Finite Group Databasebeta

← #10695781

Subgroups — #10695781

Order 768.

632 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#82789C211yesmaximal, characteristic
2C2×#5043C211yesmaximal, characteristic
3C2×#4721C211yesmaximal, characteristic
4C2×C3×#328C211yesmaximal, characteristic
5#69423C211yesmaximal, characteristic
6C2×#4452C211yesmaximal, characteristic
7#67674C211yesmaximal, characteristic
8#9973—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14#5230—21no—
15C2×#835—21no—
16C2×C3⋊#95—22no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#817—21no—
20C2×#816—21no—
21#5043—22no—
22#5043—22no—
23#4983—21no—
24#4981—21no—
25#4979—21no—
26C2×#808—21no—
27#4942—22no—
28#4721—22no—
29#4721—22no—
30C3×#328—22no—
31#4452—22no—
32C2×#483—31no—
33C2×#483—31no—
34#2659—31no—
35C2×#328—31no—
36C2×#328D311yescharacteristic
37#1445—31no—
38#1280—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C23×Dic3D412yes—
44C22×Dic6D411yescharacteristic
45C22×C3×C8—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×#217—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3D412yes—
60C2×C4⋊Dic3D411yescharacteristic
61#924—21no—
62#924—21no—
63#924—21no—
64#924—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—22no—
67C2×Dic3⋊C4—22no—
68C22×C3⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8D412yes—
71C3×#95—21no—
72C3×#95—21no—
73#835—24no—
74C3⋊#95—22no—
75C3⋊#95—22no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—44no—
79#830—21no—
80#830D412yes—
81#817—24no—
82#816—24no—
83#808—24no—
84C2×C22⋊Q8—31no—
85C2×C22⋊Q8C2×D311yescharacteristic
86C22×C4⋊C4—31no—
87#483—62no—
88#483—62no—
89#483—62no—
90#483—62no—
91C2×#106—61no—
92C2×#106—61no—
93C2×#103—61no—
94C2×#103—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8C2×D311yescharacteristic
97#398—61no—
98#398—61no—
99#395—61no—
100#395—61no—
101#380—62no—
102C2×#95—31no—
103C2×#95—62no—
104C2×#95C2×D311yescharacteristic
105#328—22no—
106#328—62no—
107C24×C3C2×D411yescharacteristic
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—42no—
111C22×C3×C4—22no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—42no—
117C22×C3×C4—41no—
118C22×C3×C4—42no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×Dic3—22no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—22no—
130C22×Dic3—42no—
131C22×Dic3—42no—
132C22×Dic3—41no—
133C22×Dic3—41no—
134C22×Dic3—42no—
135C22×Dic3—42no—
136C22×Dic3SD1614yes—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C2×C3×C8—22no—
141C2×C3×C8—42no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148#217—41no—
149#217—41no—
150C4⋊Dic3—22no—
151C4⋊Dic3—22no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—41no—
155C4⋊Dic3—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—82no—
159Dic3⋊C4—82no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—42no—
162C22×Q8—31no—
163C22×Q8D1211yescharacteristic
164C23×C4—21no—
165C23×C4—32no—
166C23×C4—61no—
167C23×C4C22×D311yescharacteristic
168C22×C8—21no—
169C22×C8—61no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4D1211yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—61no—
192#106—64no—
193#106—64no—
194#103—64no—
195#103—64no—
196C22⋊C8—21no—
197C22⋊C8—32no—
198C22⋊C8—61no—
199C22⋊C8#6712yes—
200#95—21no—
201#95—21no—
202#95—62no—
203#95—62no—
204#95—61no—
205#95—61no—
206#95—124no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—22no—
210C23×C3#12011yescharacteristic
211C23×C3#12111yescharacteristic
212C23×C3#12711yescharacteristic
213C23×C3C2×SD1612yes—
214C23×C3#13712yes—
215C3×Q8—42no—
216C3×Q8—81no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—44no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—82no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×Dic3—24no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—44no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3—44no—
248C2×Dic3—82no—
249C2×Dic3—82no—
250C2×Dic3—81no—
251C2×Dic3—81no—
252Dic6—42no—
253Dic6—81no—
254C3×C8—42no—
255C3⋊C8—42no—
256C24D3×D411yescharacteristic
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—42no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—122no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—34no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—62no—
277C22×C4—122no—
278C22×C4—122no—
279C22×C4—121no—
280C22×C4—122no—
281C22×C4—122no—
282C22×C4—121no—
283C22×C4C2×D1211yescharacteristic
284C22×C4#23511yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4C2×#6711yescharacteristic
289C2×C8—22no—
290C2×C8—42no—
291C2×C8—62no—
292C2×C8—122no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—62no—
298C4⋊C4—61no—
299C4⋊C4—62no—
300C4⋊C4—61no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C4⋊C4—121no—
304C4⋊C4—121no—
305C4⋊C4—242no—
306C4⋊C4—242no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3#39211yescharacteristic
320C22×C3#44812yes—
321C22×C3#45511yescharacteristic
322C22×C3#45611yescharacteristic
323C22×C3#45812yes—
324C22×C3#46112yes—
325C22×C3#49211yescharacteristic
326C22×C3#49711yescharacteristic
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—82no—
332C3×C4—81no—
333C3×C4—81no—
334Dic3—44no—
335Dic3—81no—
336Dic3—82no—
337Dic3—81no—
338C23—22no—
339C23—21no—
340C23—21no—
341C23#88211yescharacteristic
342C23#88411yescharacteristic
343C23D3×SD1612yes—
344C23#91812yes—
345C23#94011yescharacteristic
346Q8—42no—
347Q8—81no—
348Q8—122no—
349Q8—241no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—64no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—124no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—124no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—242no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4—242no—
381C2×C4#88211yescharacteristic
382C2×C4#88411yescharacteristic
383C2×C4#93011yescharacteristic
384C2×C4#93711yescharacteristic
385C8—42no—
386C8—122no—
387C2×C3—21no—
388C2×C3—22no—
389C2×C3—21no—
390C2×C3#160812yes—
391C2×C3#161112yes—
392C2×C3#200711yescharacteristic
393C2×C3#201311yescharacteristic
394C2×C3#202211yescharacteristic
395C22—21no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22#474212yes—
404C22#480612yes—
405C22#481812yes—
406C22#482511yescharacteristic
407C22#482611yescharacteristic
408C22#496011yescharacteristic
409C22#517511yescharacteristic
410C22#519011yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—82no—
416C4—81no—
417C4—81no—
418C4—124no—
419C4—242no—
420C4—241no—
421C4—241no—
422C3#997311yescharacteristic, complemented, Sylow
423C2—21no—
424C2—21no—
425C2—22no—
426C2#6832012yes—
427C2#6832912yes—
428C2#7003111yescharacteristic
429C2#7028211yescharacteristic
430C2#7039311yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#328C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenterC2×#328D3
hypocenterC3#9973
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
422. C3#9973semidirect

Maximal subgroups

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

subgroupindex
#827892
C2×#50432
C2×#47212
C2×C3×#3282
#694232
C2×#44522
#676742
#99733