Finite Group Databasebeta

← #10695796

Subgroups — #10695796

Order 768.

716 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 85 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#82741C211yesmaximal, characteristic, complemented
4#72136C211yesmaximal, characteristic
5#71691C211yesmaximal, characteristic, complemented
6#69972C211yesmaximal, characteristic, complemented
7#67687C211yesmaximal, characteristic, complemented
8#9976—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#5228—21no—
22#5228—21no—
23#5218—21no—
24#5207C2211yescharacteristic
25C2×#830—22no—
26C2×#830C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#5051—22no—
29#5050—21no—
30#5044—21no—
31#4970—21no—
32#4528—22no—
33#4469—22no—
34#3189—31no—
35#3007—31no—
36#3007D311yescharacteristic
37#1780—31no—
38#1472—31no—
39#1472—31no—
40#1293—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—
65#951—44no—
66C2×#216—22no—
67C2×#216—22no—
68C2×#214—22no—
69C2×C4⋊Dic3D411yescharacteristic
70#924—21no—
71#924—21no—
72#924—21no—
73#924—21no—
74C2×Dic3⋊C4—21no—
75C2×#208—21no—
76C22×C3⋊C8—21no—
77C22×C3⋊C8D412yes—
78C3×#103—22no—
79C3×#102—22no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8D412yes—
82#835—22no—
83#835—22no—
84#832—22no—
85#830—21no—
86#830—21no—
87#830—21no—
88#830—21no—
89#830—21no—
90#830—44no—
91#830D412yes—
92#810—22no—
93#807—22no—
94C22×#41C2×D311yescharacteristic
95C2×#137—21no—
96C2×#137—61no—
97C2×C22⋊Q8—31no—
98C2×C22⋊Q8C2×D311yescharacteristic
99#484—61no—
100#484—61no—
101#467—62no—
102#452—22no—
103#452—62no—
104#451—21no—
105#451—61no—
106#451—61no—
107#451—61no—
108#450—21no—
109#450—61no—
110#417—21no—
111#417—61no—
112#407—31no—
113#407C2×D311yescharacteristic
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116C2×C22⋊C8—62no—
117#324—62no—
118#324—62no—
119C24×C3—21no—
120C24×C3#4111yescharacteristic
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—21no—
136C2×C3×D4—21no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C22×C3×C4—24no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
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×C4C2×D411yescharacteristic
156C22×Dic3—21no—
157C22×Dic3—21no—
158C2×Dic6—21no—
159C2×Dic6—21no—
160C2×Dic6—42no—
161C3×Q16—44no—
162C3×SD16—44no—
163C3×M16—42no—
164C2×C3×C8—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—41no—
169C3×C22⋊C4—41no—
170C3×C22⋊C4—41no—
171#217—41no—
172#217—41no—
173#216—44no—
174#216—82no—
175#214—44no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178Dic3⋊C4—41no—
179Dic3⋊C4—41no—
180#208—42no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—42no—
184C2×C3⋊C8—42no—
185C2×C3⋊C8—41no—
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×Q16—62no—
203C2×SD16—22no—
204C2×SD16—62no—
205C2×M16—21no—
206C2×M16—61no—
207C22×C8—32no—
208C22×C8—61no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215C22⋊Q8—61no—
216C22⋊Q8—61no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4#6711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—61no—
223#107—62no—
224#103—22no—
225#103—62no—
226#103—62no—
227#103—62no—
228#102—22no—
229#102—62no—
230C22⋊C8—21no—
231C22⋊C8—32no—
232C22⋊C8—61no—
233C22⋊C8—61no—
234C22⋊C8—61no—
235C22⋊C8—61no—
236C22⋊C8—61no—
237C22⋊C8—124no—
238C22⋊C8#6712yes—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3#12111yescharacteristic
248C23×C3#12111yescharacteristic
249C23×C3#12111yescharacteristic
250C3×Q8—22no—
251C3×Q8—22no—
252C3×Q8—41no—
253C3×Q8—41no—
254C3×Q8—42no—
255C3×Q8—42no—
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—22no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—44no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—44no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—81no—
277C2×C3×C4#12011yescharacteristic
278C2×C3×C4#12011yescharacteristic
279C2×C3×C4#12011yescharacteristic
280C2×C3×C4#12711yescharacteristic
281C2×Dic3—42no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—81no—
285Dic6—42no—
286Dic6—42no—
287C3×C8—42no—
288C3⋊C8—42no—
289C3⋊C8—42no—
290C3⋊C8—81no—
291C24—21no—
292C24#23711yescharacteristic
293#41—24no—
294#41—24no—
295#41—42no—
296#41—42no—
297#41—42no—
298#41—42no—
299#41—42no—
300#41—42no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—61no—
308C2×Q8—61no—
309C2×Q8—122no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4D3×D411yescharacteristic
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332C22×C4C2×#6711yescharacteristic
333Q16—44no—
334Q16—124no—
335Q16—242no—
336SD16—44no—
337SD16—124no—
338M16—42no—
339M16—122no—
340C2×C8—22no—
341C2×C8—62no—
342C2×C8—62no—
343C2×C8—122no—
344C2×C8—122no—
345C2×C8—121no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—121no—
353C4⋊C4—121no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—121no—
357C22⋊C4—121no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—44no—
367C22×C3—42no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3—81no—
371C22×C3—81no—
372C22×C3#39211yescharacteristic
373C22×C3#47312yes—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—42no—
378C3×C4—81no—
379C3×C4—81no—
380C3×C4B(4,𝔽2)12yes—
381Dic3—81no—
382Dic3—81no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—41no—
391C23#93811yescharacteristic
392C23#93911yescharacteristic
393C23#93911yescharacteristic
394Q8—22no—
395Q8—22no—
396Q8—41no—
397Q8—41no—
398Q8—42no—
399Q8—42no—
400Q8—122no—
401Q8—122no—
402D4—22no—
403D4—22no—
404D4—41no—
405D4—41no—
406D4—44no—
407D4—44no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—81no—
423C2×C4—122no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—241no—
427C2×C4#93711yescharacteristic
428C2×C4#93711yescharacteristic
429C2×C4#94011yescharacteristic
430C2×C4#95311yescharacteristic
431C8—42no—
432C8—122no—
433C8—122no—
434C8—241no—
435C2×C3—22no—
436C2×C3—41no—
437C2×C3—41no—
438C2×C3—82no—
439C2×C3#201711yescharacteristic
440C2×C3#202011yescharacteristic
441C2×C3#202111yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—44no—
451C22—42no—
452C22—82no—
453C22—82no—
454C22—81no—
455C22—81no—
456C22#520412yes—
457C22#522311yescharacteristic
458C4—21no—
459C4—21no—
460C4—21no—
461C4—42no—
462C4—81no—
463C4—81no—
464C4—241no—
465C4—241no—
466C4#503212yes—
467C3#997611yescharacteristic, complemented, Sylow
468C2—22no—
469C2—41no—
470C2—41no—
471C2—82no—
472C2#7167111yescharacteristic
473C2#7210911yescharacteristic
474C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3007C2
socleC22×C3#392
prime radicalC22×C3×#41C22
hypercenter#3007D3
hypocenterC3#9976
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. #82741C2semidirect
5. #71691C2semidirect
6. #69972C2semidirect
7. #67687C2semidirect
467. C3#9976semidirect

Maximal subgroups

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

subgroupindex
C3×#30072
#840842
#827412
#721362
#716912
#699722
#676872
#99763