽2)12yes—
373Dic3—81no—
374Dic3—81no—
375Dic3—161no—
376C23—24no—
377C23—41no—
378C23—41no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—42no—
383C23—42no—
384C23—44no—
385C23—82no—
386C23—82no—
387C23—88no—
388C23—88no—
389C23—82no—
390C23—81no—
391C23—81no—
392C23#93811yescharacteristic
393C23#94011yescharacteristic
394C23#95012yes—
395C23#95112yes—
396C23#95311yescharacteristic
397Q8—42no—
398Q8—81no—
399Q8—122no—
400Q8—122no—
401D4—24no—
402D4—24no—
403D4—42no—
404D4—42no—
405D4—44no—
406D4—44no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—241no—
423C2×C4—241no—
424C2×C4—241no—
425C2×C4—241no—
426C2×C4—241no—
427C2×C4#93711yescharacteristic
428C2×C4#93711yescharacteristic
429C2×C4#93911yescharacteristic
430C2×C4#93911yescharacteristic
431C8—122no—
432C8—122no—
433C2×C3—21no—
434C2×C3—22no—
435C2×C3—22no—
436C2×C3—21no—
437C2×C3—84no—
438C2×C3#200811yescharacteristic
439C2×C3#201511yescharacteristic
440C2×C3#202111yescharacteristic
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—42no—
448C22—42no—
449C22—48no—
450C22—44no—
451C22—84no—
452C22—84no—
453C22—88no—
454C22—84no—
455C22—82no—
456C22—82no—
457C22#505412yes—
458C22#505911yescharacteristic
459C22#506011yescharacteristic
460C22#522311yescharacteristic
461C22#523912yes—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—82no—
466C4—81no—
467C4—81no—
468C4—241no—
469C4—241no—
470C4—481no—
471C4#503212yes—
472C3#995511yescharacteristic, complemented, Sylow
473C2—22no—
474C2—21no—
475C2—22no—
476C2—21no—
477C2—84no—
478C2#7167111yescharacteristic
479C2#7172911yescharacteristic
480C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2435C2
socleC22×C3#392
prime radicalC23×C3×D4C22
hypercenter#2435D3
hypocenterC3#9955
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
3. #82741C2semidirect
5. #71690C2semidirect
6. #69457C2semidirect
7. #67617C2semidirect
472. C3#9955semidirect

Maximal subgroups

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

subgroupindex
C3×#24352
C2×#52182
#827412
#721282
#716902
#694572
#676172
#99553