Finite Group Databasebeta

← #10695830

Subgroups — #10695830

Order 768.

722 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3007C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic
3#82808C211yesmaximal, characteristic, complemented
4#72139C211yesmaximal, characteristic
5#71689C211yesmaximal, characteristic, complemented
6#69981C211yesmaximal, characteristic, complemented
7#67686C211yesmaximal, characteristic, complemented
8#9982—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#137—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#951—21no—
13C2×#924C2211yescharacteristic
14C3×#452—22no—
15C3×#451—21no—
16C3×#450—21no—
17C3×#417—21no—
18C3×#407C2211yescharacteristic
19#5239—22no—
20#5232—21no—
21#5230—21no—
22#5228—21no—
23C2×#834—22no—
24#5218—21no—
25#5211C2211yescharacteristic
26#5211C2211yescharacteristic
27#5207C2211yescharacteristic
28#4986—21no—
29#4985—21no—
30#4979—21no—
31#4952—22no—
32#4538—22no—
33#4467—22no—
34#3007—31no—
35#3007D311yescharacteristic
36#2817—31no—
37#1782—31no—
38#1511—31no—
39#1511—31no—
40#1306—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic
47C22×C3×Q8D411yescharacteristic
48C22×C3×D4—21no—
49C22×C3×D4D411yescharacteristic
50C23×C3×C4—21no—
51C23×C3×C4C2311yescharacteristic
52C22×Dic6D411yescharacteristic
53C3×#137—44no—
54C2×C3×Q16—22no—
55C2×C3×SD16—22no—
56C2×C3×M16—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×#217—21no—
65C2×#217—21no—
66C2×#217D412yes—
67#951—44no—
68C2×#216—22no—
69C2×#214—22no—
70C2×C4⋊Dic3D411yescharacteristic
71#924—21no—
72#924—21no—
73#924—21no—
74#924—21no—
75C2×Dic3⋊C4—21no—
76C2×Dic3⋊C4—22no—
77C2×#208—21no—
78C3×#103—22no—
79C3×#102—22no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8D412yes—
82#835—22no—
83#834—44no—
84#832—22no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—21no—
89C3⋊#95—21no—
90C3⋊#95—21no—
91C3⋊#95—21no—
92#830—21no—
93#830D412yes—
94C22×#41C2×D311yescharacteristic
95C2×#137—21no—
96C2×#137—61no—
97C2×C22⋊Q8—31no—
98C2×C22⋊Q8C2×D311yescharacteristic
99#452—22no—
100#452—62no—
101#451—21no—
102#451—61no—
103#450—21no—
104#450—61no—
105#417—21no—
106#417—61no—
107C2×#99—62no—
108#407—31no—
109#407C2×D311yescharacteristic
110#400—61no—
111#400—61no—
112#397—62no—
113#395—61no—
114#395—61no—
115#381—31no—
116#381—31no—
117#328—62no—
118#328—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C3×#41—24no—
122C3×#41—24no—
123C3×#41—42no—
124C3×#41—42no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—42no—
134C2×C3×Q8—42no—
135C2×C3×D4—24no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—42no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—41no—
149C22×C3×C4—41no—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4#4111yescharacteristic
156C22×Dic3—22no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—41no—
160C2×Dic6—21no—
161C2×Dic6—21no—
162C2×Dic6—42no—
163C3×Q16—44no—
164C3×SD16—44no—
165C3×M16—42no—
166C2×C3×C8—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—41no—
170C3×C4⋊C4—41no—
171C3×C22⋊C4—41no—
172C3×C22⋊C4—41no—
173#217—41no—
174#217—41no—
175#217—42no—
176#217—42no—
177#216—44no—
178#214—44no—
179C4⋊Dic3—21no—
180C4⋊Dic3—21no—
181Dic3⋊C4—41no—
182Dic3⋊C4—41no—
183Dic3⋊C4—82no—
184#208—42no—
185C2×C3⋊C8—22no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—44no—
191C22×Q8—31no—
192C22×Q8#6711yescharacteristic
193C22×Q8#6711yescharacteristic
194C22×D4—21no—
195C22×D4#6711yescharacteristic
196C23×C4—21no—
197C23×C4C22×D311yescharacteristic
198#137—44no—
199#137—124no—
200C2×Q16—22no—
201C2×Q16—62no—
202C2×SD16—22no—
203C2×SD16—62no—
204C2×M16—21no—
205C2×M16—61no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—61no—
211C22⋊Q8—61no—
212C22⋊Q8—61no—
213C22⋊Q8—61no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—61no—
217C2×C4⋊C4—62no—
218C2×C4⋊C4#6711yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—32no—
221C2×C22⋊C4—61no—
222C2×C22⋊C4—61no—
223#103—22no—
