Finite Group Databasebeta

← #10698072

Subgroups — #10698072

Order 768.

602 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2855C211yesmaximal, characteristic
2#82741C211yesmaximal, characteristic
3#82075C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic
5#71531C211yesmaximal, characteristic
6#69510C211yesmaximal, characteristic
7#67625C211yesmaximal, characteristic
8#10325—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C42C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#391—21no—
13C3×#389—21no—
14C3×#387—22no—
15C3×#385—21no—
16C3×#381—21no—
17C3×#380C2211yescharacteristic
18#5228—21no—
19#5212C2211yescharacteristic
20#5211—21no—
21#5210—21no—
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5051—22no—
25#5050—21no—
26#5044—21no—
27#4970—21no—
28#4933—22no—
29#4929—21no—
30#4928—21no—
31#4538—22no—
32#4524—22no—
33#4452—22no—
34#3189—31no—
35#2855—31no—
36#2855D311yescharacteristic
37#1817—31no—
38#1520—31no—
39#1520—31no—
40#1289—31no—
41C22×C3×Q8D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×Dic6D411yescharacteristic
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C42—21no—
57C2×C3×C42—21no—
58C2×C3×C42—21no—
59C2×C3×C42—21no—
60C2×C3×C42—21no—
61C2×C3×C42—21no—
62C2×#217—21no—
63C2×#217—21no—
64C2×#216—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D411yescharacteristic
67#924—21no—
68#924—21no—
69#924—21no—
70#924—21no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—21no—
73C22×C3⋊C8—21no—
74C22×C3⋊C8D412yes—
75C3×#95—22no—
76C3×#95—22no—
77C3×#95—22no—
78C3×#95D412yes—
79#835—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95D412yes—
84#830—21no—
85#830—21no—
86#830—21no—
87#830—21no—
88#810—22no—
89#807—22no—
90C2×C22⋊Q8—31no—
91C2×C22⋊Q8C2×D311yescharacteristic
92C22×C42C2×D311yescharacteristic
93#484—61no—
94#484—61no—
95#467—62no—
96#451—61no—
97#451—61no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8—31no—
100#391—21no—
101#391—61no—
102#389—21no—
103#389—61no—
104#387—22no—
105#387—62no—
106#385—21no—
107#385—61no—
108#381—21no—
109#381—61no—
110#380—31no—
111#380C2×D311yescharacteristic
112#336—62no—
113#328—62no—
114#328—62no—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4Q1612yescomplemented
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×Dic3—21no—
141C22×Dic3—22no—
142C22×Dic3—21no—
143C2×Dic6—21no—
144C2×Dic6—21no—
145C2×Dic6—42no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—42no—
149C3×C4⋊C4—44no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C42—41no—
157C3×C42—41no—
158C3×C42—41no—
159C3×C42—41no—
160#217—41no—
161#217—41no—
162#217—41no—
163#217—41no—
164#216—82no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—42no—
168Dic3⋊C4—44no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—42no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8—41no—
175C22×Q8—31no—
176C22×Q8#6711yescharacteristic
177C23×C4#6711yescharacteristic
178C23×C4#6711yescharacteristic
179C23×C4C22×D311yescharacteristic
180C2×Q16—62no—
181C22×C8—32no—
182C22×C8—61no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8—61no—
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190C22⋊Q8—61no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4#6711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—61no—
202C2×C22⋊C4—61no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209#107—62no—
210#103—62no—
211#103—62no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—32no—
220#95—62no—
221#95—62no—
222#95—62no—
223#95#6712yes—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3#12011yescharacteristic
227C23×C3#12011yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3C2×Q1611yescharacteristic
230C23×C3#13511yescharacteristic
231C23×C3#13711yescharacteristic
232C23×C3#13711yescharacteristic
233C3×Q8—42no—
234C3×Q8—81no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
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×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265C2×C3×C4#12711yescharacteristic
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—42no—
273Dic6—42no—
274Dic6—81no—
275C3⋊C8—42no—
276C3⋊C8—81no—
277C24C2×#6711yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—42no—
281C2×Q8—61no—
282C2×Q8—61no—
283C2×Q8—122no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—62no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4#21612yescomplemented
302C22×C4D3×D411yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4D3×D411yescharacteristic
305C22×C4#23711yescharacteristic
306C22×C4C2×#6711yescharacteristic
307C22×C4C2×#6711yescharacteristic
308Q16—242no—
309C2×C8—62no—
310C2×C8—122no—
311C2×C8—122no—
312C2×C8—121no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—124no—
322C4⋊C4—122no—
323C4⋊C4—121no—
324C4⋊C4—121no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—121no—
330C22⋊C4—121no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3#39211yescharacteristic
348C22×C3#45111yescharacteristic
349C22×C3#45111yescharacteristic
350C22×C3#45211yescharacteristic
351C22×C3#45211yescharacteristic
352C22×C3#46711yescharacteristic
353C22×C3#47011yescharacteristic
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—42no—
358C3×C4—42no—
359C3×C4—82no—
360C3×C4—81no—
361C3×C4—81no—
362C3×C4#45812yes—
363Dic3—82no—
364Dic3—81no—
365Dic3—81no—
366C23—41no—
367C23—41no—
368C23#93711yescharacteristic
369C23#93911yescharacteristic
370C23#94211yescharacteristic
371C23C2×#21611yescharacteristic
372C23#95011yescharacteristic
373C23#95111yescharacteristic
374C23#95311yescharacteristic
375Q8—42no—
376Q8—81no—
377Q8—122no—
378Q8—241no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
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—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—122no—
413C2×C4#93711yescharacteristic
414C2×C4#93811yescharacteristic
415C2×C4#93911yescharacteristic
416C2×C4#94011yescharacteristic
417C8—122no—
418C8—241no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3#200111yescharacteristic
426C2×C3#201011yescharacteristic
427C2×C3#202211yescharacteristic
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22#504711yescharacteristic
439C22#505011yescharacteristic
440C22#505111yescharacteristic
441C22#505211yescharacteristic
442C22#522311yescharacteristic
443C22#522811yescharacteristic
444C22#523911yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—242no—
454C4—241no—
455C4—241no—
456C4#505312yes—
457C3#1032511yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—21no—
464C2#7170011yescharacteristic
465C2#7170611yescharacteristic
466C2#7171511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2855C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#2855D3
hypocenterC3#10325
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2855

Complements

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

NH ≅ G/Ntype
133. C22×C3×C4Q16semidirect
301. C22×C4#216semidirect
457. C3#10325semidirect

Maximal subgroups

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

subgroupindex
C3×#28552
#827412
#820752
#716902
#715312
#695102
#676252
#103253