Finite Group Databasebeta

← #10697199

Subgroups — #10697199

Order 768.

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

normal subgroups only

This group has 582 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
1C2×C3×#461C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic, complemented
5#71601C211yesmaximal, characteristic, complemented
6#70426C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#10184—31nomaximal, Sylow
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#932—21no—
13#5477C2211yescharacteristic
14C3×#461—22no—
15C3×#461—22no—
16C2×C3×C4⋊C8C2211yescharacteristic
17C2×C3×#103—21no—
18#5088—21no—
19#5087—22no—
20#5078—21no—
21#5019—21no—
22#5016—22no—
23#5015—21no—
24#4967—22no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30#4525—22no—
31#4483—22no—
32#3226—31no—
33#3036—31no—
34C2×#461D311yescharacteristic, complemented
35#1546—31no—
36#1480—31no—
37#1460—31no—
38#1301—31no—
39C22×C3×Q8—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C2×#235—21no—
43C2×C3×SD16—24no—
44C2×C3×SD16—42no—
45C2×C3×SD16—42no—
46C22×C3×C8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8D412yes—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#214—22no—
61#932—42no—
62#932—42no—
63C2×#213—22no—
64#930—21no—
65#930—21no—
66C2×#208—21no—
67#875—21no—
68#874D411yescharacteristic
69C4×D12—21no—
70C4×D12—21no—
71#869D411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×#103—24no—
75#830—21no—
76#830D412yes—
77#810—22no—
78#809—22no—
79#809—24no—
80#807—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C4×C3⋊C8—21no—
87C2×AGL(1,ℤ/8)—61no—
88C22×SD16—21no—
89#532—31no—
90C2×C4⋊Q8C2×D311yescharacteristic
91C2×C4×D4C2×D311yescharacteristic
92#500—61no—
93#486—61no—
94#480—62no—
95#474—62no—
96#463—62no—
97#461—22no—
98#461—22no—
99#461—61no—
100#459—61no—
101#431—31no—
102#431—31no—
103#423—31no—
104C2×C4⋊C8C2×D311yescharacteristic
105#417—61no—
106C2×#103—21no—
107#332—62no—
108D4⋊C8—62no—
109#323—62no—
110C24×C3—21no—
111C2×C3×Q8—22no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4D812yes—
123C22×C3×C4—21no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—82no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—41no—
132#235—41no—
133#235—41no—
134#235—41no—
135#235—41no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×C4×D3—41no—
140C2×Dic6—21no—
141C3×SD16—48no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157#214—44no—
158#213—44no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×#41—61no—
169C22×Q8—21no—
170C22×D4#6711yescharacteristic
171C23×C4—21no—
172AGL(1,ℤ/8)—122no—
173AGL(1,ℤ/8)—122no—
174C2×SD16—24no—
175C2×SD16—42no—
176C2×SD16—42no—
177C2×SD16—62no—
178C2×D8—62no—
179C2×M16—61no—
180C22×C8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—31no—
185C4⋊Q8#6712yes—
186#127—31no—
187#124—61no—
188#121—61no—
189#121—61no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—61no—
195C4×D4—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4#6711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C42C22×D311yescharacteristic
201#107—64no—
202C4⋊C8—21no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8#6712yes—
208#103—24no—
209#103—62no—
210#102—62no—
211#102—64no—
212C22⋊C8—32no—
213C22⋊C8—61no—
214C4×C8—61no—
215C4×C8—61no—
216C23×C3—22no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—81no—
220C23×C3—81no—
221C23×C3#12111yescharacteristic
222C22×D3—41no—
223C3×Q8—42no—
224C3×Q8—42no—
225C3×Q8—81no—
226C3×Q8—81no—
227C3×D4—22no—
228C3×D4—22no—
229C3×D4—41no—
230C3×D4—41no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—82no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4—82no—
239C2×C3×C4—81no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12711yescharacteristic
246C2×C3×C4C2×D811yescharacteristic
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
250#67—81no—
251#67—81no—
252C2×Dic3—41no—
253D12—42no—
254D12—41no—
255D12—41no—
256D12—42no—
257C4×D3—81no—
258C4×D3—81no—
259Dic6—41no—
260Dic6—41no—
261C3×C8—42no—
262C3⋊C8—42no—
263C3⋊C8—42no—
264C3⋊C8—81no—
265C24—21no—
266#41—121no—
267#41—121no—
268#41—121no—
269#41—121no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—61no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—121no—
286C2×D4#21312yes—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—121no—
293C22×C4D3×D411yescharacteristic
294C22×C4#23711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296SD16—48no—
297SD16—124no—
298D8—124no—
299M16—121no—
300M16—121no—
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—121no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—61no—
314C4⋊C4—61no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C42D3×D411yescharacteristic
320C42D3×D411yescharacteristic
321C42C2×#6711yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—42no—
326C22×C3—82no—
327C22×C3—82no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3#39211yescharacteristic
331C22×C3#46611yescharacteristic
332C22×C3#46911yescharacteristic
333C2×D3—81no—
334C2×D3—81no—
335C2×D3—81no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—82no—
340C3×C4—81no—
341C3×C4#44711yescharacteristic
342C3×C4#44711yescharacteristic
343C3×C4#44911yescharacteristic
344C3×C4#44911yescharacteristic
345C3×C4#47112yes—
346Dic3—81no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—81no—
351C23—81no—
352C23—121no—
353C23#93911yescharacteristic
354Q8—42no—
355Q8—42no—
356Q8—81no—
357Q8—81no—
358Q8—121no—
359Q8—121no—
360D4—22no—
361D4—22no—
362D4—41no—
363D4—41no—
364D4—122no—
365D4—122no—
366D4—121no—
367D4—121no—
368D4—241no—
369D4—241no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—121no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4C2×#21311yescharacteristic
383C2×C4#93211yescharacteristic
384C2×C4#93711yescharacteristic
385C2×C4#93711yescharacteristic
386C2×C4#93811yescharacteristic
387C2×C4#93911yescharacteristic
388C2×C4#94011yescharacteristic
389C2×C4#94911yescharacteristic
390C2×C4#95011yescharacteristic
391C2×C4#95311yescharacteristic
392C8—42no—
393C8—122no—
394C8—122no—
395C8—241no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—82no—
399C2×C3#199911yescharacteristic
400C2×C3#200911yescharacteristic
401C2×C3#201911yescharacteristic
402D3—161no—
403C22—21no—
404C22—21no—
405C22—42no—
406C22—82no—
407C22—82no—
408C22—81no—
409C22—81no—
410C22—241no—
411C22—241no—
412C22—241no—
413C22#503811yescharacteristic
414C22#503911yescharacteristic
415C22#522311yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—82no—
420C4—81no—
421C4—241no—
422C4#503611yescharacteristic
423C4#503711yescharacteristic
424C4#517712yes—
425C4#521711yescharacteristic
426C4#523811yescharacteristic
427C3#1018411yescharacteristic, complemented, Sylow
428C2—21no—
429C2—21no—
430C2—82no—
431C2—481no—
432C2#7168411yescharacteristic
433C2#7205911yescharacteristic
434C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#461C2
socleC22×C3#392
prime radical#82307C2
hypercenterC2×#461D3
hypocenterC3#10184
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×C3×#461C2semidirect
2. #83215C2semidirect
4. #71773C2semidirect
5. #71601C2semidirect
6. #70426C2semidirect
7. #67707C2semidirect
34. C2×#461D3semidirect
427. C3#10184semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4612
#832152
#823072
#717732
#716012
#704262
#677072
#101843