224#103—62no—
225#102—22no—
226#102—62no—
227#99—124no—
228C22⋊C8—21no—
229C22⋊C8—32no—
230C22⋊C8—61no—
231C22⋊C8#6712yes—
232#95—62no—
233#95—62no—
234#95—62no—
235#95—61no—
236#95—61no—
237#95—61no—
238#95—61no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—81no—
246C23×C3—81no—
247C23×C3#12011yescharacteristic
248C23×C3#12111yescharacteristic
249C23×C3#12711yescharacteristic
250C3×Q8—22no—
251C3×Q8—22no—
252C3×Q8—41no—
253C3×Q8—41no—
254C3×Q8—42no—
255C3×Q8—81no—
256C3×D4—22no—
257C3×D4—22no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—44no—
261C3×D4—44no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—44no—
270C2×C3×C4—44no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4#12011yescharacteristic
278C2×C3×C4#12011yescharacteristic
279C2×C3×C4#12111yescharacteristic
280C2×C3×C4#12111yescharacteristic
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—81no—
289C2×Dic3—81no—
290C2×Dic3—81no—
291Dic6—42no—
292Dic6—81no—
293C3×C8—42no—
294C3⋊C8—42no—
295C24—21no—
296C24D3×D411yescharacteristic
297#41—24no—
298#41—24no—
299#41—42no—
300#41—42no—
301#41—42no—
302#41—42no—
303#41—42no—
304#41—42no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—61no—
312C2×Q8—61no—
313C2×Q8—122no—
314C2×D4—24no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—41no—
320C2×D4—41no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—62no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—121no—
333C22×C4D3×D411yescharacteristic
334C22×C4D3×D411yescharacteristic
335C22×C4#23711yescharacteristic
336C22×C4C2×#6711yescharacteristic
337C22×C4C2×#6711yescharacteristic
338C22×C4C2×#6711yescharacteristic
339Q16—44no—
340Q16—124no—
341SD16—44no—
342SD16—124no—
343M16—42no—
344M16—122no—
345C2×C8—22no—
346C2×C8—62no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C4⋊C4—121no—
354C4⋊C4—121no—
355C4⋊C4—242no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—121no—
359C22⋊C4—121no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—42no—
371C22×C3—82no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3#39211yescharacteristic
377C22×C3B(4,𝔽2)12yes—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—42no—
383C3×C4—81no—
384C3×C4—81no—
385Dic3—82no—
386Dic3—81no—
387Dic3—81no—
388Dic3—161no—
389C23—22no—
390C23—22no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—81no—
396C23—81no—
397C23#93711yescharacteristic
398C23#93911yescharacteristic
399C23#94011yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—42no—
405Q8—81no—
406Q8—122no—
407Q8—241no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412D4—44no—
413D4—44no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—241no—
437C2×C4—241no—
438C2×C4—241no—
439C2×C4#93711yescharacteristic
440C2×C4#93811yescharacteristic
441C2×C4#93911yescharacteristic
442C2×C4#95311yescharacteristic
443C8—42no—
444C8—122no—
445C2×C3—22no—
446C2×C3—41no—
447C2×C3—41no—
448C2×C3—82no—
449C2×C3#200911yescharacteristic
450C2×C3#200911yescharacteristic
451C2×C3#202111yescharacteristic
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—81no—
465C22—81no—
466C22#503212yes—
467C22#522311yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—242no—
476C4—241no—
477C4—241no—
478C4—481no—
479C3#998211yescharacteristic, complemented, Sylow
480C2—22no—
481C2—41no—
482C2—41no—
483C2—82no—
484C2#7167111yescharacteristic
485C2#7211111yescharacteristic
486C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3007C2
socleC22×C3#392
prime radicalC22×C3×#41C22
hypercenter#3007D3
hypocenterC3#9982
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3007

Complements

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

NH ≅ G/Ntype
3. #82808C2semidirect
5. #71689C2semidirect
6. #69981C2semidirect
7. #67686C2semidirect
479. C3#9982semidirect

Maximal subgroups

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

subgroupindex
C3×#30072
#840842
#828082
#721392
#716892
#699812
#676862
#99823