Finite Group Databasebeta

← #10695759

Subgroups — #10695759

Order 768.

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

normal subgroups only

This group has 658 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
1C2×#4970C211yesmaximal, characteristic, complemented
2C2×#4968C211yesmaximal, characteristic
3#73379C211yesmaximal, characteristic
4C3×#1445C211yesmaximal, characteristic, complemented
5C2×#4452C211yesmaximal, characteristic, complemented
6C2×#4451C211yesmaximal, characteristic
7#67613C211yesmaximal, characteristic, complemented
8#9969—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×C3×#95—22no—
13C2×C3×#95C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5213—22no—
16C2×#830C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4970—22no—
19#4970—22no—
20C2×#810—21no—
21#4968—22no—
22#4968—22no—
23C2×#809—21no—
24C2×#808—21no—
25C2×#807—21no—
26#4674—21no—
27#4673—21no—
28#4672—21no—
29#4659—21no—
30#4452—22no—
31#4451—22no—
32C2×#484—31no—
33C2×#481—31no—
34#2639—31no—
35C2×#328—31no—
36C2×#327—31no—
37#1445D311yescharacteristic, complemented
38#1280—31no—
39C23×C3×C4—21no—
40C23×C3×C4—21no—
41C23×C3×C4D412yes—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×D12D411yescharacteristic
45C22×Dic6D411yescharacteristic
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D412yes—
52C2×#217—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56#930—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—22no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62#924—21no—
63#924—21no—
64#924—21no—
65#924—21no—
66C2×Dic3⋊C4—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95—44no—
74C3⋊#95—22no—
75C3⋊#95—22no—
76#830—21no—
77#830—21no—
78#830D412yes—
79#830D412yes—
80#810—24no—
81#809—24no—
82#808—24no—
83#807—24no—
84C2×C22⋊Q8—31no—
85C2×#121—31no—
86C22×C4⋊C4C2×D311yescharacteristic
87#484—62no—
88#484—62no—
89#481—62no—
90#481—62no—
91C2×#107—61no—
92C2×#106—61no—
93C2×#103—61no—
94C2×#102—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8—31no—
97#400—61no—
98#397—61no—
99#394—61no—
100#393—61no—
101#386—62no—
102C2×#95—22no—
103C2×#95C2×D311yescharacteristic
104C2×#95C2×D311yescharacteristic
105#328—62no—
106#327—62no—
107C24×C3C2×D411yescharacteristic
108C23×D3—21no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—42no—
115C22×C3×C4—42no—
116C22×C3×C4—42no—
117C22×C3×C4—42no—
118C22×C3×C4—41no—
119C22×C3×C4—42no—
120C22×C3×C4—42no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—42no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—41no—
132C2×#67—41no—
133C2×#67—42no—
134C2×#67—42no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—42no—
141C2×Dic6—21no—
142C2×Dic6—21no—
143C2×Dic6—42no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148#217—41no—
149#217—41no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—82no—
153(C2×D3)⋊C4—82no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—42no—
163C2×C3⋊C8—42no—
164C22×Q8—31no—
165C22×D4—31no—
166C22×D4—61no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4#6712yes—
170C23×C4C22×D311yescharacteristic
171C22×C8—61no—
172C22×C8—61no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177#121—61no—
178#121—61no—
179#121—61no—
180#121—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4D1212yes—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194#107—64no—
195#106—64no—
196#103—64no—
197#102—64no—
198C22⋊C8—32no—
199C22⋊C8—32no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—44no—
207#95—62no—
208#95—62no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—22no—
212C23×C3#12011yescharacteristic
213C23×C3#12111yescharacteristic
214C23×C3#12711yescharacteristic
215C23×C3#13512yes—
216C23×C3#13712yes—
217C22×D3—41no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—42no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—44no—
229C2×C3×C4—42no—
230C2×C3×C4—44no—
231C2×C3×C4—44no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—82no—
237C2×C3×C4—81no—
238C2×C3×C4—82no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4—82no—
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247#67—82no—
248#67—81no—
249#67—81no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—42no—
257D12—42no—
258D12—81no—
259Dic6—42no—
260Dic6—81no—
261C3⋊C8—42no—
262C3⋊C8—42no—
263C24—61no—
264C24C2×#6711yescharacteristic
265C2×Q8—61no—
266C2×Q8—61no—
267C2×Q8—122no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—122no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—122no—
274C2×D4—122no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4C2×D1211yescharacteristic
295C22×C4#23511yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C2×C8—62no—
301C2×C8—62no—
302C2×C8—122no—
303C2×C8—122no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—121no—
313C4⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—242no—
319C22⋊C4—242no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3#39211yescharacteristic
329C22×C3#45212yes—
330C22×C3#45311yescharacteristic
331C22×C3#45411yescharacteristic
332C22×C3#45812yes—
333C22×C3#46412yes—
334C22×C3#49511yescharacteristic
335C22×C3#49711yescharacteristic
336C2×D3—82no—
337C2×D3—82no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—81no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—82no—
346C3×C4—82no—
347C3×C4—82no—
348Dic3—81no—
349Dic3—81no—
350Dic3—81no—
351C23—22no—
352C23—21no—
353C23—21no—
354C23—121no—
355C23—122no—
356C23—122no—
357C23—121no—
358C23—121no—
359C23#93011yescharacteristic
360C23#93711yescharacteristic
361C23#94011yescharacteristic
362C23#94212yes—
363C23#95012yes—
364Q8—122no—
365Q8—241no—
366D4—122no—
367D4—241no—
368D4—242no—
369D4—241no—
370D4—241no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4#88211yescharacteristic
400C2×C4#88211yescharacteristic
401C2×C4#88411yescharacteristic
402C2×C4#88411yescharacteristic
403C8—122no—
404C8—122no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—21no—
408C2×C3#161512yes—
409C2×C3#161812yes—
410C2×C3#200611yescharacteristic
411C2×C3#201511yescharacteristic
412C2×C3#202211yescharacteristic
413D3—161no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—241no—
423C22—241no—
424C22—242no—
425C22—242no—
426C22—241no—
427C22#496011yescharacteristic
428C22#502812yes—
429C22#504712yes—
430C22#505312yes—
431C22#505911yescharacteristic
432C22#506211yescharacteristic
433C22#507711yescharacteristic
434C22#508611yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—82no—
440C4—82no—
441C4—82no—
442C4—241no—
443C4—241no—
444C4—241no—
445C3#996911yescharacteristic, complemented, Sylow
446C2—21no—
447C2—22no—
448C2—21no—
449C2—481no—
450C2#6953911yescharacteristic
451C2#6966411yescharacteristic
452C2#6968911yescharacteristic
453C2#6971512yes—
454C2#6971712yes—
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1445C2
socleC23×C3#120
prime radicalC2×#930C22
hypercenter#1445D3
hypocenterC3#9969
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
1. C2×#4970C2semidirect
4. C3×#1445C2semidirect
5. C2×#4452C2semidirect
7. #67613C2semidirect
37. #1445D3semidirect
445. C3#9969semidirect

Maximal subgroups

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

subgroupindex
C2×#49702
C2×#49682
#733792
C3×#14452
C2×#44522
C2×#44512
#676132
#99693